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Unit 3 Geometry: Understanding Congruence and Properties of Angles Introduction In this unit, students will review the basic concepts of geometry, such as lines, rays, line segments, angles, and polygons. They will use informal arguments to establish facts about the sum of the angles in a triangle, the exterior angles in a triangle, and the angles created when parallel lines are intersected by another line. Students will learn to make sketches and recognize and create counterexamples. Students will also be introduced to the idea of congruence, and they will recognize and use congruent triangles to produce informal arguments. Vocabulary. The most important terms, learned in this and previous grades, are as follows: polygon parallelogram line segment right angle equilateral vertical angles supplementary angles quadrilateral trapezoid line acute scalene alternate angles co-interior angles pentagon rhombus ray obtuse isosceles corresponding angles hexagon parallel perpendicular straight angles congruent exterior angles To help students who are struggling with these terms, use the cards from BLM Geometric Terms (pp. D-111–113). Students can match terms with the different representations of the same concept. You could also use the cards to play a memory game: Player 1 lays out the cards face down and turns over pairs at random. If the pair matches, Player 1 collects the pair; if not, Player 1 turns the cards face down again. Player 2 then takes a turn. You might also want to use this BLM as a summary sheet. Trapezoids and parallelograms. There are two possible definitions of a trapezoid: 1. A quadrilateral with exactly one pair of parallel sides 2. A quadrilateral with at least one pair of parallel sides When using the first definition, parallelograms are different from trapezoids, but when using the second definition, all parallelograms are trapezoids. Both definitions are legitimate; we use the first definition. Protractors. Students will need to use protractors in this unit. If you need additional protractors for individual students, photocopy a protractor (or BLM Protractors, p. D-114) onto a transparency and cut it out. These protractors are also convenient to use on an overhead projector. Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-1 Grids. Students will need to use grid paper throughout this unit. In addition, you will need a predrawn grid on the board for some lessons. If you do not have such a grid, photocopy BLM 1 cm Grid Paper (p. I-1) onto a transparency and project it onto the board. This will allow you to draw and erase shapes and lines on the grid without erasing the grid itself. If students are not working in grid paper notebooks, provide them with grid paper or BLM 1 cm Grid Paper. Paper folding. Many activities in these lessons involve paper folding to make, check, and examine shapes and angles. Unless otherwise noted, the starting shape is a regular 8 1/2" by 11" sheet of paper. Sometimes the starting shape is a hand-drawn circle to make sure there are no angles for students to start with or refer to. Technology: dynamic geometry software. Students are expected to use dynamic geometry software to draw geometric shapes. Some of the activities in this unit use a program called The Geometer’s Sketchpad®, and some are instructional—they help you teach students how to use the program. If you are not familiar with The Geometer’s Sketchpad®, the built-in Help Center provides explicit instructions for many constructions. Use phrases such as “How to construct congruent angles” or “How to construct a line segment of given length” to search the Index. NOTE: If you use different dynamic geometry software to complete these activities, you may need to adjust the instructions provided. Summary BLMs. Some crucial definitions, properties of shapes, and step-by-step instructions used in this unit are summarized on BLMs, for easy reference. The chart below lists the summary BLMs and the topics they cover. Summary BLMs are noted in some specific lessons, but you can use them at any point where they will support students. BLM Title Topics Covered Geometric Terms (pp. D-111–113) Geometric terms regarding shapes, lines, angles, properties, and congruence—use as a summary chart, for matching, or a memory game Using Protractors (Summary) (p. D-115) Measuring an angle, drawing an angle, drawing a line perpendicular to a given line through a point, and drawing a line parallel to a given line through a point Angle Properties (Summary) (p. D-133) Supplementary, corresponding, co-interior, alternate, and vertical angles Properties of Angles in a Triangle (Summary) (p. D-134) Sum of the angles in a triangle, exterior angle in a triangle, angles in an isosceles triangle, and congruence rules D-2 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry G8-1 Points and Lines Pages 72–73 Standards: preparation for 8.G.A.5 Goals: Students will review the basic geometric concepts and notation for points, lines, line segments, and rays. Students will review intersecting lines and line segments. Prior Knowledge Required: Can use a ruler to draw lines Can measure distances to the nearest millimeter Vocabulary: endpoint, intersect, intersection point, line, line segment, point, ray Materials: The Geometer’s Sketchpad® Review the concepts of a point, line, line segment, and ray. Draw a dot on the board. Explain that the dot represents a point. SAY: A point is an exact location. A point has no size—no length, width, or height. The dot has size, or you couldn’t see it, but real points do not. Draw on the board: line Explain that a line extends in a straight path forever in two directions. It has no ends. To show a line that extends forever in both directions, we draw arrows at both ends of the line. Lines that we draw have a thickness, but real lines do not. Draw on the board: line segment Explain that a line segment is the part of a line between two points, called endpoints. The endpoints are usually shown as dots. A line segment has a length that can be measured, but it has no width or thickness. Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-3 Draw on the board: ray Explain that a ray is part of a line that has one endpoint and extends forever in the other direction. Similar to a line, it has no length, width, or thickness. Write on the board: line segment line ray On another part of the board, draw a line segment and ask students to signal whether it is a line (by pointing to the left), a line segment (by pointing up), or a ray (by pointing right). Change the endpoints so that the object becomes first a ray, then a line, each time asking students to signal the correct term. Repeat with lines, rays, and line segments in different positions and with varying lengths. Review naming a point, line, ray, and line segment. Explain that we use capital letters to name points. Draw a point on the board, and write “A” beside it. Then draw a line through point A and mark another point on the line. Label the second point “B,” as shown below: B A SAY: To name a line, we give the names of any two points on the line. This is line AB. The order of the endpoints does not matter, so we can also name this line BA. Write on the board: line AB or BA Draw on the board: C E D SAY: Since I labeled three points on this line, we can name this line in many different ways. You can use any two points. Ask students to copy the diagram and then give all possible names for the line. (CD, DC, DE, ED, CE, EC) Draw on the board: F D-4 G Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry Explain that, since a ray always has an endpoint, we have to use that point when naming the ray and write that point first. So this is ray FG, but not ray GF. Exercises: Which point is the endpoint of the ray? Name the ray in all possible ways. a) b) Q H J R P M Answers: a) endpoint H, ray HJ or HM; b) endpoint R, ray RP or RQ Draw a line segment on the board. Explain that a line segment has two very clear endpoints, so to name a line segment you have to name both endpoints. The order in which you name the endpoints does not matter. Have students suggest labels for the endpoints and write both possible names for the line segment. (e.g., KL and LK) Draw on the board: N S O SAY: This diagram shows three different line segments. A line segment is the part of the line between the endpoints, so even though all three points are on the same line, the line segments are different. Trace the line segment NO with your finger and ask students to name it. (NO or ON) Repeat with the other two line segments. (OS or SO, NS or SN) Exercises: Name the lines, line segments, and rays in the diagram in all possible ways. Say whether each object is a line, a ray, or a line segment. a) b) Y U Z V W X A Answers: a) line segment UW or WU, ray WV; b) line YZ or ZY, line segment AZ or ZA, ray YX Introduce the intersection point. Tell students that an intersection point is a point that lines, line segments, or rays have in common. Draw on the board: Indicate the intersection point in each picture. Explain that, if the location of a point is clear, you do not need to draw a dot to show it. When two lines, rays, or line segments intersect (meet or cross), there is only one intersection point, so there cannot be any doubt about its Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-5 location. So there is no need to draw a dot in the pictures on the board. Remove the dots from the intersection points. Draw on the board: A B D C ASK: Does AB intersect CD? Explain that the answer depends on whether AB and CD are lines, line segments, or rays. As a class, check all possible combinations of lines and line segments, as shown below: A A B D C A B D C A B D C B D C Extend the lines to show intersection. (if either AB or CD is a line segment, they do not intersect; if both are lines, they intersect) ASK: Why is there no intersection when either one is a line segment? (because the positions of AB and CD mirror each other; when we make one a line and the other a line segment, if they don’t intersect in one case, they won’t intersect in the mirror of the case) Explain that the situation with rays is more complicated; the answer depends on which side of AB can be extended. SAY: Let’s suppose that the object AB or BA is a ray. Remind students that a ray is named with the endpoint first. Ask students to check all possible cases, as shown below: 1. 2. A A B D C 3. 5. C A B D 6. A C B D 4. A B D C A C B D D C 7. B 8. A C B D A C B D (1. do not intersect, 2. do not intersect, 3. do not intersect, 4. do not intersect, 5. do not intersect, 6. do not intersect, 7. intersect, 8. intersect) D-6 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry Activities 1–2 Use The Geometer’s Sketchpad® for the following two activities. (MP.5) 1. Proper use of Point and Line tools. Show students how to draw lines, line segments, and rays using the Line tool of The Geometer’s Sketchpad®. Then go through the following tasks and questions: a) Ask students to draw a line and an independent point. Ask them to move the point so that it looks like it is on the line. Then have them move or turn the line. ASK: Does the point stay on the line? (no) b) Ask students to draw a point. Then have them draw a line using that point as one of the points you draw the line through. Note that the point is highlighted when you select it; this will ensure the line passes through the point. Have students move the points and the line itself. ASK: Does the line move? (yes) Does it still pass through the point you started with? (yes) c) Show students how to mark an intersection point of two lines. Draw two lines so that they intersect. Start drawing a point and ASK: How can you tell that the point is on both lines? (both lines get highlighted) Have students construct two lines and mark the intersection point. Have them move the lines and point out that the intersection point remains on both lines. (MP.5) 2. Length of a line segment. a) Have students draw a line segment and use the Measure menu options to find its length. Then ask them to try to move the endpoints so that the length of the segment becomes 3 cm. ASK: Is it easy or hard to do? (hard) If you move the line segment around, does its length change? (no) If you move the endpoints around, does the length of the line segment change? (yes) Show students how to draw a line segment that is 3 cm long by following these steps: Step 1: Use the Number menu options to create a new parameter. Change the value to 3 and mark it as “distance.” Step 2: Mark a point. It will be one endpoint of the line segment. Select the point and the parameter. Step 3: Using the Construct menu options, construct a circle (by center and radius). Step 4: Use the Line Segment tool to create a line segment between the center of the circle and any point on the circle. Hide the circle using the right-click menu options. ASK: Will moving the endpoints change the length of the line segment now? (no) b) Have students draw a line segment and label it AB. Challenge students to draw a line segment CD that is exactly the same length as AB. Discuss the two options: 1. Use the copy and paste option to create a copy of the line segment at a given point. When you modify the original line segment, the copy does not change. 2. Measure the length of AB. Use the length of AB as the parameter in the steps in part a). Modifying the line segment AB modifies the second line segment accordingly. This method will be more useful when students work with congruent triangles. (end of activities) Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-7 Extensions 1. a) Draw two points. Label them A and B. b) Draw the line AB. c) Draw line CD so that point A is not on CD, but point B is on CD. d) Do lines AB and CD intersect? If yes, what is the intersection point? Answers: d) lines AB and CD intersect, B is the intersection point (MP.3) 2. a) Jen thinks rays DE and DF are the same ray. Is she correct? Explain. F E D b) Name one point that is on ray DF but not on ray EF. c) Ron thinks rays ED and ray FD are the same ray. Is he correct? Explain. Answers: a) yes, the rays are part of the same line, pointing in the same direction, and with the same starting point, so they are the same ray; b) point D on the ray DF is not on ray EF; c) no, point F is on ray FD, but not on ray ED, so these are not the same ray D-8 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry G8-2 Angles and Shapes Pages 74–75 Standards: preparation for 8.G.A.5 Goals: Students will review naming angles and polygons. Prior Knowledge Required: Knows what an angle is Can identify polygons Understands clockwise and counter-clockwise Vocabulary: arc, arms, endpoint, intersect, intersection point, polygon, right angle, straight angle, vertex, vertices Materials: The Geometer’s Sketchpad® Review angles. Remind students that, in mathematics, when two rays have a common endpoint, they make an angle. Draw two rays with a common endpoint. Draw the pictures below on the board. For each one, have students signal thumbs up if the picture shows an angle and thumbs down if it does not. Exercises: Does the picture show an angle? a) b) c) d) f) e) Answers: a) no, b) yes, c) no, d) yes, e) yes, f) no Remind students that the rays are called the arms of the angle, and the common endpoint is called the vertex of the angle. SAY: The plural of vertex is vertices. Have students each draw a few angles, exchange notebooks with a partner, and identify the vertex and extend the arms of the angles drawn by their partner. Explain that, when the vertex of an angle is a clear point, such as a common endpoint of two rays, you do not need to draw a dot to show the vertex. Erase the dots in the angles in parts b), d), and e) from the previous exercises, and explain that these are still pictures of angles. Naming angles. Explain that, to name an angle, you need to name three points: one on each arm and the vertex. SAY: The vertex is always the middle letter. Draw the picture on the next page and explain that you can name this angle in two ways: XYZ or ZYX, but not YXZ. SAY: To Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-9 make it clear that you are naming an angle and not anything else, such as a triangle, we use the angle sign “” before the letters. Write the two names for the angle underneath the picture. XYZ or ZYX Exercises: Name the angle in two ways. a) b) C L K B c) T M A E N Answers: a) CAB, BAC; b) KLM, MLK; c) TEN, NET Remind students that when there are several points on a ray, the ray can be named in several ways; you just need to name the endpoint of the ray first. Add a point D between points A and B in the angle in part a) of the previous exercises to illustrate the point, as shown below: C A D B Trace ray AB and SAY: We can name this ray AB or AD. This means that we can name this angle in more than two ways: CAB, BAC, CAD, and DAC. Write all four names on the board. Exercises: 1. Name the angle in all possible ways. Q Answers: AQB, BQA, AQE, EQA, KQB, BQK, KQE, EQK 2. a) Which of the following are possible names for this angle? CAT, CTP, PTA, UTA, UTC, TCP, PTU b) Write three more different names for the angle. Answers: a) CTP, PTA, UTA, UTC; b) any three of CTU, ATP, PTC, ATU D-10 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry Naming angles that are parts of other angles. Draw on the board: Explain that point A in this picture is the vertex of several angles. For example, angle BAD is the angle of the rectangle, but angle CAD also has vertex A, angle BAC has vertex A, and both angles are parts of the angle BAD. SAY: We often use arcs, a part of a circle around the vertex of an angle, to show angles that are part of other angles. ASK: What is the vertex of the angle marked with the arc in this picture? (A) Circle the letter A and the vertex itself. ASK: What angle does the arc in the picture show? (CAD) To prompt students to see the answer, thicken the rays AC and AD. You can also extend the rays and add arrows beyond the points C and D, as shown below. Then ask students to look for points on these rays. (C and D) Exercises: Copy the picture. Circle the vertex of the angle with the arc. Draw rays along the line segments the arc ends at. Then name the angle marked by the arc. a) A b) H c) M B G N D C E F P R Q Bonus: Name 4 more angles in each picture. Answers: a) ADC or CDA, b) GHF or FHG, c) MNR or RNM, Bonus: answers will vary Point out that in part c) of the previous exercises the arms of the angle MNR form a straight line. SAY: We call such angles straight angles. NOTE: To provide more practice to students who need it, mark another arc on each of the pictures in the previous exercises, as shown in the following exercises, and have them identify the new angles. Exercises: Name the angle marked by the arc. B a) A b) H G c) M N D C E F P Answers: a) DBC or CBD, b) GFE or EFG, c) MNQ or QNM Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry R Q D-11 Draw the picture below on the board and have students copy it in their notebooks: ASK: Which angle in this diagram is the angle ACB? Have a volunteer mark it on the board. Repeat with other angles, such as DCB, BAD, and BCA. Students can use different numbers of arcs or different colors of arcs to mark the angles. Naming an angle with one letter. Point out that sometimes, when there is no chance of confusion, only the vertex letter is used to name an angle. Draw on the board: Explain that, in this rectangle, there is only one possible D, but there are three possibilities for A: BAD or BAC or DAC. Review polygons. Remind students that polygons are flat shapes with sides that are line segments, so the sides are straight. Sides of polygons only meet at vertices, and exactly two sides meet at each vertex. Sides of polygons do not intersect except at endpoints. Draw on the board: polygons not polygons Naming polygons. Explain that, to name a polygon, you need to write the letters that name the vertices in order. Draw a quadrilateral on the board and have students suggest names for different vertices. Point out that the letters can be any from the alphabet and do not need to be in alphabetical order, as in the example shown below: SAY: To write a name for this polygon, you can start at any vertex. For example, I can start at F and read the letters clockwise to name this polygon FQMS. Or I could read the letters counterclockwise to name this polygon FSMQ. I cannot write M as the next letter after F. Students can check each other’s work in the next exercises. D-12 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry Exercises: Name the polygon. P a) A b) H G c) d) U W X F D Q C J E V Y K Sample Answers: a) APQD, b) CHE, c) JGFK, d) UWXYV Activity Use The Geometer’s Sketchpad® for this activity. Using the Polygon tool. Show students how to use the Polygon tool to create polygons. Draw students’ attention to the different options of the Polygon tool. Have them use the option that makes the sides of the polygon visible and selectable. Teach students to label polygons. Have them create triangle ABC, quadrilateral DEFG, and pentagon HIJKL. Show students how to measure the length of the sides of polygons. Have them measure the sides of the polygons they created. (end of activity) Extensions 1. Name all the triangles in the picture. Answers: ABC, ACE, ECD, ACD 2. a) Draw line segment FG that is 6 cm long. b) Mark point H on FG so that line segments FH and GH are the same length. c) Draw a line that intersects FG at point H. d) On the line you drew in part c), mark points J and K, so that HJ = HK = 3 cm. e) Join the points F, K, G, and J to create a quadrilateral. f) What type of quadrilateral have you produced? g) Name all the triangles and the quadrilateral in your diagram. Sample answers: a) to e) see diagram below; f) rectangle; g) triangles: FHK, FHJ, FGK, FGJ, FKJ, GKJ, GKH, GJH, quadrilateral: FKGJ K F G H J Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-13 G8-3 Perpendicular and Parallel Lines Pages 76–78 Standards: preparation for 8.G.A.5 Goals: Students will review perpendicular and parallel lines. Students will review notation for perpendicular and parallel lines, equal line segments, and right angles. Students will review properties of special quadrilaterals, such as equal sides, parallel sides, and right angles. Prior Knowledge Required: Can identify and name lines, line segments, and rays Can draw and measure line segments Can identify a right angle Can name an angle and identify a named angle Can identify and name polygons Vocabulary: endpoint, hash marks, intersect, intersection point, parallel, parallelogram, perpendiculars, quadrilateral, rhombus, square, trapezoid Materials: rectangular sheet of paper or a set square rulers BLM Geometric Terms (2) (p. D-112, optional) The Geometer’s Sketchpad® Review perpendicular lines. Explain that when two lines, rays, or line segments meet at a right angle, we say that they are perpendicular. Draw the pictures in the following exercises on the board and have students identify the perpendicular objects. Students can signal thumbs up for objects that are perpendicular and thumbs down for objects that are not. Invite volunteers to check whether the lines are perpendicular by using a corner of a page or a set square and mark the right angles with a little square. Exercises: Are the objects perpendicular? a) b) c) e) g) D-14 f) d) h) Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry Answers: a) yes, b) yes, c) no, d) yes, e) yes, f) no, g) no, h) yes Ask students where they see perpendicular lines or line segments–also called perpendiculars– in the environment. (sample answers: sides of windows and desks, intersections of streets, the frame of a poster or painting) Introduce the notation for perpendicular objects. Explain that there is a symbol people use instead of the word “perpendicular.” Add labels to the pictures from parts a), e), and d) from the previous exercises as shown below. Demonstrate on the first picture how to write the sentence AB CD, which means that line AB is perpendicular to line CD. Have students do the rest of the exercises individually. Exercises: 1. Name the perpendicular lines, rays, or line segments. a) b) c) C E Bonus: K N A B H F D T M O G P Answers: a) AB CD; b) EF FG; c) HK HM; Bonus: TO ON, TP ON, PO ON 2. Which sides look like they are perpendicular? Answers: a) AB BC, AB AD, CB CD, AD CD; b) GH FG; c) IM LM, IM JI; d) ON NR A perpendicular through a point. Explain that sometimes we are interested in a line that meets two conditions: it is perpendicular to a given line and passes through a given point. SAY: Remember that a line segment is always part of a line. We might draw a line segment in a picture, but the segment is part of a line. Draw on the board: K L N M ASK: In this picture, which line segments are perpendicular to the line NM? (KN, LM) What other line segment is here? (LN) Explain that, out of the three line segments, two are perpendicular to the line NM and two pass through the point L, but only one line segment does both: LM is perpendicular to the line NM and passes through the point L. SAY: line LM can be called “the perpendicular to line NM through the point L.” There is no other line that satisfies both conditions. Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-15 Exercises: Identify … a) the lines that are perpendicular to the line AB, b) the lines that pass through point P, and c) the line that satisfies both conditions. I Answers: a) i) EC, DH, ii) DK, EF; b) i) GF, DH, ii) DK, CI, AB; c) i) DH, ii) DK Why perpendiculars are important. Discuss with students why perpendiculars are important and where are they used in real life. For example, if construction workers need to cut wooden floorboards into perfect rectangles, they can measure the necessary length on one side of the board and make the cut at a right angle to the side of the board. In other words, they make a cut perpendicular to the board side. Review parallel lines. Remind students that parallel lines are straight lines that never intersect, no matter how much they are extended. Ask students to think about where they see parallel lines. (e.g., a double center line on a highway, the edges of construction beams) Remind students how to mark parallel lines with the same number of arrows, as shown below: Emphasize that the number of arrows and their direction has to match in each set of parallel lines, but there is no rule about where the arrows should be pointing. Reverse the arrows on sides BC and AD and explain that this picture still shows that the quadrilateral has two pairs of parallel sides. Exercises: Copy the diagram. Mark the sides that look parallel with the same number of arrows. a) b) c) d) Answers: a) D-16 b) c) d) Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry Review the symbol for parallel lines (||). Return to the previous picture of parallelogram ABCD. Explain that there is a symbol for parallel lines. Write “||” on the board. SAY: This symbol resembles the double “l” in the word “parallel.” In the picture of parallelogram ABCD, line segment AB is parallel to line segment CD, so we can write AB || CD. Write the expression on the board. ASK: What other parallel line segments are in this picture? (AD and BC) Have a volunteer write that on the board with the symbol. Ask students to label the polygons they drew in the previous exercises with letters and write the expressions showing which sides are parallel. (selected answer: KL || MN) Point out that since in part d) there are three sides parallel to each other, and the expression only refers to two sides, they will need to write each pair of three sides separately. (for shape OPQRSTU, OP || UT, UT || SR, and SR || OP) Have students exchange notebooks with a partner to check each other’s answers. Review parallelograms and trapezoids. Remind students that quadrilaterals can be classified by how many pairs of parallel sides they have. ASK: Can a quadrilateral have three parallel sides? (no) Why not? (a quadrilateral has 4 sides, so 3 parallel sides would mean only 1 side remains and it would not be able to close the shape) How many pairs of parallel sides can a quadrilateral have? (0, 1, or 2) What do we call a quadrilateral that has exactly one pair of parallel sides? (trapezoid) What do we call a quadrilateral that has two pairs of parallel sides? (parallelogram) Invite volunteers to draw a few examples of each kind on the board and mark the parallel sides. Point out that most quadrilaterals that have no parallel sides do not have a special name. Explain that in the next exercises the sides that look parallel are, indeed, parallel. Exercises: Copy the quadrilateral. Mark the parallel sides with the same number of arrows. Identify the quadrilateral as a parallelogram, trapezoid, or neither. a) b) c) d) Answers: a) parallelogram b) c) parallelogram d) neither trapezoid Review markings for equal sides. Remind students that we use hash marks to show equal sides—in other words, to show that the sides are line segments of equal length. Similar to parallel line markings, we use different numbers of hash marks when we have several groups of equal sides. Draw a rectangle on the board and show the markings for equal opposite sides. Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-17 Exercises: Mark the equal sides on the quadrilaterals in the previous exercises. Answers: a) b) c) d) Review special quadrilaterals. Draw on the board: Have students copy the shape, then measure the sides to the closest millimeter, and label equal sides with hash marks. ASK: How many equal sides does this shape have? (4) ASK: Do the opposite sides in the picture look like they are parallel? (yes) SAY: In fact, the opposite sides in this shape are parallel. This is a parallelogram. ASK: What do we call parallelograms with four equal sides? (rhombuses) What other special quadrilaterals do you know? (rectangles, squares, kites) What do you check to see if a quadrilateral is a rectangle? (if it has 4 right angles) What do you check to see if a quadrilateral is a square? (if it has 4 right angles and 4 equal sides) Do rectangles have parallel sides? (yes, opposite sides in a rectangle are parallel) Are rectangles also parallelograms? (yes) Do rectangles have any pairs of equal sides? (yes, opposite sides in a rectangle are equal) To remind students of the relationships between special quadrilaterals, draw the Venn diagram below on the board, explaining that if you had a bunch of quadrilaterals, this diagram would allow you to sort them. Quadrilaterals Parallelograms Rhombuses Trapezoids Rectangles Point out that all rhombuses and all rectangles are also placed inside the group for parallelograms. ASK: Where would all squares go on this diagram? (in the region that is common to rhombuses and rectangles) Why are squares also rhombuses and rectangles? D-18 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry (a square is a rhombus because it has 4 equal sides, and it is also a rectangle because it has 4 right angles) Have students copy the diagram so it fills a whole notebook page and draw a shape in each region of the diagram, marking the parallel sides, right angles, and equal sides. You might wish to distribute BLM Geometric Terms (2) as a summary of terms. Activities 1–3 Use The Geometer’s Sketchpad® for the following three activities. 1. Constructing perpendicular and parallel lines. a) Draw a line. Label it m. b) Mark a point not on the line m. Label the point A. c) Select the line and the point. Use the Construct menu options to construct a line n perpendicular to m through point A. d) Select line m and point A. Use the Construct menu options to construct a line p parallel to m through point A. e) Move or turn the line m. Does line n stay perpendicular to line m? Does line p stay parallel to line m? Answer: e) yes 2. Constructing a parallelogram. a) Draw a line segment AB. b) Draw a line segment BC, so that point C is not on line AB. c) Using the Construct menu options, construct a line parallel to AB through C. Then construct a line parallel to BC through A. d) Mark and label D the intersection point of the lines you drew in part c). e) Create a quadrilateral ABCD. Remember to use the option that shows the sides of a quadrilateral. What type of quadrilateral is ABCD? f) Measure the lengths of the opposite sides of ABCD. What do you notice? g) Move the vertices of ABCD around. Does the quadrilateral change? Does it stay the same type of quadrilateral? Does your observation in part f) hold when you move the vertices? Answers: e) parallelogram; f) the opposite sides are equal, AB = CD, AD = BC; g) the quadrilateral changes, but it stays a parallelogram and the opposite sides remain equal 3. Constructing a rectangle. a) Draw a line segment AB. b) Using the Construct menu options, construct a line perpendicular to AB through A. c) Construct a line perpendicular to AB through B. Mark a point C on it. d) Construct a line perpendicular to BC through C. e) Mark and label D the intersection point of the lines you drew in parts b) and d). f) Create a quadrilateral ABCD. What type of quadrilateral is ABCD? g) Measure the lengths of the opposite sides of ABCD. What do you notice? h) Move the vertices of ABCD around. Does the quadrilateral change? Does it stay the same type of quadrilateral? Does your observation in part f) hold when you move the vertices? Answers: f) rectangle; g) the opposite sides are equal, AB = CD, AD = BC; h) the quadrilateral changes, but it stays a rectangle and the opposite sides remain equal (end of activities) Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-19 Extensions 1. Explain that a plane is a flat surface with no thickness. It extends forever along its length and width. Parallel lines in a plane will never meet no matter how far they are extended in either direction. Have students find a pair of lines not in the same plane that never meet and do not intersect, but are not parallel. To prompt students to see the answer, sketch a cube on the board and suggest that students look at the edges of the cube. Sample answer: The thickened edges of the cube above are an example of parts of lines that are not in a single plane, are not parallel, and never intersect. 2. Draw … a) a hexagon with three parallel sides b) an octagon with four parallel sides c) a heptagon (7-sided polygon) with three pairs of parallel sides d) a heptagon with two sets of three parallel sides e) a polygon with three sets of four parallel sides f) a polygon with four sets of three parallel sides Sample answers: D-20 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry (MP.3) 3. Discuss with students whether there can be more than one line perpendicular to a given line through a given point and whether such a perpendicular always exists. You can use the diagrams below to help students visualize the answers: Answer: Imagine a line rotating around point P, as illustrated in the first diagram. At one moment, it is perpendicular to AB, and then, as you continue to rotate the line, it will stop being perpendicular to AB. You can also look at a line that is perpendicular to AB and slowly move it from one end to the other. At one moment, it will touch point P, becoming the perpendicular to AB through point P. Since P has no real width, the line will then immediately leave P. So there cannot be more than one such line. Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-21 G8-4 Measuring and Drawing Angles Pages 79–81 Standards: preparation for 8.G.A.1, 8.G.A.5 Goals: Students will measure and construct angles. Prior Knowledge Required: Knows what an angle is Can identify right angles Can identify and name angles and polygons Understands the concept of measurement Vocabulary: acute, base line, degree, endpoint, intersect, intersection point, obtuse, opposite angles, origin, parallelogram, perpendicular, protractor, reflexive angles, rotation, scale Materials: pair of scissors or board compass for demonstration several pieces of transparencies with angles drawn on them overhead projector and markers protractors rulers The Geometer’s Sketchpad® BLM Using Protractors (Summary) (p. D-115) geoboards and elastics (see Extension 1) Review that the size of an angle is the amount of rotation between the arms. Remind students that, in mathematics, the size of an angle is how much you need to turn one arm extending from the vertex to get to the other arm extending from the vertex. In other words, the size of an angle is the measurement of the rotation between the angle’s arms when the arms are rays. Show students a pair of scissors or a board compass. Point out that the blades of the scissors rotate around a peg, which is like a vertex. Hold the scissors so that one blade is horizontal at all times. Open the scissors a little bit and then open them more and more to show how the angle increases as the top blade rotates away from the horizontal blade. Trace a small angle on the board and open the scissors wider than that angle; then show how the smaller angle on the board “fits inside” the larger angle of the open scissors. Emphasize that the more you rotate the blades, the more you open the scissors, and the wider the angle becomes. You can also demonstrate the rotation using your arms; stand so that students look at your side while you stretch your arms forward, one arm horizontal and the other raised somewhat. The line of your shoulders acts as a vertex, and your arms act as the two arms of an angle. If you keep one arm horizontal and swing the other up or down, you widen the angle between the arms. The more you swing the one arm, the greater the angle. D-22 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry Draw on the board: SAY: When the angle is small, the arms are only open a little bit. When the angle is large, the arms are open a lot. Explain that we draw an arc to show how much the arms open or how much we turn one arm to get to the other. Size of the angle does not depend on the length of the arms in the picture of the angle. Students often mistake the area between the arms of an angle for the size of the angle. To help students who have this misconception, remind them that the angle is how much we need to turn from one arm of the angle to the other. For any of the following activities that compares a pair of angles, have students place a pencil on one of the arms of the angle and rotate it to get to the other arm. Repeat with the second angle. The angle that needs more rotation is the larger angle. For the size of the angle, it does not matter whether the arms of the angle in the picture are short or long; the arms are rays and they can be extended as much as we need. Draw the two angles below on two separate pieces of transparency and project them on the board: NOTE: If an overhead projector is unavailable, you can do the following explanations by drawing the angles on the board. You will have to skip the direct comparison and rely on students estimating the size of the angles by eye. Explain that although these angles look different and one picture looks larger than the other, the angles are in fact the same size. In both these angles, you need to turn one arm the same amount of rotation to get to the other arm. Slide one of the transparencies on top of the other to show that the angles are the same. Point out that, just as we do not change a ray by extending it, we do not change an angle by extending its arms. Extend the arms on the smaller picture to show that the angles are exactly the same. Draw the two angles below on two separate pieces of transparency and project them on the board: Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-23 ASK: Which angle is larger? Which angle requires you to rotate, or turn, more from one arm to the other? (the one on the left) Remind students that, even though the arc in the angle on the right looks larger, this does not mean that the angle is larger; the sides of the angle are rays so they can be extended, and that does not change the angle. Again, slide the transparencies one on top of the other to compare the angles. Have a volunteer extend the arms of the angles. Students should clearly see that the angle on the left is larger. Repeat with the pair of angles shown below: Draw the pairs of angles in the exercises below, one pair at a time, on the board. For each pair, have students decide which angle is larger, the angle on the left or the angle on the right. They can signal the answers for each pair by pointing their thumbs to the left or right. Exercises: Which angle is larger? a) b) c) Answers: a) left, b) left, c) right Review degrees. Draw on the board: SAY: These angles look to be about the same size. ASK: How can I check which angle is larger? (measure them) In what units do we measure angles? (degrees) Remind students that an angle that measures 1 degree is a very small angle. Have students look at the picture in the top box on AP Book 8.1 p. 79 to see an angle that measures 1 degree. Explain that we divide a full turn into 360 equal divisions and call them degrees. A right angle measures 90 degrees. Draw on the board: D-24 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry Explain that this arc shows a full turn or full rotation around the intersection point of the lines. ASK: What do you know about the lines in this picture? (they are perpendicular) How many right angles does the picture show? (4) SAY: There are four right angles in a full turn or rotation, so each right angle is a quarter turn. When you have a right angle with an arm that points straight up (trace it on the picture) and an arm that points directly to the right (trace it on the picture), you need to turn one arm a quarter turn to get to the other arm. ASK: If a full turn is 360 degrees, how many degrees are in a quarter turn? (90 degrees) How do you know? (360 ÷ 4 = 90) Introduce the notation for degrees. Explain that writing the word “degrees” takes time, so people often use a symbol instead. The symbol is a small raised circle that is written after the number. Write “90°” on the board as you SAY: for example, the measure of a right angle is written as 90°. Introduce acute and obtuse angles. Explain that angles that are smaller than a right angle are called acute angles, and angles that are larger than a right angle are called obtuse angles. You might point out the connections to these words: acute means “sharp” and obtuse means “blunt” or “not pointed.” Before assigning the following exercises, point out that even though a 1 angle is very small, some angles can have a measure between two whole degree measures, so we can use decimals or fractions to show degree measures. Decimals are more common. Have students signal their answers in the next two exercises by making the letters A or O with their hands. Exercises: 1. Is the angle with this measurement acute or obtuse? a) 35 b) 95 c) 78 d) 129 e) 90.25 Answers: a) acute, b) obtuse, c) acute, d) obtuse, e) obtuse, f) acute 2. Is the angle acute or obtuse? a) b) c) d) f) 88.9 e) Answers: a) acute, b) obtuse, c) obtuse, d) acute, e) obtuse For each angle in Exercise 2 above, have students also say whether they expect the measure of each angle to be more than 90° or less than 90°. (less for acute, more for obtuse) Review protractors. ASK: What do we use for measuring angles? (a protractor) Have students examine their protractors and say how they are similar to rulers and how they are different. Draw attention to the fact that a protractor has two scales. Explain that having two identical scales going in different directions allows you to measure the angles from both sides, but this also means that you need to decide which scale you will use each time. Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-25 Draw the picture on the left below on the board and explain that this protractor is like the protractor in Question 3 on AP Book 8.1 p. 79. Explain that a good way to decide which scale to use is to look at which scale starts with a zero on one of the arms of the angle. ASK: Which scale has a zero on one of the angle arms, the inner scale or the outer scale? (the outer scale) Cover or cross out the inner scale. ASK: What is the measure of the angle? (150) Is this angle an acute angle or an obtuse angle? (obtuse) How do you know? (it is more than 90) Repeat with the picture below on the right. Have students complete Questions 3 and 4 on AP Book 8.1 pp. 79–80. Placing protractors on angles. Point out the base line and the origin on a protractor (see gray box on AP Book 8.1 p. 80). Have students find the base line and the origin on their own protractors. Draw an angle on the board and demonstrate how to place a protractor correctly so that the base line lines up with one arm of the angle and the origin is at the vertex. Point out that this is similar to placing a ruler with the 0 at the beginning of the object you are measuring. Have students use rulers to draw an acute angle in their notebooks and ask them to place their protractors correctly. Circulate in the classroom to check that all students have done so. Then have students measure the angle they drew. Repeat with an obtuse angle. Have students exchange notebooks with a partner and measure each other’s angles to check their work. Angles in a polygon. Draw a parallelogram on the board. Explain that we can measure angles inside shapes. SAY: The measure of an angle inside the shape is the amount of rotation between one arm (one side of the shape) and another arm (a different side of the shape). Draw an arc to emphasize the angle between the sides, as shown below: Point to the parallelogram on the board as you remind students that parallelograms have four sides with two pairs of parallel sides and pairs of opposite sides that are equal in length. Have students each a use ruler to draw a large parallelogram that is not a rectangle in their notebooks, measure all four angles in the parallelogram, then exchange notebooks with a partner and check their partner’s answers. Opposite angles in a parallelogram are equal. Discuss students’ findings from measuring the angles in the parallelogram above. ASK: Did you find any angles that had equal measurements? (yes) Which angles were equal? (angles opposite each other in the shape, or opposite angles) D-26 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry Did everyone have the same parallelogram? (no) Did everyone get equal opposite angles? (yes) Will all parallelograms have equal opposite angles? (yes) Point out that the sides of a shape are often too short to conveniently measure the angle. However, the sides of the shape are actually the arms of an angle starting at the vertex and extending as rays, so you can extend these rays farther. Extend two sides of the parallelogram you drew on the board. Point out that the parallelogram is inside the angle—it is part of the space between the arms. Have a volunteer measure the angle. Drawing angles. Model drawing an angle of 60 step by step, emphasizing the correct positioning of the protractor: Step 1: Draw a ray. Step 2: Place the protractor on the ray. Line up the base line of the protractor on the ray. Line up the origin of the protractor on the endpoint of the ray. Step 3: Follow the scale that has a zero on the ray. Find the mark for 60. Make a mark beside the protractor at the 60. Step 4: Remove the protractor. Use a ruler to draw a ray from the endpoint to the mark. Exercises: 1. a) Draw a line segment 7 cm long. Label it AB. b) Use point A as a vertex. Draw an angle of 40. Extend the second arm of the angle so that it is at least 12 cm long. c) Use point B as a vertex. Draw an angle of 105 so that the second arm of the angle intersects the ray you drew in part b). d) Label the intersection point of the rays C. Measure the angle ACB. Answer: d) ACB = 35 2. a) Draw a line segment 6 cm long. Label it DE. b) Use point D as a vertex. Draw an angle of 53. Extend the second arm of the angle and mark point F on it so that DF = 10 cm. c) Draw the line segment EF. d) Measure the sides and the angles of triangle DEF. Answer: d) DEF = 90, DFE = 37, EF = 8 cm Have students exchange notebooks and measure the sides and the angles of the triangles their partners drew to check each other’s answers. Constructing a perpendicular through a point. Draw a line on the board and mark a point P outside it. Model the steps for constructing a perpendicular through a point that is not on the line: Step 1: Place the protractor so that the line passes through the origin and the mark for 90. Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-27 Step 2: Slide the protractor so that the flat side touches the point P. Step 3: Draw a new line along the flat side of the protractor through point P and across the original line. Circulate among students to ensure that they are using the protractors and rulers correctly as they work on the exercises below. Exercises: Draw a pair of perpendicular lines that are on a slant—in other words, they are neither vertical nor horizontal—and a point not on the lines. Draw perpendiculars to the slant lines through the point. What quadrilateral have you constructed? Answer: rectangle Bonus: Draw a slant line and a point not on the line. Using a protractor and a ruler, draw a square that has one side on the slant line and one of the vertices at the point you drew. Answer: Draw a perpendicular through the given point to the given line. Measure the distance from the point along the perpendicular to the line. Then mark a new point on the given line that is that same distance from the intersection as the given point. Draw a perpendicular to the given line through this point as well. Finally, draw a perpendicular to the last line through the given point. Alternatively, when drawing the second last perpendicular, mark a point on that perpendicular that is the same distance from the given line as the given point is, and on the same side of the given line as the given point. Join this new point to the given point. In Activity 1, below, students will draw and measure angles to reinforce the understanding that the size of the angle does not depend on the length of the arms. They will learn to draw angles of a given measure and understand the need to follow the specific procedures for doing so. In Activity 2, students will practice measuring angles of polygons. In Activity 3, students will learn to construct right triangles, including the need to use specific tools. In Activity 4, students will construct a parallelogram and review the fact that its opposite angles are equal. Activities 1–4 Use The Geometer’s Sketchpad® for the following four activities. (MP.5) 1. Drawing and measuring angles. a) Teach students to draw and measure angles by following these steps: Step 1: Draw ray AB. Step 2: Draw ray AC. Step 3: Mark and label point D on AB and point E on AC. Step 4: Select points in this order: D, A, E. Use the Measure menu options to measure angle DAE. Have students move points D and E. Draw students’ attention to the fact that the points stay on the rays they were drawn on and that the angle measure does not change. Have students move D-28 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry points A, B, or C. ASK: Does the angle measure change? (yes, because the placement of the rays change) Do points D and E stay on the rays? (yes) Students will need this angle for parts b) and d), below. b) Have students modify the angle they drew in part a) to try to make the angle measure exactly 30. ASK: Is this hard or easy to do? (hard) Have students move the angle around to see that the measure changes again. c) Teach students to draw an angle of 30 by following these steps: Step 1: Draw ray FG. Step 2: Select vertex F. Use the Transform menu options (Mark center) to mark F as the center of rotation. Step 3: Select the ray. Use the Transform menu options to rotate the ray around the marked center. When selecting the angle of rotation, remember to mark it as degrees. Have students measure the angle. They will need to create points on the arms. Have students move the angle and the points on the arms to see how it changes. ASK: Does the angle measure change? (no) d) Teach students to draw an angle equal to DAE from part a) by following these steps: Step 1: Draw and label point J away from angle DAE. Construct a ray JK. Step 2: Mark point J as a center of rotation. Step 3: Select the measure of DAE. Use the Transform menu options (Mark angle) to mark this measure as the angle of rotation. Step 4: Select ray JK. Use the Transform menu options (Rotate) to rotate the ray around J by the selected angle. Mark a point on the new ray and label it M. Have students measure angle KJM. Have them try to modify both angles DAE and KJM to see that the angle measurements stay equal even when angle DAE changes. (MP.5) 2. Measuring angles of polygons. Have students draw polygons and measure the sizes of the angles and the lengths of the sides of these polygons. Have students check that the angle measurements they find make sense. For example, the software sometimes measures angles in the wrong direction, producing an answer less than 180° for reflexive angles (e.g., angles such as ABC in the quadrilateral below). D A B C Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-29 (MP.5) 3. Constructing right triangles. Have students draw a triangle using the Polygon tool. Ask them to move the points around to make it look like a triangle with a right angle. Remind students this is called a right triangle. Then ask them to measure the angles of the triangle and to check whether it is indeed a right triangle. ASK: Is it easy to draw a perfect right triangle this way? (no) If you move the points around, does the triangle remain a right triangle? (no) Have students think about how they draw a right triangle on paper. ASK: What instruments do you use and why? (a protractor, to make sure the triangle has a right angle) What could we use instead of protractors in The Geometer’s Sketchpad®? (perpendicular lines) Have students draw a right triangle in The Geometer’s Sketchpad® by following these steps: Step 1: Draw a line segment AB. Step 2: Draw a line perpendicular to AB through point B. Step 3: Mark a point anywhere on the perpendicular line you drew and label it C. Step 4: Use the Polygon tool to construct a triangle ABC. Have students check that triangle ABC remains a right triangle even if the vertices are moved around. 4. Angles in a parallelogram. a) Draw and label a line segment AB. From B, draw another line segment so that they form an angle. Label the second line segment BC. b) Draw a line parallel to AB through C. c) Draw a line parallel to BC through A. d) Mark the intersection point of the lines you drew in parts b) and c). Label it D. e) Use the Polygon tool to create a quadrilateral ACBD. What type of quadrilateral is ACBD? f) Measure the angles in ACBD. What do you notice about the angles that are opposite? g) Move the vertices of ACBD. Does the type of quadrilateral change? Do the angles change? Do the opposite angles stay equal? Answers: e) parallelogram, f) opposite angles are equal, g) the type of quadrilateral does not change, the angles change, the opposite angles stay equal (end of activities) For a summary related to angles and using protractors, you can provide students with BLM Using Protractors (Summary). Extensions 1. Students can use geoboards and elastics to make right, acute, and obtuse angles. When students are comfortable doing that, they can create figures with the following given angles. a) a triangle with 3 acute angles b) a quadrilateral with 0, 2, or 4 right angles c) a quadrilateral with 1 right angle d) a shape with 3 right angles e) a quadrilateral with 3 acute angles D-30 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry Sample Answers: Students can also try to create polygons similar to those created on geoboards using protractors and rulers. (MP.4) 2. a) What is the angle between the hands of an analog clock at 3:00? Can you tell without using a protractor? b) A minute hand rotates from one mark to the next, every minute. What is its angle of rotation in 1 minute? c) How many degrees does the hour hand turn in 1 hour? In 1 minute? d) The time is 12:24. How much did the hour hand turn from the vertical position (12)? e) The time is 12:24. How much did the minute hand turn from the vertical position (12)? f) What is the angle between the hands on an analog clock at 12:24? g) Find the angle between the hands on an analog clock at 1:36 and again at 3:48. Answers: a) At 3:00 the minute hand points to the 12, so straight vertical. The hour hand points to the 3, so straight horizontal. The angle is 90. b) An hour is 60 minutes and a whole circle is 360. Each minute the minute hand turns 360 ÷ 60 = 6. c) An hour hand makes a full 360 turn in 12 hours, so it turns 360 ÷ 12 = 30 every hour. Each hour is 60 minutes, so the hour hand turns 30 ÷ 60 = 0.5 every minute. d) The hour hand turned 24 × 0.5 = 12. e) The minute hand turned 24 × 6 = 144. f) The angle between the hands at 12:24 is 144 − 12 = 132. g) The angle between the hands at 1:36 is 216 − 48 = 168 (48 for the hour hand because 30 for 1 hour + 36 × 0.5 for minutes = 18), and the angle between the hands at 3:48 is 288 − 114 = 174. 3. Use The Geometer’s Sketchpad® to create: a) a triangle with each side equal to 3 cm and each angle equal to 60 b) a quadrilateral with each side equal to 3 cm and each angle equal to 90 c) a pentagon with each side equal to 3 cm and each angle equal to 108 d) a hexagon with each side equal to 3 cm and each angle equal to 120 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-31 G8-5 Sum of the Angles in a Triangle Pages 82–83 Standards: 8.G.A.5 Goals: Students will informally establish that the angles in a triangle add to 180° and use this to find the angles in a triangle. Prior Knowledge Required: Can identify and name angles and polygons Can measure and construct angles Knows that angle measures are additive Vocabulary: endpoint, intersect, intersection point, reflexive angles, straight angle Materials: BLM Using Protractors (Summary) (p. D-115, optional) protractors The Geometer’s Sketchpad® (optional) BLM Sum of the Angles in a Triangle and a Quadrilateral (p. D-116) BLM Sum of the Angles in a Polygon (p. D-117, see Extension 1) NOTE: To have students discover the fact that interior angles of a triangle add to 180, you can either work through the introduction below or do Activity 1. Discovering the sum of the angles in a triangle. Draw an angle on the board. Remind students how to place a protractor correctly to measure an angle. Have a volunteer measure the angle on the board. Remind students how to construct an angle of a given measure. For example, you might ask students to draw two angles, 35 and 123, exchange notebooks with a partner, and check each other’s work. To review drawing angles, students could refer to BLM Using Protractors (Summary). Have students construct a triangle with angles 50 and 30, and measure the third angle in the triangle. (100) Repeat with a triangle with angles 90 and 20. (70) Have students add the angles in both triangles. ASK: What do you notice? (the three angles always add to 180) Ask students to construct a triangle of their choice, measure the angles, and add them. Have partners to exchange notebooks and check each other’s work. ASK: Did everyone get the same triangle? (no) Did everyone get angles adding to 180? (yes) If anyone answers no, have them check the measurements to find the mistake. SAY: Angles in a triangle always add to 180. D-32 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry Activity 1 Use The Geometer’s Sketchpad® for this activity. Sum of the angles in a triangle. Have students construct a triangle using the Polygon tool. Then have them measure the angles of the triangle. Remind students to check that each angle measure makes sense: they need to be smaller than 180. Show students how to add the angles in the triangle using the Number menu option: Use the Calculate option in the Number menu options. In the calculation window, write an expression to add the angle measures. Click on each angle measure to make them appear in the expression. Add the three angle measures. Have students move the vertices of the triangle around and watch how the angles change, but the sum remains 180. (MP.5) Point out that, when we use protractors, we cannot measure angles with great precision. Software measures angles with better precision, but it displays the rounded answer. If you add the answers by hand, you might not get 180. The software adds the angles before rounding them. (end of activity) Finding the measure of the angles using the sum of the angles in a triangle. Draw a triangle on the board and write the measure of two of the angles in the triangle. ASK: How can I find the measure of the third angle? (subtract: 180 minus the sum of the other two angles) Work through the first exercise below together and then have students work individually. Exercises: Find the missing angle in the triangle. a) b) c) A D 45 53 C 64 65 B E 47 d) S H F G 45 Answers: a) 51, b) 80, c) 90, d) 44 K 115 U 21 N (MP.2) Finding the measure of the angles using information presented symbolically. Explain that we can usually use an equal number of small arcs to show that the angles in a diagram are equal. For example, we could change the diagram in part c) in the exercises above by erasing the measure of one of the angles, G or H, and marking the two angles with an arc. Show the change on the diagram, as shown below: H 45 K G Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-33 Draw on the board: M 76 L K ASK: What do the angles in this triangle add to? (180) What is the sum of angles K and L? (104) How do you know? (180 − 76 = 104) What does the diagram tell us about the sizes of angles K and L? (they are the same) What is the size of each of these two angles? (52) How do you know? (104 ÷ 2 = 52) Before assigning the next exercises, remind students that right angles are labeled with a small square and that a right angle measures 90. Exercises: Find the missing angle in the triangle. a) b) A D E C 63 c) A 32 F 55 C N B Answers: a) 27, b) 58, c) 70 SAY: Suppose a triangle has two equal angles. One of the angles in this triangle is 90. What are the sizes of the other two angles? ASK: Can the equal angles be 90 each? (no) Why not? (the equal angles would add to 180, leaving no room for the third angle) Invite a volunteer to draw the triangle on the board and mark the measures of the angles. Then have students find the size of the two missing angles in the triangle. (45) (MP.2, MP.3) Present a similar problem. SAY: A triangle has two equal angles. One of the angles in this triangle is 50. ASK: What are the sizes of the other two angles? Give students several minutes to think and then ASK: How is this problem different from the previous problem? (the given angle is an acute angle, not a right angle) Can a triangle have two angles of 50? (yes) What is the third angle then? (80) Sketch the triangle on the board and ask volunteers to mark the angles on the picture. Then draw on the board: 50 ASK: Can this situation happen? (yes) What are the measures of the other two angles? (65) Sum of the angles in a quadrilateral. Ask students to draw a quadrilateral, paying no attention to side lengths. Ask them to measure the angles in the quadrilateral and add them together. Have students exchange notebooks and check each other’s answers. Discuss the results: students should see that the angles in each quadrilateral add to 360 even though the D-34 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry quadrilaterals are different. You can repeat Activity 1 above using The Geometer’s Sketchpad®, but drawing quadrilaterals instead of triangles as an alternative to measuring with a protractor. Before having students work with the software, draw on the board: D B A C Explain that angles such as angle B are called reflexive angles. Point out that, with angle B, to turn from one arm to the other arm you need to make more than half a turn, so B measures more than 180. Point out to students that, if they create a quadrilateral with one reflexive angle, the software measures the angle incorrectly, and the sum of the angles changes, producing an incorrect result. They should avoid modifying quadrilaterals so that reflexive angles appear. You can also work through the informal proof of the sum of the angles in triangles and quadrilaterals in Activity 2. Activity 2 Remind students that an angle that is made by two rays going in opposite directions creating a line is called a straight angle and measures 180. Remind them also that a full turn measures 360. Draw on the board: ASK: What is the sum of the angles in this picture? (360) As a prompt, place a pencil along one of the arms and SAY: If an arm rotates through all of these angles, it will return back to this position, making a full turn. Turn the pencil to illustrate the rotation. Give students BLM Sum of the Angles in a Triangle and a Quadrilateral and have them cut out the triangle. Have them cut off the angles along the lines marked on the triangle and place them together to create a straight angle. Then have them answer the questions. (1. b) yes, c) 180, 180) Repeat with the quadrilateral, this time showing that the angles together make 360. Then have them answer the questions. (2. c) 360, 360) (end of activity) (MP.3) Proving that sum of the angles in a quadrilateral is 360 using logic. SAY: I would like to use logic to explain why the sum of the angles in a quadrilateral is 360. Draw on the board: A D B C Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-35 Ask students to think of how they could explain why the sum of the angles in ABCD is 360. Have them discuss their explanations in pairs and then in groups of four. Debrief as a class. The discussion should include the following: The sum of the angles in each triangle is 180. Some angles in the triangles are the same angles as in the quadrilateral (B and D); other angles combine to become angles in the quadrilateral: DAC + CAB = DAB DCA + ACB = DCB Thus the sum of the angles in the quadrilateral is the same as the sum of the angles in the two triangles combined. Extensions (MP.7) 1. Have students work through BLM Sum of the Angles in a Polygon. Selected Answers: b) Number of Expression for Sum of Number Triangles the Sum of Interior Polygon of Created by Interior Angles Angles Sides Diagonals Quadrilateral 4 2 180 × 2 360 Pentagon 5 3 180 × 3 540 Hexagon 6 4 180 × 4 720 Heptagon 7 5 180 × 5 900 Octagon 8 6 180 × 6 1080 Nonagon 9 7 180 × 7 1260 Decagon 10 8 180 × 8 1440 n-sided polygon n n−2 180 × (n − 2) 180 × (n − 2) 2. In the diagram below, triangle EFG is a right triangle and FGH = FHG. Also, EGF = 26 and EFH = 42. Find the measure of GHF. E 42 F H 26 G Solution: From the sum of the angles in triangle EFG, EFG = 180 − (90 + 26) = 64. Then HFG = 64 − 42 = 22. From the sum of the angles in triangle FGH, FGH + FHG = 180 − 22 = 158. Since FGH = FHG, they both measure 158 2 = 79. D-36 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry G8-6 Triangles Pages 84–85 Standards: 8.G.A.5 Goals: Students will classify triangles by the number of equal sides and by the size of angles. Students will use the properties of triangles to find the size of angles in the triangles. Prior Knowledge Required: Can identify and name angles Can measure angles Knows that angles in a triangle add to 180º Vocabulary: acute-angled triangle, acute triangle, equilateral, intersect, intersection point, isosceles, obtuse-angled triangle, obtuse triangle, right-angled triangle, right triangle, scalene Materials: BLM Triangles for Folding (p. D-118) scissors blank paper cut into the shape of a hand-drawn circle protractors BLM Geometric Terms (pp. D-111–113, see Extension 2) Introduce classification of triangles by angles. Draw a few different triangles on the board, including acute, obtuse, and right triangles. Number the angles in each as 1, 2, and 3. Have students identify the largest angle in each triangle by raising the number of fingers equal to the number in the angle. Circle the label for the largest angle and ask students to identify the angle as acute, obtuse, or a right angle. Explain that triangles are classified by the size of the largest angle. Triangles in which the largest angle is acute are called acute-angled or acute triangles; triangles in which the largest angle is a right angle are called right-angled or right triangles; and triangles in which the largest angle is obtuse are called obtuse-angled or obtuse triangles. Have students identify each triangle on the board as acute, right, or obtuse. NOTE: Students might be familiar with a different form of classification from earlier grades: a triangle that has an obtuse angle is an obtuse triangle, and a triangle that has a right angle is a right triangle. Triangles that have no obtuse and no right angles have three acute angles, and so are called acute triangles. If the issue arises, discuss with students that the definitions are essentially equivalent. ASK: Can a triangle can have more than one obtuse or more than one right angle? (no) Why not? (the sum of the triangle’s three angles would be more than 180) Point out that this means that the largest angle in a triangle that has an obtuse angle is the Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-37 obtuse angle, so the triangle will be called obtuse by both definitions. The same argument applies to the right triangle. Finally, if the largest angle is acute, then all three angles in the triangle are acute, and the triangle is an acute triangle by both definitions. Classifying triangles by angles. Draw on the board: Ask students to identify the largest angle. (angle 1) ASK: What type of triangle is this? Students are likely to say this is a right triangle. Invite a volunteer to check using a protractor. The triangle is an acute triangle because the largest angle is 85°. SAY: This is why it is important to always check the measure of an angle that looks like a right angle. Write on the board: right triangle acute triangle obtuse triangle Draw the triangles in the exercises below one at a time and have students signal the answer to the exercises by pointing in the direction of the type of triangle (left for acute triangles, up for right triangles, or right for obtuse triangles). After students signal the answer for part d), have a volunteer check that the angle that looks like a right angle is indeed a right angle. Exercises: Classify the triangle as acute, right, or obtuse. a) b) c) d) Answers: a) acute, b) obtuse, c) acute, d) right Using the sum of the angles in a triangle to classify triangles. Remind students that angles in a triangle add to 180. Review finding the measure of the third angle from the measures of the other two. Draw on the board: C A 41 44 B Ask students to find the measure of angle C. (95) Have a volunteer explain how they found the answer. (180 − (41 + 44) = 180 − 85 = 95) ASK: Which angle is the largest angle in the D-38 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry triangle? (C) Is angle C a right, acute, or obtuse angle? (obtuse) What type of triangle is ABC? (obtuse triangle) ASK: How could you figure out that ABC is an obtuse triangle if I have only given you the measures of angles A and B, without drawing the triangle? (use the same method: find the measure of the third angle, decide which angle is the largest, and then check what type of angle it is) SAY: Triangle DEF has one angle of 35 and another angle of 45. ASK: What is the measure of the third angle? (100) How do you know? (180 − (35 + 45) = 100) Have a volunteer write the calculation on the board. ASK: What is the largest angle in the triangle, 35, 45, or 100? (100) What type of triangle is DEF? (obtuse triangle) Exercises: What type of triangle is triangle GHI? a) G = H = 47 b) G = 50, I = 47 c) I = 63, H = 27 d) I = H = 59 e) G = H = 36.5 f) G = 90, I = 35 Selected solution: a) I = 180 − (47 + 47) = 86, so the largest angle is I and it is an acute angle, so triangle GHI is an acute triangle Answers: b) acute, c) right, d) acute, e) obtuse, f) right (MP.8) SAY: In one of the questions in the previous exercises you could find the answer without finding the third angle. ASK: Which question was that? (part f) How could you solve this problem without finding the size of the third angle? (G is 90, a right angle, so it has to be the largest angle in the triangle, and thus the triangle is a right triangle) Introduce classification by triangle side lengths. Explain that another way to classify triangles is by using side lengths. Remind students that we mark sides of equal length with equal numbers of hash marks. SAY: Triangles with at least two equal sides are called isosceles triangles. Triangles that have no equal sides are called scalene triangles. Exercises: Classify the triangles as isosceles or scalene. a) b) c) d) e) Answers: a) isosceles, b) scalene, c) isosceles, d) scalene, e) isosceles Introduce equilateral triangles. Explain that shapes with all sides equal are called equilateral. For example, a rhombus is an equilateral quadrilateral. Triangles with three equal sides are called equilateral triangles. ASK: Are equilateral triangles isosceles or scalene triangles? (isosceles triangles) Is one of the triangles in the previous exercises equilateral? (yes) Which one? (part e)) Classify triangles using both classifications. Explain that you can label triangles using both classifications, according to the largest angle and by the number of equal sides. For example, if you fold a rectangular sheet of paper along the diagonal, you get a right scalene triangle. Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-39 Exercises: 1. Classify the triangles in the previous exercises using both classifications. Answers: a) acute isosceles triangle, b) obtuse scalene triangle, c) acute isosceles triangle, d) right scalene triangle, e) acute equilateral triangle 2. On grid paper draw: a) a right isosceles triangle c) an obtuse isosceles triangle Sample answers: a) b) b) a right scalene triangle d) an acute scalene triangle c) d) Activity 1 Isosceles triangles also have equal angles. Remind students that they can use paper-folding to check if sides or angles of triangles are equal. Give students BLM Triangles for Folding and have them cut out the triangles. Have students fold the triangles so that they can see which are isosceles and which are scalene. Point out that this folding also compares two of the angles in the triangle. Have students label each triangle with its type and mark the equal angles with the equal number of arcs. (isosceles triangles: A, D, F, H, I; scalene triangles: B, C, and E; triangle H is an equilateral triangle; triangle G is a right scalene triangle) ASK: What do you notice about the angles in isosceles triangles? (there are at least two equal angles) Are there equal angles in scalene triangles? (no) (end of activity) SAY: In an isosceles triangle, the angles between the equal sides and the third side are always equal. Explain also that, if a triangle has two equal angles, it is always an isosceles triangle. Draw on the board: B A C ASK: Which sides are equal in this triangle? (AB and BC) Point out that the equal angles are both adjacent to the third, unequal side of the triangle. Size of the angles in an equilateral triangle. ASK: Are there any equilateral triangles in your collection from BLM Triangles for Folding? (yes, triangle H) What do you notice about its angles? (they are all equal) How do you know? (when we fold the triangle to check the angles, we find that any 2 angles are the same and the sides are the same length; the same is true when we check the next side) D-40 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry (MP.2) SAY: I would like to find the size of the angles in an equilateral triangle without actually measuring them. ASK: How could we do it? Give students a few minutes to think and then discuss the solution in pairs. PROMPT: What is the sum of the angles in any triangle? (180) What do we know about the angles in an equilateral triangle? (they are all equal) What is the size of one angle? (180 ÷ 3 = 60) Have students measure the angles in triangle H from BLM Triangles for Folding to check. Finding angles in isosceles triangles. Work through the first two problems in the following exercises as a class. Then have students solve the rest of the exercises individually. Exercises: Find the missing angles in the isosceles triangle. a) b) c) A D W 63 C B V 124 E d) 24 F U R 60 O N Sample solution: a) A = C = 63, so B = 180 − (63 + 63) = 180 − 126 = 54 Answers: b) E = D = 28, c) U = 24, W = 132, d) R = N = 60 Activities 2–4 2. Creating right angles. Show students a piece of paper cut in the shape of a hand-drawn circle. Ask them how they could make a right angle from this piece of paper. Have students draw a circle by hand, cut it out, and use it to test their ideas. To prompt students to see the solution, remind them that there are 360° in a full turn, and they need 90°. ASK: What fraction of the whole turn is a right angle? (a quarter turn) How can you fold the paper into four equal parts? (fold the shape in half, and then fold it a second time so that the crease is folded against itself) Point out that students have made a right angle, which they can use to distinguish between right, acute, and obtuse angles. All they need to do to classify angles is compare the angle in question to the right angle they created. Students can use the right angle they created as the square corner they use to distinguish between right, acute, and obtuse angles in Question 2 on AP Book 8.1 p. 84. 3. Have students use the right angle made from the circle-shaped piece of paper from Activity 2 to create a square. If necessary, remind students that they can use the circle to make a right angle and ask students about the defining properties of squares. (MP.2) 4. Have students create a square from a large rectangular sheet of paper following the steps below: Step 1: Fold the short side of the paper down onto the long side to create a right isosceles triangle. The extra part will be a rectangle. Step 2: Fold the extra part of the page—a rectangle—over the triangle. Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-41 Step 3: Unfold the paper and cut off the rectangle. (see diagram below) Now have students create a special triangle. Fold the square in half vertically (not diagonally) so that a crease divides the square into two rectangles. (see a), below) Fold the top right corner of the square down so that the top right vertex (or corner) touches the crease. The vertex should be slightly above the bottom edge. (see b), below) Mark the point where the corner touches crease and trace a line along the folded-over edge of the square with a pencil. (see c), below) Unfold. (see d), below) With the top left corner of the square, repeat steps b) to d). The top left corner will touch the crease at the same point, which is the vertex of your new triangle. (see e), below) Cut the triangle out along the traced lines. What type of triangle have you created? Explain. Answer: I started with a square and ensured in step b) that the sides of the triangle are equal to the side of the square. So it is an equilateral triangle. (end of activities) Extensions (MP.1) 1. Remind students that the longest side in a triangle is always shorter than the sum of the other two sides. For example, you cannot make a triangle with sides 1 cm, 2 cm, and 3 cm. How many triangles with sides that are a whole number of centimeters in length are there for the perimeter? For each triangle, say what type of triangle it is. a) perimeter = 3 cm b) perimeter = 4 cm c) perimeter = 5 cm d) perimeter = 7 cm e) perimeter = 8 cm f) perimeter = 9 cm Answers: a) 1 equilateral triangle with all sides = 1 cm; b) no triangles; c) 1 isosceles triangle with sides of 2, 2, and 1 cm; d) 2 isosceles triangles with sides of 3, 2, and 2 cm or 3, 3, and 1 cm; e) 1 isosceles triangle with sides of 3, 3, and 2 cm; f) 3 triangles: an equilateral triangle with all sides = 3 cm, 1 isosceles triangle with sides of 4, 4, and 1 cm, 1 scalene triangle with sides of 2, 3, and 4 cm (MP.7) 2. Remind students that a Venn diagram can group shapes that have similar and different properties—for example, by showing shapes that have one property or the other property, both properties, or neither property. Remind them that shapes that have neither D-42 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry property are placed outside of both groups. Write these properties on the board for use on Venn diagrams: Group 1 Acute Right Obtuse Group 2 Scalene Isosceles Equilateral Ask students to pick a property from each column and make a Venn diagram about triangles using both properties. Point out that, in some cases, a Venn diagram will have an empty region. Also, remind them that equilateral triangles are also isosceles, so when they are sorting triangles using “isosceles,” equilateral triangles should be within this group. Students can refer to BLM Geometric Terms as needed. When students finish, ask them to try to draw an example triangle in each region of their Venn diagram. If they cannot manage to produce a triangle in one of the regions, ask them to explain to a partner what problems they encountered and to think together whether a triangle in that region is possible. Selected answers: Triangles Right Equilateral There is no intersection between the two groups because equilateral triangles have all angles equal to 60 and so cannot have a right angle. The triangle on the far right is an acute scalene triangle, so is not a right triangle and not equilateral. (MP.2) 3. Classify triangle ABC if АВ + ВС = АВ + АС = ВС + АС. Solution: АВ + ВС = АВ + АС, so BC = AC, and АВ + АС = ВС + АС, so AB = BC, and AB = BC = AC, and the triangle ABC is equilateral. Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-43 G8-7 Making a Geometric Sketch Pages 86–89 Standards: 8.G.A.5 Goals: Students will make quick sketches for problems, identifying the relevant information and solving problems using a sketch. Prior Knowledge Required: Can identify equal sides, equal angles, and right angles Is familiar with standard markings for equal sides, equal angles, and right angles Can classify triangles Knows that isosceles triangles have equal angles adjacent to the unequal side Can name angles and polygons and can identify a named angle or polygon Knows that the sum of the angles in a triangle is 180 and the sum of the angles in a quadrilateral is 360 Vocabulary: acute-angled triangle, acute triangle, endpoint, equilateral, intersect, intersection point, isosceles, obtuse-angled triangle, obtuse triangle, right-angled triangle, right triangle, sketch Review. Remind students how to mark equal sides, equal angles, and right angles. Review with students the classification of triangles according to the number of equal sides and the size of the angles and the properties of special quadrilaterals related to sides and angles. You might wish to draw the table below on the board, which summarizes this classification: Quadrilateral Properties parallelogram 2 pairs of equal parallel sides, equal opposite angles rectangle parallelogram, 4 right angles rhombus parallelogram, 4 equal sides square rectangle, rhombus, 4 right angles, 4 equal sides trapezoid exactly 1 pair of parallel sides Remind students that the angles of an equilateral triangle are equal to 60, and the angles that equal sides of an isosceles triangle make with the third side are equal. Making a sketch. Explain that a sketch is a quick drawing made without using instruments such as a ruler or protractor. SAY: Knowing how to make a sketch is an important math skill. Sketches can help us organize information, visualize described objects, see relationships, explore ideas, and solve problems. D-44 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry Have students complete Questions 1–15 on AP Book 8.1 pp. 86–89. You can use the exercises below as extra practice for students who work more quickly than others or for students who are struggling with a particular concept or step. Drawing diagrams accurately. Exercises: 1. Sketch an obtuse isosceles triangle and an acute isosceles triangle. Sample Answers: 2. Which sketch is better, A or B? A. B. A. A. A. B. B. B. Answers: a) B, b) A, c) B, d) A 3. Sketch the figures. a) line segment AC with AB = BC b) a rectangle 2 cm by 6 cm c) a rhombus with angles 20° and 160° d) triangle KLM with sides 5 cm, 3 cm, and 3 cm Sample Answers: 6 cm a) b) c) A B C 2 cm 3 cm d) 20 160 L K 5 cm M Drawing the shapes as generally as possible and not adding extra information accidentally. Explain that we need to draw shapes matching the description as precisely as we can, but it is important not to add something that is not given. For example, if we know a shape is a rectangle, we should not sketch a square. Though the square is a rectangle, squares have equal adjacent sides, not just equal opposite sides. Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-45 Exercises: (MP.6) Which sketch adds more information than in the description, A or B? Explain. a) Parallelogram ABCD has diagonals AC and BD. A. B B. B C A C D A D b) Quadrilateral EFGH has perpendicular diagonals EG and FH. F F A. B. E E G H G H Answers: a) Sketch B gives too much information—it meets the criteria but shows a rhombus, not a general parallelogram; b) Sketch A gives too much information—it meets the criteria but seems to have equal sides, so it is a rhombus Solving problems that require making different sketches. Exercises: 1. A parallelogram is made from two isosceles triangles, each with two sides of 3 cm and one side of 5 cm. What is the perimeter of the parallelogram? Make two sketches to show the different placement of the triangles in the parallelogram. Then solve the problem. Answer: 3 cm 3 cm 5 cm Perimeter = 4 × 3 cm = 12 cm 5 cm Perimeter = 2 × 3 cm + 2 × 5 cm = 16 cm 2. A parallelogram is made from two isosceles triangles that have an angle of 98 each. What are the angles of the parallelogram? Can the angle of 98 be one of the angles between one of the equal sides and the third side? Make two sketches to show the different placement of the triangles in the parallelogram. Then solve the problem. Answers: The 98 angle is an obtuse angle, so it has to be the angle between the two equal sides of the isosceles triangle: 98 41 98 Angles: 98, 98, 82, 82 D-46 Angles: 41, 41, 139, 139 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry Ignoring irrelevant information. Exercises: Make a sketch for the problem (without solving it). Ignore the unnecessary information. A traffic island has the shape of a parallelogram with one of the angles 65º. The island contains three shrubs and a circular flowerbed 1 m wide. What are the sizes of the angles of the traffic island? Bonus: Solve the problem. Answers: 65 Bonus: 65º, 115º, 65º, 115º Adding information that can be deduced. Exercises: Add to the sketch other information you can deduce. Then solve the problem. In triangle ABC, AB = BC and ∠A = 50. Point D is on the line segment AC, so that BD AC. What is the size of CBD? Answer: B A 50 50 D C CBD = 180 − (90 + 50) = 40 Solving problems by making sketches. Exercises: a) The shorter side of a parallelogram is 5 cm. The longer side is 2 cm longer than the shorter side. What is the perimeter of the parallelogram? b) A square is cut into two identical parts and rearranged to make a rectangle. The short side of the rectangle is 6 cm. How long is the long side of the rectangle? c) A square is cut into two identical parts and rearranged to make a triangle. What are the angles of the triangle? Answers: a) 24 cm; b) 24 cm; c) 45, 45, 90 Extension (MP.1) Make a sketch to find the answer. Ron drew rhombus KLMN and its diagonal NL. He measured NML = 96. a) What is the size of NKL? b) What is the size of KNL? Explain how you know. Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-47 Answers: K N L 96 96 M a) Since KLMN is a rhombus, it is a parallelogram, its opposite angles are equal, and NML = NKL = 96. b) Since KLMN is a rhombus, KL = KN, so triangle KNL is isosceles, and NLK = KNL. From the sum of the angles in a triangle, NLK + KNL = 180 − 96 = 84, so NLK = KNL = 42. D-48 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry G8-8 Counterexamples Pages 90–92 Standards: preparation for 8.G.A.5, 8.G.B.7 Goals: Students will recognize and create counterexamples. Prior Knowledge Required: Can identify polygons Can identify vowels and consonants Can identify even and odd numbers Is familiar with standard markings for equal sides, equal angles, and right angles Can classify triangles Knows that isosceles triangles have equal angles adjacent to the unequal side Can classify quadrilaterals Knows the basic properties of special quadrilaterals Vocabulary: acute triangle, counterexample, equilateral, false, isosceles, obtuse triangle, right triangle, true Materials: Anno’s Hat Tricks by Akihiro Nozaki and Mitsumasa Anno (see Extension 2) BLM Sudoku—Warm-Up (pp. D-119–120, see Extension 3) BLM Sudoku—Introduction (pp. D-121–122, see Extension 3) BLM Sudoku—Another Strategy (p. D-123, see Extension 3) BLM Sudoku—Advanced (p. D-124, see Extension 3) BLM Always, Sometimes, Never (p. D-125, see Extension 4) NOTE: Recognizing and using counterexamples is a mathematical practice (MP.3) that needs to be explicitly taught. Students will use what they learn in this lesson to solve problems in other lessons. Introduce the term counterexample. Draw on the board: All circles are shaded. Have a volunteer identify which circle shows that the statement is not true. (the white circle) Draw on the board: Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-49 Repeat with two different statements about the picture: All squares have a horizontal side. (third from the left square) All squares are striped. (far left square) SAY: An example that proves a statement false is called a counterexample to the statement. In other words, a counterexample shows that the statement is not true. Exercises: (MP.3) 1. Which shape is the counterexample to the statement? A. B. C. D. a) All triangles are shaded. b) All triangles have a horizontal side. Bonus: All triangles are isosceles. Answers: a) C, b) B, Bonus: C (MP.3) 2. Find a counterexample for the statement. a) All animals that live in water are fish. b) All animals with two legs are humans. Sample answers: a) whales are mammals that live in water, b) birds have two legs and are not human Bonus: Make up your own false statement, and have a partner find a counterexample. Make sure the statement is true in some cases so that your partner needs to work to find a counterexample. For example, “All people in my class wear glasses” is a better statement than “All people in my class are aliens.” Recognizing when a statement does not apply to all examples. Write and draw on the board: All circles are shaded. OR If the shape in the picture is a circle, it is shaded. Explain that the sentences mean the same thing, but they use two different ways to say it. ASK: What are these statements about? (circles) Underline all the circles. Emphasize that the statements refer only to the circles; whether any of the other shapes are shaded or not is irrelevant. ASK: Are all circles shaded? (no) Have a student circle the counterexample. (E) Erase the underlining and the circling and repeat with the statements below, starting by first underlining the relevant shapes. Emphasize in each case that the sentence is only about the shapes you underline; the shapes that are not underlined are not relevant. All squares in the picture are big. (underline squares, counterexample: D) All squares in the picture are shaded. (underline squares, counterexamples: D, F) All big squares in the picture are shaded. (underline large squares, counterexample: F) All small circles in the picture are shaded. (underline small circles, counterexample: E) D-50 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry Exercises: 1. Use the picture of circles and squares above. List the shapes the statement is talking about. Then name the counterexample for the statement. a) All shaded shapes are circles. b) All white shapes are small. c) All small shapes are white. d) All small white shapes are squares. Answers: a) shaded shapes: A, B, C, counterexample: B; b) white shapes: D, E, F, counterexample: F; c) small shapes: C, D, E, counterexample: C; d) small white shapes: D, E, counterexample: E 2. Use the picture of circles and squares above. Name the counterexample for the statement. a) All shaded shapes are big. b) All white shapes are squares. c) All big shapes are squares. d) All big shapes are shaded. Answers: a) C, b) E, c) A, d) F 3. Name the counterexamples from the list. a) All numbers are either positive or negative. 9, −4, 9.6, 1/2, 0, −3.82 b) All right triangles are isosceles. A. B. C. D. c) If you add two numbers, the sum is larger than both addends. 3+4 5+0 6 + (−2) 0.4 + 1.2 Answers: a) 0; b) B, D; c) 5 + 0, 6 + (−2), −4 + 9 E. −4 + 9 Review the word “vowel” if necessary. The letters a, e, i, o, u, and sometimes y are vowels. Have students signal the answer to the following exercises with thumbs up for yes and thumbs down for no. Exercises: Is “Tom” a counterexample for the statement? a) All names have two vowels. b) All names have three letters. c) All names have four letters. d) All boys’ names start with D. e) All names are boys’ names. f) All names read the same backward and forward. Answers: a) yes, b) no, c) yes, d) yes, e) no, f) yes Finding counterexamples. Exercises: 1. Find a counterexample to both statements in the previous exercises for which “Tom” is not a counterexample. Find one example that works as a counterexample to both statements at the same time. Sample answer: Sara Bonus: Explain why there cannot be a counterexample to all six statements. Hint: Look at parts d) and e). Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-51 Answer: A counterexample to part d) has to be a male name. It cannot be a counterexample to part e) because the counterexample to that cannot be a boys’ name. Therefore, there cannot be a counterexample that disproves all six statements at the same time. Explain that, in geometry, you often need to show a counterexample to explain why the statement is false. A statement is often true about some shapes and not true about other shapes. For example, the statement “In a parallelogram, all sides are equal” is true about some parallelograms (rhombuses), but not true about all parallelograms. So, for a counterexample, we need to draw a parallelogram that does not have all equal sides. Write the statement on the board and invite a volunteer to draw a parallelogram that does not have all equal sides. Explain that, in the next exercises, students will need to tell what type of shapes they will be looking at to find a counterexample. For example, you would look at parallelograms to find a counterexample for the statement about parallelograms. Looking at hexagons would not make sense. Point out that two of the parts in the following exercises are exactly the same. They are just worded differently because geometric statements are often worded in different ways: some of them look like “all ___ are ____,” and others look like “if___, then____.” You might also point out that these statements are general statements about types of shapes, but specific shapes can provide counterexamples. Exercises: What shapes is the statement talking about? a) All rectangles are squares. b) All triangles have at least one right angle. c) All isosceles triangles have an angle of 60. d) All triangles with a 60 angle are equilateral. e) If a triangle has an angle of 60, then it is equilateral. f) If a quadrilateral has at least two right angles, then it is a rectangle. g) In all rectangles, the longer side is twice as long as the short side. h) In all trapezoids, the opposite sides are not equal. Bonus: If a quadrilateral has two right angles, it is a trapezoid or a parallelogram. Answers: a) rectangles, b) triangles, c) isosceles triangles, d) triangles with a 60 angle, e) triangles with a 60 angle, f) quadrilaterals with two right angles, g) rectangles, h) trapezoids, Bonus: quadrilaterals with two right angles SAY: To find a counterexample to the statement in part a) in the previous exercises, we need to look at rectangles. ASK: What must be special about the rectangle for it to be a counterexample of the statement? (it must be a rectangle that is not a square) SAY: All these sentences have two parts. When we are looking for a counterexample, we are looking at a shape the sentence is talking about in the first part, but the second part of the sentence is not true about that shape. D-52 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry Exercises: Describe the counterexample for each statement in the previous exercises. Answers: a) a rectangle that is not a square, b) a triangle that has no right angles, c) an isosceles triangle that has no 60 angle, d) a triangle with a 60 angle that is not equilateral, e) a triangle with a 60 angle that is not equilateral, f) a quadrilateral with two right angles that is not a rectangle, g) a rectangle where the longer side is not twice as long as the short side, h) a trapezoid with equal opposite sides, Bonus: a quadrilateral with two right angles but that is not a rectangle and not a trapezoid NOTE: If students struggle with the previous exercises, have them circle the two parts in the sentence first. Give students time to find the counterexamples following the descriptions they wrote. Encourage students to make a sketch of the shape they are looking for, as shown in the selected answers below: f) right trapezoid h) isosceles trapezoid Bonus: a quadrilateral that has two right angles that are opposite True or false? Explain that the next task is going to be harder. You are going to give students statements that might be either true or false, and they need to decide which ones are true and which ones are false, and provide a counterexample to the false statements. Exercises: Is the statement true or false? Provide a counterexample if it is false. a) All quadrilaterals have sum of interior angles equal to 360. b) All even numbers have a digit 4. c) All multiples of 5 have the ones digit 5 or 0. d) All positive numbers are greater than 1. e) If a triangle is isosceles, it has at least two equal angles. f) All quadrilaterals have at least two equal angles. Bonus: If a parallelogram has a right angle, it is a rectangle. Answers: a) true, b) false, counterexample: 2, c) true, d) false, counterexample: 0.5 or 1, e) true, f) false, counterexample: , Bonus: true Bonus: For each part, say whether the statement is true or false. Provide a counterexample if it is false. If the boxes are filled in with whole numbers, then the answer is a whole number. a) + b) – c) × d) ÷ Answers: a) true, b) true, c) true, d) false; counterexample: 3, 5 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-53 Extensions (MP.1) 1. Make up a statement so that the given word or number is a counterexample. a) the word “run” b) the number 8 Sample answers: a) all words have 4 letters, b) all numbers are prime (MP.4) 2. To help students practice making logical deductions, go through the book Anno’s Hat Tricks by Akihiro Nozaki and Mitsumasa Anno with the class. Work through the book over several days, a few pages at a time. The book is suitable for Grades 5–12 and students will let you know when the logic becomes too tough. (MP.7) 3. Have students complete BLM Sudoku—Warm-Up, BLM Sudoku—Introduction, and BLM Sudoku—Another Strategy. These BLMs introduce students to sudoku puzzles, which require substantial logical thinking to solve. Students can complete BLM Sudoku— Advanced for an added challenge. 4. Explain that a whole number is divisible by another whole number if the answer in the division is a whole number. For example, 12 is divisible by 1, 2, 3, 4, 6, and 12. Have students complete BLM Always, Sometimes, Never, which provides practice with logical thinking. D-54 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry G8-9 Congruence Pages 93–95 Standards: 8.G.A.2 Goals: Students will identify congruent shapes with emphasis on triangles. Students will use equality between corresponding angles and sides to find angles and sides of congruent triangles. Students will write congruence statements. Prior Knowledge Required: Can measure angles and sides of polygons Is familiar with notation for equal sides and angles Can name angles and polygons Is familiar with the symbol for angle Can classify triangles Vocabulary: congruence statement, congruent (), counterexample, isosceles, scalene Materials: paper shapes for demonstration tracing paper for demonstration scissors overhead projector rulers protractors transparency of BLM 1 cm Grid Paper (p. I-1) Introduce congruent shapes. SAY: Congruent shapes have the same size and shape, so if you put one shape on top of the other, they should match exactly. Explain that you might need to flip or turn the shapes to make them match. Attach the shapes below to the board: SAY: These shapes are congruent. I can turn the shape on the left a quarter turn clockwise and place it on top of the other shape to check. Demonstrate this with the shapes on the board. Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-55 SAY: Now we can see that they match exactly. Move the shapes apart and place them on the board as shown below: ASK: What do I need to do with the second shape to get it to match the first shape? (turn it 180 clockwise or counterclockwise and move over the first shape) Invite a volunteer to check. Repeat with the pair of shapes shown below, which require a flip through a vertical line instead of a turn: Attach the shapes below to the board: ASK: Is this new shape, at right, congruent to the others we have just seen? (no) Why not? (both of the shapes are made from a row of 4 squares with another square attached to the side, but in the shape on the left, the additional square is attached to one of the two middle squares; in the shape on the right, it is attached to the end square.) Explain that the only things that matter in congruence are size and shape. Position, color, pattern, and thickness of lines do not matter. For the exercises below, draw the four shapes for each part on a grid. Students can signal their answers by raising the number of fingers that corresponds to the correct answer. NOTE: Students might use shading as a clue. Remind them that shading does not matter. Exercises: Which shape is congruent to the top shape, 1, 2, or 3? a) b) 1. D-56 2. 3. 1. 2. 3. Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry c) 1. 2. 3. Answers: a) 1, b) 2, c) 2 For each of the previous exercises, ask students to describe informally how they can get the congruent shape from the original shape. (sample answers: a) rotate 90 counterclockwise, b) rotate 180 clockwise or counterclockwise or flip upside-down, c) rotate 180 clockwise or counterclockwise) Congruent triangles have sides of the same length. Explain that, for the rest of the lesson, we will be dealing with triangles. Draw on the board: Triangle 1 Triangle 2 B 7 cm A 4 cm C 5 cm Triangle 3 H E 4 cm D 5 cm 6.4 cm 4 cm 7 cm F I J 5 cm ASK: Which two triangles are congruent? Have students signal the answer. (1 and 2) Use tracing paper to copy Triangle 2 and cut the tracing out. Show how the cutout matches Triangle 1 exactly. Point out that you need to flip the triangle horizontally to get from 2 to 1. SAY: These two triangles have sides of the same length: both have a side that is 4 cm long, a side that is 5 cm long, and a side that is 7 cm long. ASK: How is the third triangle different from the other two triangles? (Triangle 3 does not have a side 7 cm long; Triangles 1 and 2 are obtuse triangles, but Triangle 3 seems to be a right triangle) Emphasize that to explain why two triangles are not congruent, you can say that one of them has a side of a different length than the other. Congruent triangles have to have sides that are the same lengths. Remind students that they can use the same number of hash marks to show line segments (such as sides) of equal length. Invite a volunteer to mark the sides that are the same length in Triangles 1 and 2 with the same number of hash marks. Ask students to write which side in Triangle 1 is equal to which side in Triangle 2. (AB = EF, BC = ED, AC = FD) Keep the triangles on the board for later. NOTE: Even though it is good practice to use the same order for corresponding vertices in the triangles (so AB = FE, not AB = EF in the example), the equality described above is between line segments, so writing the vertices in the opposite order is not a mistake. Model the good practice for students, but there is no need to emphasize it at this point of the lesson. Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-57 Exercises: The triangles are congruent. a) Sketch the triangles. Mark the equal sides with hash marks. E B i) ii) iii) W D F H A C K M X Y V U L I G Z b) Write which side in one triangle is equal to which side in the other triangle. Selected answers: b) i) DE = GH, DF = GI, EF = HI; ii) AB = KL, BC = LM, AC = KM; iii) UV = ZY, VW = YX, UW = ZX To check that the equal sides are identified correctly, explain that you can imagine placing one of the triangles on top of the other so that the sides match. ASK: What do you need to do with triangle ABC to get triangle KLM? (flip it upside-down) What side will be placed on top of what side? (AB on top of KL, BC on top of LM, AC on top of KM) Repeat with the other two pairs of triangles, which need, in part i), a slide to place GHI on top of triangle DEF, and in part iii), a 90 rotation clockwise to get from UVW to XYZ. Congruent triangles have angles of the same size. Return to the earlier example of Triangles 1, 2, and 3. Demonstrate using the cutouts that Triangles 1 and 2 have angles that match exactly. Remind students that equal angles are marked with the same number of arcs. Invite a volunteer to mark the equal angles in triangles ABC and DEF. (A = F, B = E, C = D) Exercises: The triangles are congruent. a) Sketch the triangles. Mark the equal angles with matching arcs. E i) ii) W iii) V D C Y F B A M H I K J U Z X L b) Write which angle in one triangle is equal to which angle in the other triangle. Answers: b) i) D = I, E = H, F = J; ii) U = X, V = Y, W = Z; iii) A = K, B = M, C = L For the pairs of triangles in the previous exercises, discuss strategies for finding the equal angles. Concentrating on the size of the angles is one strategy. Students should also try to imagine putting the triangles one on top of each other. Discuss what needs to be done to one triangle to put it on top of the other: in part i), slide triangle IHJ on top of triangle DEF, in part ii), rotate triangle XYZ 90 clockwise and shift it to place it on top of triangle UVW, in part iii), flip triangle KLM horizontally and shift it up to place it on top of triangle ABC. D-58 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry Introduce the triangle and congruence symbols. SAY: We know that triangles UVW and XYZ in the exercises you have just finished are congruent. Write on the board: Triangle UVW is congruent to triangle XYZ. Remind students that mathematicians often use symbols to shorten the notation. Explain that there is a special symbol for triangles, just as there is a special symbol for angles. SAY: Instead of writing “triangle ABC,” we can write ABC. Rewrite the sentence on the board: UVW is congruent to XYZ Explain that we also have a special symbol we can use to show congruence. Draw “” on the board. SAY: Just as we can write AB = CD instead of “AB is equal to CD,” we can write ABC EFG instead of “ABC is congruent to EFG.” Point out to students that this symbol is similar to the “approximately equal to” sign (≈). (MP.6) Writing congruence statements. Explain that the congruence sign means more than the equal sign does. SAY: Mathematicians have agreed to write congruence statements with the congruence symbol and in an order that shows the matching sides and matching angles in the congruent shapes. ASK: If you want to place triangle UVW on top of triangle XYZ, what would you need to do? (rotate triangle UVW 90 counterclockwise) SAY: This will help us to write the congruence statement. Write on the board: UVW ___ ___ ___ ASK: Which vertex will U be placed on top of? (vertex X) Point out that when students wrote the angle equalities, they wrote U = X, so this matches their earlier work. Write “X” in the first blank. Repeat with the other two vertices to get the congruence statement UVW XYZ. Repeat with the pair of triangles from part iii) in the previous exercises. (ABC KML) Emphasize that when writing the congruence statement, we must write the letters for the second triangle exactly in the order of vertices that shows congruence to the first triangle. (MP.6) Exercises: Write a congruence statement for the triangles. a) b) Q E D F A M S T P U R N Answers: a) DEF ANM, b) PQR TUS Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-59 Identifying equal elements from congruence statements. Write on the board: ABC WUT SAY: This congruence statement gives me all the information I need to say which sides and angles are equal in this pair of triangles. Have a volunteer sketch two identical scalene triangles on the board and label one of them ABC. ASK: According to the congruence statement, which angle in WUT is equal to A? (W) Label the angle in the second triangle. Continue with the rest of the angles and invite volunteers to label the angles in the second triangle. ASK: From the congruence statement, which side in WUT is equal to side AB? (WU) How do you know? (A, B and W, U are the first two letters in the triangles in the congruence statement) Record the equality on the board. ASK: Does this fit the sketch? (yes) Continue with the other two pairs of sides. (BC = UT, AC = WT) (MP.7) Exercises: Use the congruence statement RAT COB to write the equalities between the sides and the angles of the triangles. Answers: RA = CO, AT = OB, RT = CB, R = C, A = O, T = B Identifying equal sides and writing congruence statements. Have students copy the three triangles in the exercises below in their notebooks. Point out that the triangles look very similar and might be congruent. Remind students that to check if the triangles are congruent, students need to imagine placing the triangles one on top of the other, trying to make them match exactly. Then they should measure the sides and the angles of the triangles and see if the matching sides are equal and if the matching angles are equal. If some of the elements do not match, the triangles are not congruent. This is similar to looking for a counterexample: if you find just one pair of sides or angles that should be equal but are not equal, this is enough to say that the shapes are not congruent. Exercises: Which triangles are congruent? Use a ruler and a protractor to check. Write the congruence statement. O T A R C M B N E Answers: ARC BNE Bonus: Draw two congruent isosceles triangles and label them so that OWL TAC. Write another congruence statement for the same triangles. Sample answer: OWL CAT L O W D-60 T C A Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry (MP.7) Using congruence statements to find sides and angles of congruent triangles. Draw on the board: UVW DEF W V 20 cm 30 34.6 cm 40 cm U Have students sketch a similar diagram in their notebooks. SAY: These triangles are congruent. This means that they have angles of the same size and sides of the same length. I would like to find the size of the angles and the lengths of the sides in both triangles. ASK: What do we know from the congruence statement? (UV = DE, VW = EF, UW = DF, U = D, V = E, W = F) Have a volunteer write the equations on the board. Ask students to label the second triangle using the equations. SAY: I need to turn the second triangle 90 clockwise to get it to the same position as the first triangle. Redraw the second triangle as shown below, and have students check that their answers are correct. W E V F E 20 cm 40 cm F 30 34.6 cm D U D Ask students to transform information they know from both diagrams to the new picture of triangle DEF. Students should be able to label all the side lengths and two of the angles. (UV = DE = 40 cm, VW = EF = 20 cm, UW = DF = 34.6 cm, U = D = 30, W = F = 90) ASK: How can we find the size of the third angle in each triangle? (the angles add to 180, so the third angle in both triangles, V and E, is 180 − (90 + 30) = 60) PROMPT: What is the sum of the angles in a triangle? (180) Extensions (MP.6, MP.7) 1. Write a congruence statement for the quadrilaterals. B M C A D K L N Answer: ABCD NLKM Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-61 (MP.1) 2. Draw two non-congruent figures with: a) the same perimeter b) the same area Bonus: the same perimeter and the same area Sample answer: Bonus: c) the same shape (MP.1, MP.3) 3. In a triangle ABC, the sides AB and AC are equal. a) Sketch the triangle. b) Mark point D on the side BC so that BD = CD. Draw the line segment AD. c) The perimeter of ABC is 48 cm. The perimeter of ABD is 36 cm. Find the length of the line segment AD. Solution: A B D C The perimeter of ABC is AB + BC + AC = 48 cm. Since AB = AC, and BC = 2 × BD, AB + BC + AC = 2 × AB + 2 × BD = 48 cm So AB + BD = 24 cm. The perimeter of ABD = AB + BD + AD = 36 cm 24 cm, So AD = 36 cm − 24 cm = 12 cm. D-62 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry G8-10 Supplementary and Vertical Angles Pages 96–98 Standards: 8.G.A.1b, 8.G.A.5 Goals: Students will identify supplementary and vertical angles, investigate their properties, and use these properties to solve problems. Prior Knowledge Required: Can measure angles and sides of polygons Is familiar with notation for equal sides and equal angles Can name angles and polygons Is familiar with the symbol for angle Can classify triangles Vocabulary: adjacent angles, intersect, intersection point, straight angle, supplementary angles, vertical angles Materials: transparency overhead projector Introduce supplementary angles. SAY: Supplementary angles are a pair of angles that add to 180°. If A and B are supplementary angles, we can say that A supplements or is a supplementary angle to B. Have students signal the answers to the following exercises by showing thumbs up for yes and thumbs down for no. Exercises: Are the angles supplementary? a) b) 45 c) 63 135 129 115 51 d) e) 35 155 148 42 f) 83.5 96.5 Answers: a) yes, b) no, c) yes, d) no, e) no, f) yes Explain that sometimes we can label angles with a single small letter inside the angle. SAY: This does not work when there are many angles or when they overlap, but when we are dealing with Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-63 a few angles and they don’t overlap, we can use small letters or numbers. Draw the picture below on the board and trace each angle separately while naming it: a c b d Remind students that straight angles have arms that make a straight line, and the measure of a straight angle is 180°. SAY: Angles a and c make a straight angle. Are angles a and c supplementary? (yes) What other pairs of supplementary angles do you see in this picture? (a and b, d and b, c and d) Finding the measure of a supplementary angle. Ask students to write an equation that shows what it means that a and b are supplementary. (a + b = 180°) Mark ∠a in the picture as 115° and have students rewrite the equation using the measure of a. (115° + b = 180°) ASK: What is the measure of b? (180° − 115° = 65°) Mark the measure in the picture and have students find the measure of c the same way. Then challenge them to find the measure of ∠d and to explain the solution. (d = 115) Keep the pictures from the following exercises on the board for use later in the lesson. Exercises: Find the measures of the missing angles. a) b) f e c d = 123 b a c = 24 d c) g k h m = 86 Answers: a) a = c = 57, b = 123; b) f = d = 156, e = 24; c) h = k = 94, g = 86 Introduce adjacent and vertically opposite angles. Draw on the board: a b Explain that two angles that share an arm, or have an arm in common, such as angles a and b, are called adjacent angles. Ask students to identify pairs of adjacent angles in one of the pictures from the previous exercises. Explain that, when two lines intersect, as in the pictures in the previous exercises, the angles that are not adjacent and that are formed by the intersecting lines are called vertically opposite angles, or vertical angles for short. ASK: In the picture for part a), what angle and angle d form a pair of vertical angles? (b) SAY: We can also say that angle b is vertical to angle d and angle d is vertical to angle b. D-64 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry Exercises: 1. Are the angles adjacent or vertical? 1 2 6 3 5 4 a) 1 and 2 b) 2 and 5 c) 3 and 6 d) 1 and 6 e) 1 and 4 f) 4 and 5 Answers: a) adjacent, b) vertical, c) vertical, d) adjacent, e) vertical, f) adjacent 2. Identify the pairs of vertical angles in the pictures in the previous exercises. Answers: a) a and c, b and d, b) f and d, e and c, c) h and k, g and m Draw on the board: a) b) c) 1 A 2 3 1 2 Explain that there are no vertical angles in these pictures because there are no intersecting lines, only rays with a common endpoint. Point at each picture in turn and ASK: Are there adjacent angles? If so, which angles are adjacent? (a) no, b) yes, all three angles are adjacent to each other, c) yes, 1 and 2 are adjacent) (MP.3) Vertical angles are equal. ASK: What do you notice about the measures of vertical angles? (they are always equal) Have students think of how they can explain why this always happens. Work through the following proof as a class, using the picture below. Have students suggest what each next step should be and what to put in each blank. c d b a a + b = _____, so a = ______________ (180, 180 − b) b + c = _____, so c = ______________ (180, 180 − b) So a = ____ (c) Using rotation to show that vertical angles are equal. Draw the picture below on a transparency and project it on the board: 2 1 3 4 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-65 ASK: If I turn the picture, will the angles change? (no) Press a pencil tip to the intersection point on the transparency to create a pivot for turning the transparency. Rotate the transparency 180° to show how the rotated image coincides exactly with the original image. Another option is to use two identical transparencies: keep one fixed and rotate the other. Have students identify the amount of rotation. (180, half turn) Show the rotation several times if necessary. ASK: Which angles are rotated onto which angles? (1 becomes 3, 2 becomes 4) What does this tell us about the sizes of 1 and 3? (they are equal) What about 2 and 4? (they are equal) Point out that another way to look at the sizes of vertical angles is to think of an angle as the amount of rotation you need to get from one arm to the other. The amount of rotation you need to get from the lower arm of 1 to its upper arm is the same as the amount of rotation you need to get from the upper arm of 3 to its lower arm, because the arms of both angles are parts of the same lines. To help students see that, draw arrows demonstrating the amount of rotation, as shown below: 2 1 3 4 Keep the pictures from the following exercises on the board for later use. Exercises: Find the missing angle measures using vertical angles. a) b) c) c b 103 a 21 56 93 d e n 51 1 2 87 45 130 Answers: a) a = 103, b = 21, c = 56; b) d = 87, e = 93, n = 51; c) 1 = 130, 2 = 45 Draw on the board: SAY: I want to know which angle is vertical to the marked angle. I’m only interested in the lines that show the arms of this angle, so I’m going to highlight them. Do so by making the lines thicker, as shown below in the diagram on the left. SAY: Now the drawing makes it clear where the vertical angle is. The angle vertical to the marked angle is a combination of two smaller angles. Draw an arc on the vertical angle, as shown below in the diagram on the right. D-66 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry Exercises: Which two angles, when added together, make the angle that is vertical to the angle marked with an arc? a) b) c) 4 1 1 2 3 2 3 3 4 2 4 1 5 Answers: a) 2 and 3, b) 2 and 3, c) 3 and 4 Finding angle measures using both supplementary and vertical angles. Return to the pictures from the exercises on the previous page. SAY: In some pictures we are given more information than we need to find the angle measures. Let’s look at part a). Erase the 56 and replace it with the label d. ASK: How can we find the measure of this angle from the other two angles we are given? (180 − 103 − 21 = 56) How do you know? (angle d and the two given angles make a straight angle, so they all add to 180) Repeat with 93 in part b). SAY: In part c) we have several angles we do not know. Add the labels 3 to 7 to the unmarked angles, as shown below: 7 6 4 45 2 5 3 1 130 SAY: We found the measures of angles 1 and 2. ASK: If you know the measure of angle 1, how can you find the measure of angle 3? (subtract the measure of angle 1 from 180) How do you know? (angles 1 and 3 are supplementary angles) What is the measure of 3? (180 − 130 = 50) Repeat with angles 4 and 5. (4 = 5 = 135) Then ask students to find the measure of angle 6 and explain how they know. (180 − (45 + 50) = 85) PROMPT: Are some of the angles here angles in a triangle? (angles 2, 3, and 6) What do they add to? (180) Exercises: Find the measures of all angles in the picture using the given angle measures. a) b) c) 45° 58° 154° 95° 112° Answers: a) 135 45° 45 135 112 68 68 112° 121° b) c) 58° 32 58° 90 32 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry 26 154 121 59 154° 26 59 121° 95 85 85 95° D-67 Extensions 1. Find x. x 2x Answer: x + 2x = 180, x = 60 (MP.8) 2. Imagine n lines that all pass through the same point. Label the angles in order, clockwise, from 1 to 2n. What angle is vertical to 1? Hint: Try different values of n and make a sketch. Solution: 3 4 5 Number of lines n Sketch 6 5 Angle vertical to 1 1 4 2 3 7 6 4 8 1 5 4 5 (MP.2) 3. Write an equation and solve for x. a) b) 2x 3x 4x + 15° 9 8 2 3 10 1 2 3 4 7 6 5 6 c) x 3x 60° n + 1 x 5x Answers: a) 2x + 3x = 4x + 15, 5x = 4x + 15, so x = 15°; b) 3x = x + 60, 2x = 60, so x = 30°; c) x + 90 = 5x, 90 = 4x, so x = 22.5° D-68 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry G8-11 Congruence Rules Pages 99–101 Standards: 8.G.A.2 Goals: Students will develop and use rules for congruence of triangles. Prior Knowledge Required: Can measure angles and sides of polygons Is familiar with notation for equal sides and equal angles Can name angles and polygons Is familiar with the symbols for angle, triangle, and congruence Can identify congruent triangles Can write a congruence statement for two triangles Knows that the sum of the angles in a triangle is 180 Can classify triangles Vocabulary: angle-side-angle (ASA), congruence rule, congruence statement, congruent, conjecture, corresponding angles, corresponding sides, corresponding vertices, counterexample, isosceles, side-angle-side (SAS), side-side-side (SSS) Materials: BLM Investigating Congruence (pp. D-126–127, optional) straws of different lengths and 2 pipe cleaners for each student (optional) scissors (optional) The Geometer’s Sketchpad® (optional) BLM Congruence Rules on The Geometer’s Sketchpad® (pp. D-128–130, optional) Corresponding sides and angles. Remind students that, when we want to check whether shapes are congruent, we ask ourselves if we could place the shapes one on top of the other so that they match exactly. SAY: When we place the shapes one on top of the other, the sides of the different shapes that will sit on top of one another are called corresponding sides. The vertices that will sit one on top of the other are called corresponding vertices. And the angles of the different shapes that will sit one on top of the other are called corresponding angles. If the shapes are congruent, corresponding sides and angles will be equal. Draw on the board: SAY: These two triangles are congruent. Some of the equal sides are marked, and a pair of equal angles is also marked. ASK: If we imagine placing one triangle on top of the other, what Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-69 do we need to do so they will match? (turn the top triangle clockwise a little) Are the sides marked with thick lines corresponding sides? (no) Point to different sides of the triangle on the bottom and ASK: Does this side correspond to the thick side of the triangle on the top? Have students signal yes or no. Students can also signal the answers to the exercises below. After each question, have a volunteer explain the answer. Exercises: Are the two thick sides corresponding sides? a) b) c) Answers: a) yes, b) no, c) yes) Introduce the idea of congruence rules. SAY: Congruent triangles have three equal corresponding sides and three equal corresponding angles. However, we do not always need to check all six pairs of elements to decide that two triangles are congruent. Today we will be looking for shortcuts—ways to check fewer pairs of angles and sides. Draw on the board: 22 45 22 45 ASK: If we check two pairs of angles and find that these two pairs of angles are equal, do we need to check that the third pair of angles is equal? (no) How do you know? (the three angles of a triangle always add to 180) What is the measure of the third angle in both triangles? (113) SAY: This means we do not have to check all six elements of a pair of triangles; checking three sides and two angles will be enough. Now let’s see if we can check even fewer. SSS, SAS, ASA rules. Have students investigate one of the congruence rules (side-side-side, side-angle-side, or angle-side-angle) in Activity 1 using The Geometer’s Sketchpad®. Alternatively, have students investigate all three rules using BLM Investigating Congruence. Activity 1 Use The Geometer’s Sketchpad® for this activity. Divide students into groups of three. Students will work on the construction individually, each using one page of BLM Congruence Rules on The Geometer’s Sketchpad®, sharing the results with the group. Students should tell which elements of the construction could be modified. (For example: I could modify the first triangle any way I want by moving any of the vertices. I could only move around the second triangle by moving vertex E, and I could only turn the triangle by moving vertex D. When I tried to move vertex D, it would only go along a circle, because it was constructed so that ED has a fixed length.) Students might need to reflect the triangles they created during the activity to place them on top of each other. In this case, have students place the triangles so that they share a side and look like mirror images of each other. Then have students select the common side and use the D-70 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry Transform menu option to declare the side a mirror line; they need to choose the option “Mark mirror” in the Transform menu options. Now if they select one of the triangles and reflect it using the Transform menu options, they will be able to see that the triangles match exactly. In their groups of three, have students match each BLM with the congruence rule it seems to be showing—side-side-side (SSS), side-angle-side (SAS), or angle-side-angle (ASA). Have them label the BLM with the full name of the congruence rule. For example, page 2 shows the sideangle-side (SAS) rule by keeping two side lengths and the angle between them constant, which forces the triangle to be fixed. (end of activity) Summarize the congruence rules on the board (as in the “Congruence Rules for Triangles” box on AP Book 8.1 p. 99). Emphasize that the order of elements in the congruence rules is important: in the side-angle-side (SAS) rule, the equal angles have to be between the corresponding equal sides; in the angle-side-angle (ASA) rule, the equal sides have to be between the corresponding equal angles. Exercises: Identify the congruence rule that tells you that the triangles are congruent. B S E H a) b) c) F I X Y K A C V d) e) O J L G D M R U S A f) Z A G P T U L W N Q Answers: a) ASA, b) SAS, c) ASA, d) SAS, e) SSS, f) ASA B T O D Remind students that, in a congruence statement, the corresponding vertices match. So for example, in part a) above, the congruence statement is ABC FDE, because if you try to turn triangle DEF and place it on top of triangle ABC, vertex A corresponds to vertex F (and A = F), vertex B corresponds to vertex D (and B = D ), and vertex C corresponds to vertex E (and C = E). Exercises (MP.6) Write the congruence statements for the pairs of triangles in parts b)–f) of the previous exercises. Answers: b) GHL JKI, c) BAT DGO, d) UVW POQ or UVW PQO, e) LMN TSR, f) USA YZX Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-71 Bonus: In part d) above, write two different congruence statements with the letters in the first triangle in the same order. Can you do this for another pair of triangles in this exercise? Draw another pair of triangles for which you can do this and write two congruence statements. Sample answer: UVW POQ and UVW PQO No, you cannot write a statement like for another pair of triangles in the exercise. ABC DEF and ABC FED B A E C D F Using congruence rules to show congruence. Draw and write on the board: B E A A = D = 45 C = F = 70 BC = EF = 30 cm C D F Ask students to sketch the triangles in their notebooks and label the equal sides and angles in the triangles. Have a volunteer label the equal angles and sides on the board, as shown below: (MP.6) ASK: Do you think these triangles could be congruent? (yes) Is there a congruence rule that tells us that these triangles are congruent based on what we know now? (no) Why not? (because the equal sides in the triangles are not between the corresponding equal angles) PROMPT: What elements are marked as equal in each of these triangles? (two angles and a side) Is the side between the corresponding equal angles? (no) SAY: So this situation does not fit the angle-side-angle (ASA) congruence rule. ASK: Is the order of equal, or matched, elements the same in both triangles? (yes, the equal sides are BC = EF = 30 cm, the angles opposite those sides match: A = D = 45, and there is another match: C = F = 70) If you try to place triangle DEF on top of triangle ABC to make the triangles match, would the equal sides fall one on top of the other? (yes) What about the angles? (yes) SAY: So these triangles have two pairs of corresponding equal angles and one pair of corresponding equal sides. I would like to use a congruence rule to show the triangles are D-72 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry congruent, but we can’t use the angle-side-angle (ASA) rule yet because the equal sides are not between the corresponding equal angles. Maybe we can deduce some more information to see if the rule applies. ASK: What pair of sides would we need to know are equal to use the ASA congruence rule? (AC, DF) Do we know that these sides are equal? (no) What pair of angles would we need to know are equal to use the ASA congruence rule? (B = E) ASK: How can you find the measure of B from the rest of the angles of the triangle? Have students write down the expression for the measure of the angle. (B = 180° − (45 + 70) = 65) Repeat for E. (E = 180° − (45 + 70) = 65) ASK: Are B and E equal? (yes) Write on the board: B = E = 65 Mark angles B and E as equal on the picture and ASK: Can we now use a congruence rule? (yes) Which rule? (angle-side-angle, ASA) Invite a volunteer to circle the equalities between sides and angles that allow us to use the ASA rule. ASK: So, based on this additional information, can we say that the triangles are congruent? (yes) (MP.3, MP.6) Exercise: Explain why the triangles ABC and KML are congruent. A = K = 35 C = L = 120 AB = KM Answer: B = 180 − (35 + 120) = 25, M = 180 − (35 + 120) = 25, so B = M. B = M, A = K, AB = KM, so with the ASA rule, the triangles are congruent. Two pairs of equal angles and one pair of equal sides do not always mean that the triangles are congruent. Draw on the board: Q E P R D F P = D = 90 Q = E = 56 PQ = DF = 36 cm (MP.3, MP.6) ASK: Do these triangles have two pairs of equal angles? (yes) Do they have a pair of equal sides? (yes) Can we apply one of the congruence rules? (no) Why not? (answers will vary, but students will likely point out that the triangles do not look congruent) Are the third angles equal in these triangles? (yes) How do you know? (in both triangles the size of the third angle is 180 − (90 + 56) = 34) Invite a volunteer to label the equal angles and sides in the triangles. ASK: Why can we still not use congruence rules here? (in triangle PQR, the side PQ Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-73 is opposite R = 34, but in triangle DEF the side DF is opposite E = 56) PROMPT: The equal side is opposite one of the angles in both triangles. Which angle? (in triangle PQR, the side PQ is opposite R, in triangle DEF the side DF is opposite E) What is the size of the opposite angles in each triangle? (R = 34, E = 56) Emphasize that this means the order of equal sides and equal angles is different in these two triangles, so the congruence rule does not apply. SAY: And indeed, the triangles do not look congruent at all! They have the same shape, but they are obviously different sizes. (MP.3, MP.6) Exercise: Explain why the ASA congruence rule cannot be used for these two triangles. B Y C A A = X = 40 B = Y = 85 AB = XZ = 3 cm Z X Answer: In triangle ABC, the given side is between angles A and B, which have the measures 40 and 85. In triangle XYZ, the side XZ that is equal to AB is between X = 40 and Z = 180 − (40 + 85) = 55. The order of equal pairs of angles and equal sides is different in the two triangles, so the ASA rule cannot be applied. The triangles are not congruent. Bonus: Use a ruler and a protractor to draw ABC and XYZ with the measurements given. AAA is not a congruence rule. Write on the board: If two triangles have three pairs of corresponding equal angles, then the triangles are congruent. True or false? (MP.3) Have students vote on the question above. (the statement is false) PROMPT: Think of the triangles you saw in the previous exercise. Have students draw a counterexample to the statement. Students who wish and have time could draw more than one counterexample. SSA is not a congruence rule. SAY: We checked what happens with three pairs of angles, three pairs of sides, two pairs of angles and one pair of sides, but we did not check what happens when two triangles have two pairs of corresponding equal sides and one pair of corresponding equal angles. Let’s see if the order matters in the side-angle-side (SAS) congruence rule. Draw on the board: E B A D-74 C D G F Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry (MP.3) SAY: I drew a copy of triangle ABC and called it DEF. Then I drew an isosceles triangle EFG, so that EG = EF. Now look at the triangles ABC and DEG. ASK: Which angles are equal? (A = D) Which sides are equal? (AB = DE, BC = EG) Are the equal angles opposite corresponding equal sides? (yes, A is opposite BC, D is opposite EG) Are the triangles congruent? (no) Which description identifies the equal elements, in order, in your triangles: side-angle-side or side-side-angle? (side-side-angle, SSA) Of the two descriptions, which is a congruence rule and which is not? (side-angle-side, or SAS, is a congruence rule) How do the triangles I drew help you to decide? (they are a counterexample to this statement) Emphasize that the order of equal corresponding elements matters in triangles. The pair of equal angles has to be between the pairs of equal corresponding sides for the triangles to be congruent. (MP.3) Exercise: On grid paper, draw a counterexample to SSA. Use a ruler. Sample answer: Using congruence rules to find sides or angles in congruent triangles. Draw on the board: W V 17.3 cm 20 cm 30 T 10 cm R 30 17.3 cm S U SAY: I want to find the size of the angles and the lengths of the sides in both of these triangles. Let’s see if these triangles are congruent. If the triangles are congruent, we will know that the lengths of all the sides in both triangles match even though we haven’t measured them. Have students sketch the triangles and label all the known information. Then ask them to label the angles that are the same with arcs and the equal sides with the same number of hash marks. Point out that we do not know yet if the triangles are congruent, and so we cannot, for example, mark angles T and V as equal. ASK: How many pairs of equal sides have you marked? (1 pair, UW = RS) How many pairs of equal angles do we see? (2 pairs, W = R, U = S) Which congruence rule would we like to apply? (angle-side-angle, or ASA) Is the pair of equal sides between the corresponding equal angles? (yes) SAY: Then we can apply the congruence rule ASA. Have students write the congruence statement and the rest of the angle and side equalities. Then ask them to find the Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-75 lengths of all the sides and angles. (see answers below) Ask volunteers to explain how they found the answers. UVW STR UV = ST = 20 cm VW = TR = 10 cm V = T = 180 − (90 + 30) = 60 Remind students that, when several lines or rays meet at a point and there are overlapping angles, we use three letters to label an angle with the vertex label always in the middle. Draw the picture in the exercises below on the board, mark different angles with arcs one at a time, and ask students to name them. Exercises: Explain why the triangles are congruent. Then write the congruence statement and find the missing angle measures. A = 43 B D E = 83 AB = CD = 3 cm AC = CE = 3.7 cm BC = DE = 2.5 cm A E C Answers: The triangles have 3 pairs of corresponding equal sides, so by the side-side-side, or SSS, rule they are congruent, and ABC CDE. Therefore, A = DCE = 43, BCA = E = 83, and B = D = 180 − (43 + 83) = 54. Extensions (MP.1, MP.3, MP.7) 1. In the diagram below, AC BD, AO = CO, and BO = DO. B A O C D a) Copy the diagram and label the equal sides and right angles. b) Use triangles AOB and COD to explain why AB = CD. c) Explain why AD = BC. d) Use triangles AOB and COB to explain why AB = CB. e) What type of quadrilateral is ABCD? Selected solutions: b) Since AC BD, AOB = COD = 90 and we know that AO = CO and BO = DO. This means we have two pairs of corresponding equal sides and a pair of corresponding equal angles between them, so by the SAS rule AOB COD. Therefore AB = CD. d) Since AC BD, AOB = COB = 90 and we know that AO = CO. The side BO is in both triangles, so we have another equal side in both triangles. This means we have two pairs of D-76 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry corresponding equal sides and a pair of corresponding equal angles between them, so by the SAS rule AOB COB. Then AB = CB. e) ABCD is a rhombus. (MP.1, MP.3, MP.7) 2. Maria drew 2 right triangles as shown below, with PQ = TR and QT = RS. She thinks that the points P, T, and S are on the same line. Is she correct? Explain. Hint: Find the measure of PTS. P T Q S R Solution: Maria is correct. Since PQ = TR and QT = RS and the angles between the corresponding pairs of equal sides are equal, Q = R = 90, the triangles are congruent by the SAS rule. This means PTQ = S. From the sum of the angles in a triangle we know that: S = 180 − (90 + RTS) = 90 − RTS. PTS = PTQ + QTR + RTS = S + 90 + RTS = 90 − RTS + 90 + RTS = 180 The angle PTS is a straight angle, so the points P, T, and S are on the same line. (MP.1, MP.3) 3. ABC and DEF are both isosceles triangles. A = D and AB = DE. Are ABC and DEF always congruent? Explain. Hint: Make a sketch that includes all the information you have been given. Try making more than one sketch using a different position for the equal angles. Answer: The triangles do not have to be congruent. Sample counterexample: F B A E D C (MP.3) 4. Sketch a counterexample to show why the statement is false. ABC has AB = BC = 7 cm and DEF has DE = EF = 7 cm. So ABC ≅ ΔDEF. Sample answer: E B 7 cm 7 cm A C D F Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-77 G8-12 Congruence (Advanced) Pages 102–103 Standards: 8.G.A.2, 8.G.A.5 Goals: Students will use informal arguments involving congruent triangles to solve problems. Prior Knowledge Required: Can name angles and polygons Can measure angles and sides of polygons Is familiar with notation for equal sides and angles Can classify triangles Knows that the sum of the angles in a triangle is 180 Is familiar with the symbols for angle, triangle, and congruence Can identify congruent triangles Knows the SAS, ASA, and SSS congruence rules Can write a congruence statement for two triangles Vocabulary: ASA (angle-side-angle), congruence rule, congruence statement, congruent, conjecture, corresponding angles, corresponding sides, corresponding vertices, counterexample, isosceles, midpoint, SAS (side-angle-side), SSS (side-side-side), supplementary angles, vertical angles Materials: BLM Two Pentagons (p. D-131) scissors Review congruence rules and properties of isosceles triangles. Remind students that congruence rules are shortcuts that allow us to determine whether or not triangles are congruent by checking only three elements (sides or angles). ASK: What three elements could we use? (3 sides, 2 sides and 1 angle, 1 side and 2 angles) Can we use any three elements? (no; for example, we can’t use 3 angles) Remind students that the order of the elements is important. Draw on the board: Remind students that, in an isosceles triangle, the angles between the equal sides and the third side are equal. SAY: Both these triangles are isosceles right triangles. ASK: What is the size of the angles that are not marked? (45) How do you know? (angles in a triangle add to 180, and D-78 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry one angle is 90, so the other two angles add to 180 − 90 = 90; since they are equal, they measure 90÷ 2 = 45 each) Mark the angles in the triangles on the board as 45. (MP.3) Point to the triangles and SAY: These triangles have all angles the same size, and they have some sides that are equal. ASK: Are they congruent? (no) Why not? (they are different sizes) Why can you not apply any of the congruence rules here? (the sides that are equal in both triangles are in different places in relation to the equal angles: the equal side is between the 45° angles in the smaller triangle but between the 90° angle and one of the 45° angles in the other) Remind students that the equal sides need to be adjacent to the equal angles for the triangles to be congruent using the side-angle-side (SAS) rule. ASK: When you have two pairs of equal sides and a pair of equal angles, where does the equal angle have to be for the triangles to be congruent? (in both triangles, the equal angle has to be between the 2 pairs of equal sides) Draw on the board: E B A C D G F Point out that the two triangles have two pairs of equal sides and a pair of equal angles, but the triangles are not congruent. SAY: Even though the order of the equal elements is the same in both triangles—angle-side-side—the triangles are still not congruent because the congruence rule requires the order side-angle-side. Remind students that the third congruence rule is the side-side-side rule: if two triangles have all sides of the same length, they are congruent. Also remind students that a congruence statement lists the vertices of the triangles so that you can say which angle is equal to which angle and which side equals which side. For example, if ABC ≅ PQR, then we know that A = P, B = Q, AB = PQ, and so on. Exercises: Which congruence rule tells that the two triangles are congruent? Write the congruence statement. a) b) Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-79 W Bonus: V X U Answers: a) ASA, GHI ≅ JLK; b) SAS, MOQ ≅ PRN; c) ASA, ABC ≅ FDE; Bonus: SSS, UVW ≅ WXU Identifying congruent triangles and explaining why they are congruent when triangles have a common side or vertex. Explain that sometimes you see triangles that share a side or a vertex. Draw on the board: B A D C Have students copy the picture. SAY: To prove that triangles ABD and BCD are congruent, you need to use the fact that the common side BD belongs to both triangles and therefore makes a pair of equal sides. ASK: What sides are equal in these triangles? (AD = CD, BD = BD) Point out that we know that AD and DC are equal, so we can describe D as at the midpoint in the line segment AC—in other words, D divides the line segment in half. Write the equalities between the sides on the board and have students write them in their notebooks. ASK: What angles are equal in these triangles? (ADB = CDB) PROMPT: What is the size of angle ADB? (90) Have students write the equality. ASK: What congruence rule can you apply? (SAS) What do you need to check to be sure the rule applies? (that the matching angle is between the sides listed in the equalities) Is this the case? (yes, angles ADB and CDB are between the sides AD and DB, and CD and DB) Have a volunteer write the congruence statement on the board and have students write it in their notebooks. (ABD ≅ CBD) Leave this picture on the board for later use. Exercises: The pairs of equal sides and angles are marked in the diagram. Which congruence rule can you use to prove that the triangles are congruent? a) b) c) d) Answers: a) ASA, b) SAS, c) SSS, d) SSS Draw two intersecting lines on the board and remind students which angles are vertical angles, and that vertical angles are equal. D-80 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry (MP.6) Exercises: a) Write the equalities between the sides and the angles in the triangles. B B i) ii) O C A C D iii) A D Bonus: L W Z O K M X Y N b) Which congruence statement can you use to show the triangles are congruent? c) Write the congruence statement. Answers: a) i) AD = CD, AB = CB, BD = BD; ii) AO = CO, BO = DO, COD = AOB; iii) KO = MO, KON = MOL, OKN = OML; Bonus: ZY = XW, ZYW = XWY, YW = WY b) i) SSS; ii) SAS; iii) ASA; Bonus: SAS c) i) ABD ≅ CBD; ii) ABO ≅ CDO; iii) KNO ≅ MLO; Bonus: YZW ≅ WXY ASK: In part ii), how do you know the angles COD and AOB are equal? (they are vertical angles, and vertical angles are equal) What other pair of vertical angles did you see in this exercise? (KNO and MLO) Using congruence. Return to the picture below, from earlier in the lesson. B A D C AD = CD BD = BD ADB = CDB ABD ≅ CBD Remind students that they showed that the triangles are congruent using the side-angle-side congruence rule. Ask students to write the rest of the equalities between the sides and the angles of the triangles. (AB = BC, A = C, ABD = CBD) ASK: What have we just written about the larger triangle, ABC? (the triangle is isosceles) To prompt students to see the answer, trace the larger triangle with a finger and ask students to classify it. Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-81 Add the information in the exercises below to the triangles from Exercise 2, above. Students might need a prompt to identify ADB as a right angle in part a). If so, ask them what the measure of angle ADC is, and what they know about the two angles with vertex D. When students have finished their work, discuss solutions as a class. Exercises: Find all the missing sides and angles that you can. a) b) c) B B L 7 in 5 cm A O 60 70 C 64 A 6m O 51 K M C D D N Answers: a) C = 64, ADB = CDB = 90, BC = 5 cm; b) CD = 7 in, COD = 60, OCD = 70, OBA = ODC = 50; c) LM = KN = 6 m, KON = MOL = 51, MLO = KNO = 39 Bonus: Use the picture to prove that, in an isosceles triangle, the angles between the equal sides and the third side are equal (or A = C). B A C D Answer: AB = CB, BD = BD, and ABD = CBD, so by the SAS congruence rule, ABD ≅ CBD. From the congruence statement we know that C = A, so in ABC, AB = BC. Using notation shortcuts in diagrams. Draw the picture below on the board and have students copy it: G F H FE = GH EFH = GHF E SAY: We have a quadrilateral EFGH that is made from two triangles with a common side, another pair of equal sides, and equal angles. I would like to see what properties this quadrilateral has. ASK: What can you tell about triangles EFH and FGH? (they are congruent) How do you know? (they have two pairs of equal sides and a pair of equal angles between them, so by the SAS congruence rule the triangles are congruent) Ask students to write the congruence statement for the triangles. (EFH ≅ GHF) Then ask them to write the equalities D-82 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry for the sides and the angles that follow from the congruence statement. (EH = GF, FEH = HGF, EHF = GFH) Point out that listing all the angle names takes a long time, and it is often hard to see which angle name is equal to which angle on the diagram. In addition, if you are talking about several angles, it becomes difficult to use arcs. SAY: In such cases, we often mark the size of the angles and sides with letters, using the same letter for angles of the same size. Label EFH = a, FEH = b, and EHF = c, and have students to label all the equal angles on the diagram with these letters. Have a volunteer do the same on the board. The picture will look like this: b F G c a a c H b E SAY: Now it is clear from the picture that the quadrilateral also has equal opposite angles. Angles E and G are equal, but so are angles EFG and GHE. ASK: Using letters, what are angles EFG and GHE equal to? (a + c) ASK: What type of a quadrilateral does EFGH seem to be? (a parallelogram) Point out that parallelograms have equal opposite sides and equal opposite angles, but students have not proven that the quadrilateral has parallel opposite sides. The fact that EFGH is a parallelogram remains a conjecture—something we think is true but have not proved using logic. Explain that students will be able to prove this conjecture later in this unit. (MP.3, MP.7) Exercise: Prove that quadrilateral PQRS has equal opposite angles. Use the shortcut notations for the angles. Q P R S Answer: PQ = RS, QR = SP, and PR = PR (common side), so by the SSS congruence rule, PQR ≅ RSP. From the congruence statement, the corresponding angles in the triangles are equal, as labeled below. Q c P a b b a S R c This means Q = S and QPS = a + b = SRQ, so the opposite angles in the quadrilateral PQRS are equal. Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-83 Bonus: XYZW is a rhombus with line segment XZ. a) Sketch the situation. b) Prove that YXZ = WXZ = YZX = WZX. Answers: a) Y X Z W b) XY = XW, YZ = WZ, XZ = XZ, so by the SSS congruence rule, XYZ ≅ XWZ. From the congruence statement, YXZ = WXZ and YZX = WZX. Since XYZ is isosceles, YXZ = YZX, so all angles are equal. Congruence in other polygons. Remind students that they can talk about congruence of any pair of polygons. SAY: If you can place two polygons one on top of the other and they match exactly, they are congruent. Explain that, in polygons with more than three sides, the order of the equal sides and angles is even more important than in triangles. It is not enough to say that the shapes have all sides the same lengths and that all angles are the same size; the equal sides and angles have to match in order. (MP.3) Activity Give students BLM Two Pentagons. Have students work individually to cut out the pentagons and compare the sides and the angles to answer the questions on the BLM. When students are finished, ask the class Question a) and have them signal the number of right angles on each polygon to check their answer. (3 each) Read Question b) to the class, and have them hold up the folded shapes so that they can show that the remaining four angles are all equal. Ask the class Questions c) to h) and have students to signal their answers to each one so that you can check the whole class at the same time. Students can signal thumbs up if their answer is “yes” and thumbs down if their answer is “no.” (c) yes, d) yes, e) no, f) no, g) yes, h) no) Have students answer Question i). Pair students who are struggling with students who show greater understanding of the material to compare their answers (the former can coach the latter as they come up with a common answer). Repeat with groups of four and groups of eight, and then have the groups share their answers with the whole class. (The order of sides and angles matters. For example, in Pentagon A the obtuse angles are adjacent, but in Pentagon B the obtuse angles are separated by a right angle. The order of the equal sides is not the same on these pentagons, so they are not congruent.) (end of activity) Explain that, when we place Pentagons A and B from the BLM one on top of the other, we define the order in which we will check the angles and sides. If we place the pentagons so that at least one pair of sides or angles matches, we cannot make all the remaining corresponding sides and angles match. For example, you can match the shapes so that two angles and the D-84 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry side between them correspond, but the other sides adjacent to the angles are not equal because the order of the sides in each shape is different. SAY: When you check for congruence, you either need to try all the possible combinations or show why congruence is impossible. In this case, congruence is impossible because all the right angles are adjacent in Pentagon A but not in Pentagon B. Extensions 1. Look again at the pentagons on BLM Two Pentagons. Which one has greater area? Take the necessary measurements to check. Answer: Pentagon A (MP.1, MP.3) 2. In the quadrilateral ABCD, АВ = CD and ВС = AD. Copy the picture. Answer the questions to explain why the point O divides the line segments BD and AC in half. (In other words, explain why O is the midpoint of both AC and BD.) a) ABD ≅CDB by the _____ congruence rule, so ABD = ________ Label the equal angles with letter a. b) ABC ≅ CDA by the _____ congruence rule, so BAC = ________ Label the equal angles with letter b. c) Shade AOB and COD. d) AOB ≅_________ by the ________ congruence rule, so BO = ______ and AO = _______, so the point O divides the line segments ______ and ______ in half. Answers: a) SSS, CDB; b) SSS, DAC; d) COD, ASA, DO, CO, AC, BD a b b a Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-85 (MP.4) 3. Tina wants to measure the distance from point X to point Y, but a pond stops her from walking directly between the points. She finds a point, Z, from which she can walk to both X and Y in a straight line. Pond Y X Z Copy the sketch and follow the steps below to see how Tina solves the problem. a) Tina walks from X to Z and counts her steps. She continues in a straight line from Z, walking the same number of steps. She labels the point she stops at W. Draw the point W on the sketch. Mark the equal distances. b) Tina repeats the task in part a), walking from Y through Z, to find point U so that YZ = ZU. Draw the point U on the sketch. Mark the equal distances. c) Tina measures the distance UW. Explain why this distance is the same as the distance between X and Y. Answers: Pond X Y Z U W c) XZ = ZW and YZ = ZU as constructed. Also, XZY = WZU because they are vertical angles. So XZY ≅WZU by the side-angle-side (SAS) congruence rule. From the congruence statement, XY = WU. D-86 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry G8-13 Exterior Angles of a Triangle Pages 104–105 Standards: 8.G.A.5 Goals: Students will discover, prove, and use the fact that the exterior angle in a triangle equals the sum of the non-adjacent interior angles. Prior Knowledge Required: Knows that the sum of the angles in a triangle is 180° Can identify supplementary and vertical angles Knows that vertical angles are equal Can draw and measure with a ruler and a protractor Can identify and construct parallel lines using a protractor Vocabulary: conjecture, exterior angle, interior angle, parallel, supplementary angles, vertical angles Materials: protractors The Geometer’s Sketchpad® Introduce exterior angles. Draw on the board: ASK: What do you know about the measures of angles a, b, and c? (they add to 180) What do you know about angles c and x? (they add to 180) What are angles like c and x called? (supplementary angles) Ask students if anyone knows what “exterior” means. (outer, on the outside) Explain that an angle such as angle x, created by extending one of the sides outside the triangle, is called an exterior angle of the triangle because it is outside the triangle. The angles inside the triangle are called interior angles. Looking for a pattern in the measures of exterior and interior angles. Mark the measures of a and b in the triangle on the board as 50°and 57°, respectively. Ask students to find the measure of c. (73°) ASK: How do you know? (180 − (50 + 57) = 73) What do you know about c and the exterior angle, x? (they are supplementary angles; they add to 180°) Have Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-87 students find the measure of x. (107°) Draw the table below on the board and fill in the first column with the information from this triangle: a b x 50 57 107 Ask students to each draw a triangle in their notebooks and label the angles a, b, and c. Then ask them to extend one of the sides of the triangle that make angle c beyond the vertex so that the exterior angle is x. Have students measure the angles a, b, and x, and write the measures in the table. Then have them exchange notebooks and repeat the exercise with the triangles drawn by their peers. (MP.7) Ask students to look for a pattern in their tables and have them formulate a conjecture about the sizes of the angles. Have students pair up and to improve the conjecture they have written, using the words “exterior” and “opposite.” Students can improve their conjecture again in groups of four. Have all groups share their conjectures with the class. (2 interior angles add to the measure of the exterior angle that is the supplementary angle of the third interior angle) You might point out that the angles that add to the exterior angle are opposite the third angle in the triangle. The activity below allows students to check their conjecture using The Geometer’s Sketchpad®. Activity Use The Geometer’s Sketchpad® for this activity. Checking that the exterior angle equals the sum of the interior angles opposite to it. a) Construct a triangle ABC and measure its angles. b) Draw a ray BC and mark a point D on the ray, outside the triangle. c) Measure ACD. d) Using the Number menu option, calculate the sum of the measures of ABC and BAC. You can click on the angle measures to make them appear in the calculation windows. e) What do you notice about the answers in parts c) and d)? f) Modify the triangle. Did your answer to part e) change? Answers: e) the answers are the same, f) no (end of activity) (MP.3) Proving the conjecture for the size of exterior angle. Return to the triangle above and erase the angle measures. ASK: How can you find the measure of angle c from the measure of angle x? (c = 180 − x) Write the equation on the board. ASK: How can you find the measure of angle c from the measure of angles a and b? (c = 180° − (a + b)) Write the second equation underneath the first, as shown below: c = 180 − x c = 180° − (a + b) D-88 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry SAY: The expressions on the left side of these equations are the same. This means the expressions on the right side are the same too. ASK: How are the expressions on the right side of the equal sign the same in both equations? (something is subtracted from 180) Have a volunteer circle the parts that are subtracted. ASK: What can you say about the subtracted parts? (they are the same) Have students write the equation showing this. (x = a + b) Point out that students have now proved their conjecture about the external angle using logic. Summarize on the board: An exterior angle of a triangle equals the sum of the two angles opposite to it in the triangle. x = a + b Use the measure of the exterior angle to find missing angles. Work through the examples below as a class. Then have students work individually on the following exercises. 61 38 115 x x 68 x = 68 + 38 = 106 x = 115 − 61 = 54 Exercises: Find the measure of angle a. a) b) Bonus: a 70 45 a 32 127 a 47.5 58 Answers: a) 90, b) 82, Bonus: 22.5 SAY: Now you will use what you know about exterior angles, vertical angles, and supplementary angles to find the missing angle measures. Draw on the board: d 58 c b a 15 ASK: Which angles are the exterior angles for this triangle? (c and the angle labeled 58) What is the measure of angle a? (43) How do you know? (58 − 15 = 43) Have students find the rest of the angles in the picture, and then have volunteers explain the solutions. (b = 122, supplementary to 58 or using sum of the angles in a triangle; c = 58, supplementary to b or vertical to 58; d = 122, supplementary to 58 or vertical to b) Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-89 Exercises: Find the measures of angles x and y. y x x y 52 71 x 72 x 72 41 y Answers: a) x = 54.5°, y = 109°; b) x = 36°, y = 144°; c) x = 128°, y = 93° Extensions (MP.3) 1. Josh thinks that the exterior angle in a triangle is larger than each of the interior angles not supplementary to it. a) Make a sketch of a triangle with an exterior angle. b) Is Josh’s statement true for your triangle? c) Is Josh’s statement true for any triangle? Explain. d) Ted thinks the exterior angle is larger than any angle in a triangle. Is Ted’s statement true? Explain. Answers: c) Yes, Josh’s statement is true for any triangle. If x is the exterior angle, x = a + b, all the measures are positive numbers, and the measure of x, being the sum, is larger than any of the addends—in other words, larger than any two interior angles that can add to the exterior. d) Ted’s statement is not true. Counterexample: a 70 47.5 The angle adjacent to the exterior angle is an obtuse angle, so it is larger than 70; in this case it is 110 and it is the largest of the three interior angles and larger than the exterior angle. (MP.3) 2. The angles QPR and PQR in triangle PQR are acute. Point S is on the line PQ so that RS PQ. Jenny thinks that point P can be between the points Q and S. Is she correct? Explain. Hint: Make a sketch placing point P between Q and S. Look at triangle PRS. What can you say about the size of QPS? Answer: Jenny is not correct. R Q P S If point P is between Q and S, and RS PQ, triangle PRS is a right triangle with external angle QPR. The external angle QPR has to be larger than PSR = 90, so QPR is an obtuse angle. D-90 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry However, we are given that angle PQR is acute. An angle cannot be both acute and obtuse, so point P cannot be between points Q and S. (MP.3) 3. Remind students that, in an isosceles triangle, the angles between the equal sides and the third side are equal. Then present them with the following problem. Grace thinks ABC is a right angle. Is she correct? Explain without measuring. B y A x 110º D C Solution: Angle C is equal to angle x, so 2x = 110 because ADB is the exterior angle for triangle BDC. So x = 55. Triangle ABD is an isosceles triangle, so A = y, and from the sum of the angles in triangle ABD, 2y = 180 − 110 = 70, so y = 35. ABC = x + y = 55 + 35 = 90. Grace is correct: ABC is a right angle. Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-91 G8-14 Corresponding Angles and Parallel Lines Pages 106–108 Standards: 8.G.A.5 Goals: Students will use informal arguments to establish facts about corresponding and co-interior angles at parallel lines. Prior Knowledge Required: Can identify parallel lines using a protractor Can draw and measure with a ruler and a protractor Can identify supplementary and vertical angles Knows that vertical angles are equal Knows what counterexamples are Vocabulary: co-interior angles, conjecture, corresponding angles, counterexample, parallel, supplementary angles, vertical angles Materials: transparencies overhead projector rulers protractors The Geometer’s Sketchpad® Introduce corresponding angles. Draw on the board: 3 5 7 1 2 4 6 8 SAY: We are going to look at different situations when two lines intersect a third line. One situation is when angles create a pattern like in the letter F. We call these corresponding angles. NOTE: The term “corresponding angles” here is not the same as the corresponding angles in congruent triangles—in other words, not the angles that match up in congruent shapes. SAY: In the case of corresponding angles among intersecting lines, the letter F pattern can be flipped from side to side or rotated in a circle. For example, angles 4 and 8 in the picture are corresponding angles. Trace the letter F in the picture with your finger. Then copy the picture D-92 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry three more times, trace the letter F in each picture, and have students find all four pairs of corresponding angles, as shown below: 1 3 5 7 1 3 6 5 8 7 1 3 5 7 2 4 6 8 1 2 3 4 5 6 7 8 2 4 2 4 6 8 If students do not see that the upside-down or reflected pattern resembles an F, copy the picture to a transparency, highlight the letter F, and turn it over or rotate it so that students see the pattern. Exercises: List the corresponding angles. a) 5 1 6 2 3 4 7 8 b) a b c d k m n u Answers: a) 1 and 3, 2 and 4, 5 and 7, 6 and 8; b) a and k, b and m, c and n, d and u Corresponding angles for parallel lines are equal. Have students draw a pair of parallel lines by using the opposite sides of a ruler. Ask them to place the ruler across the two lines they drew and draw a third line intersecting both lines. Have them measure all eight angles created this way with protractors and identify which angles are corresponding angles. ASK: What do you notice about the measures of corresponding angles? (they are equal) Did everyone create angles of the same size? (no) Did everyone get the same result: corresponding angles in a pair of parallel lines are equal? (yes) The activity below allows students to check this result with The Geometer’s Sketchpad®. Activity 1 Use The Geometer’s Sketchpad® for this activity. Checking that corresponding angles for parallel lines are equal. a) Draw a line. Label it AB. Mark a point C not on the line and construct a new line parallel to AB through C. b) Draw a line through points B and C. Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-93 c) On the line parallel to AB, mark point D on the same side of BC as point A. d) On the line BC, mark a point E outside the line segment BC, as shown below: not here here B C or here e) Name two corresponding angles in the diagram. f) Measure the corresponding angles you named. What do you notice? Make a conjecture. g) Move the points A, B, C, or D around. Are the corresponding angles always equal? (MP.5) h) Move point E so that the angles you measured stop being equal (e.g., move point E to be between B and C). Look at the pattern the angles create. Are they corresponding angles? Does this create a counterexample to your conjecture from c)? Explain. Answers: f) corresponding angles at parallel lines are equal; g) yes; h) When angles become not equal, they are not corresponding angles anymore. This does not create a counterexample, because the statement talks about corresponding angles and the unequal angles are not corresponding. (end of activity) Practice finding measures of corresponding angles at parallel lines. Draw on the board: x 137 SAY: The lines are parallel. ASK: What do you know about angle x and the angle that measures 137? (they are corresponding angles, so they are equal) What is the measure of angle x? (137) Exercises: Find the measure of the corresponding angles. a) b) 23 70 x c) a Bonus: x 101 54 y 60 65 y x Answers: a) x = 70, b) a = 23, c) x = 101, y = 54, Bonus: x = 60, y = 65 D-94 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry Find measures of angles between intersecting lines using corresponding, vertical, and supplementary angles. Review what vertical angles are and the fact that they are equal. Remind students what supplementary angles are. Then return to part c) in the previous exercises and work as a class to fill in all the missing angle measures. (see answers below) 79 101 79 x 101 79 101 79 54 126 y 126 54 126 126 54 Exercises: Find all the angles in the picture. a) b) 66 73 c) 123 a Bonus: c b 65 60 53 123 y x d Answers: a) 66 11466 b) 73 107 90 90 114 66 114 66 114 66 c) b c 123 57 123 57 127 53 a 57 123 57 123 53 127 d 107 73 73 107 90 90 90 90 Bonus: 55 65 60 65 65 55 120 60 115 65 120 60 65 115 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-95 (MP.3) When corresponding angles are equal, lines are parallel. Write on the board: When lines are parallel, corresponding angles are equal. SAY: We have seen that this statement is true. Let’s change the order in this sentence. Continue writing on the board: When corresponding angles are equal, lines are parallel. Point out that when we change an order in a sentence, the meaning changes. For example, the sentence “All boys are people” is true, but the sentence “All people are boys” is not true. ASK: Do you think the statement on the board is true? You might want to have students vote. Ask students to draw a pair of lines that are not parallel and then a third line that intersects them. Have them pick a pair of corresponding angles in their drawing and measure them. ASK: Did anyone get a pair of equal corresponding angles? (no) Explain that this shows that the statement “When corresponding angles are equal, lines are parallel” is likely to be true, but does not prove it. SAY: However, mathematicians have proven that the conjecture is true. Write “true” beside the statement on the board. NOTE: If your class is ready for the challenge, you can have them do Extension 1 to prove the statement. Using corresponding angles to identify parallel lines. Explain that the statement “When corresponding angles are equal, lines are parallel” allows you to tell which lines are parallel and which are not. Return to the picture in part c) of the previous exercises and ask students to circle a pair of corresponding angles for each pair of lines, using different colors. Point out that, to make identifying lines easier, you used small letters to name the lines in this picture. Then ask students to identify which lines are parallel. (b and c) Activity 2 Have students work in pairs to complete this activity. a) Draw two pairs of lines, one parallel and the other not but looking like it might be. (For example, draw a line and then place a ruler along the line as if to draw another line along the parallel side of the ruler. Then rotate the ruler very slightly.) Do not indicate which pair is which. b) For each pair of lines, draw a third line intersecting the first and second lines. c) Exchange your paper with a partner. Measure and compare the corresponding angles to identify the pair of parallel lines. (end of activity) Introduce co-interior angles. Explain that, when two lines intersect a third line, there can be other patterns of angles, which also have special names. Draw on the board: 3 5 7 D-96 1 2 4 6 8 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry Explain that angles that create a pattern like in the letter C are called co-interior angles or same-side interior angles. For example, angles 4 and 6 in the picture are co-interior angles. Trace the letter C in the picture with your finger. Then copy the picture and have students find the second pair of co-interior angles. (3 and 5) Trace the letter C in the second picture, as shown below: 1 3 5 7 2 1 4 3 6 5 8 7 2 4 6 8 In parallel lines, co-interior angles are supplementary. Draw on the board: 138 Point to different angles in the picture and ASK: Is this angle co-interior with the angle given? Have students signal the answer with thumbs up or thumbs down. When students have identified the correct angle as co-interior, ASK: How can we find its measure? To prompt students to see the answer, label the angles as shown below and ask them to find angle x before finding angle y. (the lines are parallel, so the corresponding angles are equal, so x = 138; angles x and y are supplementary, so y = 180 − 138 = 42) 138 x y ASK: Do you think this will work for any pair of parallel lines that have a third line intersecting them? (yes) What do co-interior angles add to when lines are parallel? (180) In other words, what can we call co-interior angles at parallel lines? (supplementary angles) Write on the board: When lines are parallel, co-interior angles are supplementary so they add to 180. Exercises: Find the missing angle measures. a) b) Bonus: y x y z 52 121 b 70 x a Answers: a) x = 128, y = 52; b) a = 121, b = 59, Bonus: x = y = 130, z = 70 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-97 Extensions (MP.3) 1. SAY: Let’s see what happens if we have two lines that intersect. Can we still have equal corresponding angles? Draw on the board: x y x SAY: It looks like these two lines have equal corresponding angles. Maybe I did not draw them perfectly, but I’ve marked the angles I think are equal with an x. Circle the x at the bottom and SAY: I see a triangle in this picture. ASK: What do we call the angle I circled for that triangle? (external angle) What do we know about the measure of the external angle in a triangle? (the measure of the external angle equals the sum of the measures of the interior angles opposite it) How can we write it down using the letters in the diagram? (x = x + y) What does this say about the measure of angle y? ( y = 0) Can that happen? (no) SAY: If the lines have equal corresponding angles they cannot intersect, because if they do we get a 0 angle between them. In mathematics, when you suppose something happens and you logically arrive at nonsense, you conclude that your original idea was incorrect. So in this case, we proved using logic that, if two lines meet a third line making equal corresponding angles, the lines are parallel. (MP.3) 2. Use co-interior angles to explain why opposite angles in a parallelogram are equal. Answer: B A C D AB || CD and angles B and C are co-interior angles, so C = 180 − B. AD || BC and angles B and A are co-interior angles, so A = 180 − B. This means C = A. Similarly, AB || CD and angles B and C are co-interior angles, so B = 180 − C. AD || BC and angles C and D are co-interior angles, so D = 180 − C. This means B = D. 3. Demonstrate the equality between corresponding angles using translation—a slide. Make two copies of the picture below on transparencies. Place the transparencies one on top of the other and show students that they are identical. Then slide one transparency down the other so that the line KQ slides along the line KL to the position of LR, so that MKQ slides to match MLR. This means MKQ = MLR. D-98 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry Have students complete the statements below. a) KQ is parallel to ____ and RLK and QKM are _______________ angles, so QKM = RLK = 70°. b) QKM and TQS are corresponding angles and QKM = 70° = TQS, so ____ is parallel to _____. c) The quadrilateral QRLK is a ____________________. Answers: a) LR, corresponding; b) KL, RS; c) parallelogram (MP.1, MP.3, MP.7) 4. Draw a parallelogram. Measure each angle of the parallelogram and draw a ray that divides each angle into two equal angles. Extend each ray so that it intersects two other rays. What geometric shape can you see in the middle of the parallelogram? Use the sum of the angles in the shaded triangle to explain why this is so. Answer: The shape in the middle is a rectangle. The acute angles of the triangle are both half of the angles of a parallelogram. The adjacent angles of a parallelogram add to 180, so their halves add to 90. Because the sum of the angles in a triangle is 180, the third angle in each triangle is a right angle. Since there are three other triangles like the shaded triangle in the parallelogram, the shape in the middle has four right angles and must be a rectangle. (MP.1, MP.3) 5. A restaurant has many windows of unusual shapes. One of them is the trapezoid shown below. A 60 7 cm 120 10 cm D 5 cm 114 66 B 6.2 cm C The restaurant owner, Bill, calls a window company to order a replacement for this window. He tells the company representative that the window is a trapezoid with AB parallel to CD, and he gives some more information. For each combination of information below, say whether Bill is giving enough information to create a unique trapezoid of the exact shape and size wanted. If so, provide directions for constructing the trapezoid if they are easy. If there is no unique trapezoid, draw two trapezoids with the given specifications. Hint: Draw a line parallel to AD through point C. It separates the trapezoid into a quadrilateral and a triangle. Label the new line EC and label the triangle EBC. What type of quadrilateral have you created? Can you construct this quadrilateral and the leftover triangle given the additional information? Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-99 a) window is a trapezoid with AB parallel to CD; angles A and B and sides AB and DA b) window is a trapezoid with AB parallel to CD; angles A and B and sides AB and CD c) window is a trapezoid with AB parallel to CD; angles A and B and sides BC and DA d) window is a trapezoid with AB parallel to CD; sides AB, BC, CD and angle B e) window is a trapezoid with AB parallel to CD; sides AB, BC, CD, DA and angle A f) window is a trapezoid with AB parallel to CD; the four sides, AB, BC, CD, DA Answers: a) Yes, the information Bill gives creates a unique trapezoid. Construct line AB and two rays to form angles A and B, and then draw the side DA = 7 cm. Draw a line parallel to AB through D and label it DC. The ray creating angle B intersects with line DC at C and finishes the unique trapezoid. b) Yes. To create the combined shape, create the triangle before the quadrilateral (a parallelogram), but determine the triangle sides and angles from the parallelogram. The parallelogram AECD will have equal opposite sides, so CD = AE = 5 cm and AD = EC. Since AD || EC, A and CEB are corresponding angles at parallel lines and they are equal (60 each). Thus, in triangle CEB, CEB = 60, CBE = B = 66, and side EB = 5 cm (because AB = 10 cm and AE = DC = 5 cm, AB − AE = EB, thus 10 − 5 = 5 cm). Start by constructing the triangle. Then extend side BE beyond point E 5 cm to point A. Complete the parallelogram by drawing angle A = 60 for line AD and drawing a line CD parallel to AB through point C. c) No. Without knowing the length of side AB, we don’t know where to construct angle B, so this leaves both sides AB and CD of unknown lengths. A 60 7 cm 120 x cm D y cm 114 66 B 6.2 cm C d) Yes. Construct the side AB and angle B. Construct the side BC. Since CD AB, C = 180 − B = 114, so construct angle C so that CD is 5 cm. Join points A and D to create the fourth side. e) Yes. Construct line AB, angle A, and line AD. Since ABCD is a trapezoid, D is a co-interior angle with A, so D = 180 − A = 120. Construct D and side CD (5 cm). Join points B and C. f) Yes. To create the combined shape, create the triangle before the parallelogram, but determine the triangle sides from the parallelogram. The parallelogram AECD will have equal opposite sides, so AE = DC and AD = EC. Start the triangle with the sides EB = 5 cm (AB − EB = AE = DC = 5 cm), BC = 6.2 cm, and EC = 7 cm (AD = 7 cm = EC). Extend side BE beyond point E 5 cm to point A. From point C, extend line CD 5 cm parallel to line AB and join AD (7 cm) to complete the parallelogram. D-100 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry G8-15 Alternate Angles and Parallel Lines Pages 109–111 Standards: 8.G.A.5 Goals: Students will use informal arguments to establish facts about alternate angles at parallel lines. They will use these facts to informally prove that angles in a triangle add to 180º. Prior Knowledge Required: Can identify parallel lines Can identify supplementary, vertical, corresponding, and co-interior angles Knows the properties of supplementary, vertical, corresponding, and co-interior angles Can draw and measure with a ruler and a protractor Knows what counterexamples are Vocabulary: alternate angles, co-interior angles, conjecture, corresponding angles, counterexample, parallel, straight angle, supplementary angles, vertical angles Materials: rulers transparencies overhead projector protractors The Geometer’s Sketchpad® Introduce alternate angles. Draw on the board: 1 2 4 3 5 7 6 8 Explain that alternate angles are angles that create a pattern like in the letter Z. For example, angles 3 and 6 in the picture are alternate angles. Trace the letter Z on the picture with your finger. Copy the initial picture, trace the letter Z, and have students find the other pair of corresponding angles, as shown at right below: 1 3 5 7 8 6 2 3 4 5 7 1 2 4 6 8 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-101 If students do not see that the reflected pattern resembles a letter Z, copy the picture onto a transparency, highlight the letter Z, and turn it over so that students see the pattern. Exercises: List the alternate angles. a) 5 1 6 2 b) 3 4 7 8 a c b d k m n u Answers: a) 6 and 3, 2 and 7; b) d and k, b and n Alternate angles for parallel lines are equal. Have students draw a pair of parallel lines by using the opposite sides of a ruler. Ask them to place the ruler across the lines they drew and draw a third line intersecting both the first and second lines. Have them identify both pairs of alternate angles and measure them. ASK: What do you notice about the measures of alternate angles? (they are equal) Did everyone create angles of the same size? (no) Did everyone get the same result: alternate angles in a pair of parallel lines are equal? (yes) ASK: Are alternate angles always equal? Have students repeat the exercise above, this time starting with a pair of lines that are not parallel. Students can also do Activity 1 below to discover that parallel lines create equal alternate angles. Part e) encourages students to pay close attention to what they see and to what changes on the screen. Activity 1 Use The Geometer’s Sketchpad® for this activity. Discovering that parallel lines create equal alternate angles. a) Draw a line. Label it AB. Mark a point C not on the line, and construct a line parallel to AB through C. b) Draw a line through points B and C. c) On the line parallel to AB, mark point D on the other side of BC from point A. d) Name two alternate angles in the diagram. e) Measure the alternate angles you named. What do you notice? Make a conjecture. f) Move the lines or the points A, B, or C around. Are the alternate angles always equal? (MP.5) g) Pull point D to the other side of the line BC to stop the angles you measured being equal. Look at the pattern the angles create. Are they alternate angles? Does this create a counterexample to your conjecture from e)? Explain. Answers: e) alternate angles at parallel lines are equal; f) yes; g) When angles stop being equal, they are not alternate angles anymore. This does not create a counterexample because the statement talks about alternate angles and the unequal angles are not alternate. (end of activity) D-102 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry Proving that alternate angles at parallel lines are equal. To review supplementary, vertical, corresponding, and co-interior angles, draw the picture below on the board and have students find all the missing angle measures. Ask them to explain how they found each angle measure. Encourage multiple explanations. (sample answers: b is co-interior with the given angle, so its measure is 180 − 40 = 140; a and the given angle are corresponding angles, so they are equal; angles a and b are supplementary angles, so b = 180 − 40 = 140) a b c 40 d m n u ASK: Which angles in this picture are alternate angles? (d and the given angle, b and n) Are they equal? (yes) Are alternate angles at parallel lines always equal? (yes) How can we explain using logic that these angles are equal? Replace the angle measure with the letter k and have students explain why angles k and d are equal. PROMPT: Which angle is corresponding with k? (a) What do we know about corresponding angles? (they are equal) What do we know about angles a and d? (they are equal) Why? (they are vertical angles) Find measures of angles between intersecting lines using alternate, corresponding, vertical, and supplementary angles. (MP.7) Exercises: Find all the angles in the picture. a) b) 54 62.4 b c) a 119.77 c Bonus: 61.23 d 119.77 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry 55° 65° D-103 Answers: a) 54 126 54 b) 126 54 126 54 126 54 90° 62.4 117.6 90 90 117.6 62.4 62.4 117.6 c) b 119.77 c 119.77 60.23 a 60.23 60.23 119.77 Bonus: 118.77 65° 60.23 61.23 61.23 118.77 119.77 d 90 90 90 90 120° 125° 55° 55° 60° 120° 60° 65° 125° Using alternate angles to identify parallel lines. Remind students that when they learned about corresponding angles, they looked at two statements: When lines are parallel, corresponding angles are equal, and when corresponding angles are equal, lines are parallel. Write on the board: When lines are parallel, alternate angles are equal. ASK: Is this true or false? (true; we have proved it using logic) Write “true” beside the statement. ASK: If we change the order in this statement, what statement will we get? (when alternate angles are equal, the lines are parallel) Write that statement on the board and explain that you want to investigate whether this statement is true. ASK: What do you need to do? (draw a pair of lines with equal alternate angles and check whether they are parallel) How can we check if lines are parallel? (For example, if the corresponding angles are equal, then the lines are parallel) Draw on the board: x x (MP.3) SAY: These lines have equal alternate angles. Which other angles do we know are equal to x? (vertical angles to the given ones) Have a volunteer mark the angles and explain why they are equal to x. ASK: What other types of angles can we see in this picture? (corresponding angles) Have another volunteer circle a pair of corresponding angles. ASK: Are the corresponding angles equal? (yes) What do we know about lines that have equal corresponding angles? (they are parallel) SAY: We have just proved that these two lines are parallel, so we have just proved that the statement is true. Write “true” beside the second statement on the board. D-104 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry Practice identifying parallel lines using the equality between alternate angles. Return to the picture in part c) in the previous exercises and ask students to circle a pair of corresponding angles for each pair of lines, using different colors. Ask students to identify which rays are parallel. (rays b and c) Activity 2 Have students work in pairs to complete this activity. a) Draw two pairs of lines, one pair that is parallel and the other not but looking like it might be. (For example, draw a line and then place a ruler along the line as if to draw another line along the parallel side of the ruler, but then rotate the ruler very slightly.) Do not indicate which pair is which. b) For each pair of lines, draw a third line intersecting the first and second line. c) Exchange your paper with a partner. Measure and compare the corresponding angles to identify the pair of parallel lines. (end of activity) (MP.3) Proving that sum of the angles in a triangle is 180. Explain that the properties of alternate angles allow us to prove using logic that the sum of the angles in a triangle equals 180. Point out that, even though we discovered this fact and used it many times, we have not proven it using logic. Draw a triangle on the board and label the angles a, b, and c as shown below. Explain that you want to use alternate angles, so you are drawing a line parallel to one of the sides, through the opposite vertex. Draw the line, as shown below: b a c ASK: Are there any equal alternate angles in this picture? (yes) Is there an angle alternate to angle a? (yes) Point to different angles and have students signal thumbs up or thumbs down to show if this is the angle alternate with a. Repeat with c. Point to the three angles at the top of the picture and ASK: What type of angle do the three angles make? (straight angle) What is the measure of a straight angle? (180) Ask students to write an equation that shows that the three angles add to 180. (a + b + c = 180) ASK: What did we need to prove to show that angles in a triangle add to 180? (a + b + c = 180) SAY: We have just proven precisely that fact. Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-105 Extensions 1. Demonstrate the equality between corresponding angles using transformations. Make two copies of the picture below on transparencies. Place the transparencies one on top of the other, and show students that they are identical. Have students identify a pair of alternate angles (e.g., QKL and NLK). Mark the angles with arcs. Q O a) Slide the line KQ down along the ray KL to the position of LR so that MKQ slides to MLR. This means MKQ = MLR. The angles MLR and NLK are vertical angles, so they are equal. But MLR = MKQ, so MKQ = NLK. b) Press a pencil to act as a pivot to the point O. Rotate the top transparency 180 around point O to show how angle QKL becomes NLK. (MP.3) 2. Sketch parallelogram ABCD. A D B C Fill in the blanks to prove using logic that opposite sides of a parallelogram are equal. a) Parallel lines AB and _____ intersect with the third line DB. ABD and ____ are alternate angles at parallel lines, so ABD = _____. Label these angles with the same number of arcs. b) Parallel lines AD and _____ intersect with the third line DB. ADB and ____ are alternate angles at parallel lines, so ADB = _____. Label these angles with the same number of arcs. c) Triangles ABD and ______ are congruent by the _____ congruence rule because they have two pairs of equal corresponding _____________ and a common ______________. d) ABD ______, so AB = ____ and AD = ______. Answers: a) CD, CDB, CDB; b) BC, CBD, CBD; c) CDB, ASA, angles, side; d) CDB, CD, CB D-106 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry G8-16 Solving Problems Using Angle Properties Pages 112–113 Standards: 8.G.A.2, 8.G.A.5 Goals: Review of the material learned in the unit. Prior Knowledge Required: Can identify parallel lines Can identify supplementary, vertical, co-interior, corresponding, and alternate angles Knows the properties of supplementary, vertical, co-interior, corresponding, and alternate angles Can draw and measure with a ruler and a protractor Knows what counterexamples are Knows the properties of triangles and special quadrilaterals Can identify congruent triangles Can apply the congruence rules for triangles Vocabulary: alternate angles, co-interior angles, congruent, corresponding angles, counterexample, diagonal, equilateral, isosceles trapezoid, isosceles triangle, kite, parallel, right trapezoid, right triangle, scalene, supplementary angles, vertical angles Materials: scissors BLM Triangles for Making Quadrilaterals (p. D-132) BLM Angle Properties (Summary) (p. D-133, optional) BLM Properties of Angles in a Triangle (Summary) (p. D-134, optional) This lesson serves as a review and combination of all the material learned in this unit. The exercises can be assigned or you can work through them as a class to review the angle properties. Exercises: Find the missing angles. a) A B b) c) 37° x x 84° z D y w x u C Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry a 50° y z y 46° b 72° z D-107 d) e) 34° F e E d z d d 57 G c w f H x y 54° u 20° v Answers: a) x = 53, y = 37, z = 53; b) u = 134, w = 46, x = 84, y = 130, z = 50; c) a = x = 62, b = 118, y = 44; d) c = f = 63, d = 60, e = 120; e) w = y = 126, x = 20, z = 34, u = v = 63 Have students explain how they found the angles in the previous exercises. Then discuss what other additional information, such as types of quadrilaterals, parallel lines, or congruent triangles we can deduce from the diagrams in the exercises. For example, in part a) lines AB and CD are parallel because the co-interior angles ABC and BCA, both right angles, add to 180, so ABCD is a trapezoid. In part d) triangle EFG is an equilateral triangle because all of its angles are equal, but triangle FGH is an acute scalene triangle because it has no equal angles. We can also deduce that lines EF and GH are not parallel. In part e) we have two pairs of congruent triangles by the angle-side-angle (ASA) congruence rule, so we can mark other equal line segments. One pair of triangles has angles 34, 20, and 126 and a common vertex, and the other pair has a common side and angles 34, 83, and 63. Write and draw on the board: A quadrilateral with two pairs of equal adjacent sides is called a kite. A trapezoid with equal opposite sides is called an isosceles trapezoid. A trapezoid with two right angles is called a right trapezoid. NOTE: The activity below helps students produce counterexamples to various statements about quadrilaterals. Keep the above definitions on the board to help students during the activity. Make sure students know that a line connecting the opposite vertices of a quadrilateral is called a diagonal. (MP.3) Activity Cut out the triangles from BLM Triangles for Making Quadrilaterals. Check the white triangles for sides and angles equal to sides or angles of the gray triangle. Mark the equal sides and angles on all triangles, on both sides of each triangle. D-108 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry Combine the gray triangle with one of the other triangles in different ways so that they share a side—in other words, two sides line up to form the diagonal for a new shape. Use the triangles you cut out to answer the questions. a) Which triangle combined with the gray triangle allows you to make a parallelogram? Use equal angles to explain why the shape you produced is a parallelogram. Make more than one parallelogram from this pair of triangles. Make a quadrilateral that is not a parallelogram and explain how you know it is not a parallelogram. b) Use two different right triangles to produce a quadrilateral that has exactly one pair of equal opposite angles and that is not a kite. c) Find all triangles that allow you to make a trapezoid with the gray triangle. Explain why the shape you produced is a trapezoid. What kind of trapezoid is it? d) Combine the two triangles you picked in part c). Can you make a trapezoid from these two triangles? Explain your answer. Is it the same type of trapezoid as in part c)? Answers: a) A, you can make three different parallelograms (one of them a rectangle) and a kite. Sample explanation for parallelogram: a b b a Triangle A is congruent to the gray triangle, and both are right triangles. This means that the acute angles in both triangles add to 90, or b = 90 − a. The adjacent angles in the quadrilateral are then a and b + 90 = 90 + (90 − a) = 180 − a. The adjacent angles add to 180 and they are co-interior angles for the opposite sides of the quadrilateral. This means the opposite sides of the quadrilateral are parallel. The kite is not a parallelogram because parallelograms have equal opposite sides, and the kite has opposite sides that are not equal. c) Triangles E and F allow you to make a right trapezoid with the gray triangle. They both have at least one angle that is the same as the smallest angle of the gray triangle. Sample explanation: b a a a The angles marked with an arc are equal, and they are alternate angles for the top and the bottom side of the quadrilateral. When alternate angles are equal, the lines are parallel, so the top and the bottom sides of the quadrilateral are parallel. The angles a and b adjacent to the two angles just discussed are alternate angles for the other two sides and they are not equal, so these sides are not parallel. The quadrilateral is a trapezoid. We also know that a + b = 90, so the top left angle of the trapezoid is a right angle. So the trapezoid has two right angles and is a right trapezoid. d) Triangles E and F together can produce a trapezoid, but it is not a right trapezoid. The explanation is similar to the one in part c). (end of activity) Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry D-109 You might wish to provide students with BLM Angle Properties (Summary) and BLM Properties of the Angles in a Triangle (Summary) as summaries of material learned in this unit. Extensions (MP.3) 1. a) What is the sum of the angles in a quadrilateral? b) Draw a quadrilateral with exactly three equal angles (see two samples below). Is it a special quadrilateral? c) Can a quadrilateral with exactly three equal angles be a parallelogram? A trapezoid? Use what you know about the sum of adjacent angles in a parallelogram or a trapezoid to explain your answer. d) A special quadrilateral has exactly three equal angles. What type of special quadrilateral is it? Answers: a) 360 b) Answers will vary, most likely no. c) No for both. Explanation: Suppose the shape is a trapezoid. Angles adjacent to a non-base side of a trapezoid add to 180° because they are co-interior angles at parallel sides. At least two of the three equal angles have to be adjacent to the same non-base side, so they have to add to 180 and so have to be 90°. If the three equal angles are all right angles, the fourth angle is 360° − 3 × 90° = 90°. This shape has four equal angles and not three, so a quadrilateral with exactly three equal angles cannot be a trapezoid. The explanation for a parallelogram is very similar. d) The only type of special quadrilateral that is not a trapezoid or a parallelogram is a kite. (MP.3) 2. a) Draw a line AC and mark a point B on it. Draw a line segment BD intersecting AC. b) Using a protractor, draw rays BE and BF so that ray BE divides ABD into two equal angles and ray BF divides DBC into two equal angles as shown below: c) Find the measure of EBF without using a protractor. Explain your answer. Verify your answer using a protractor. Answer: c) EBF = 90. Explanation: ABE = EBD, DBF = FBC, and ABE + EBD + DBF + FBC = 180 However, ABE + EBD + DBF + FBC = EBD + EBD + DBF + DBF = (EBD + DBF) + (EBD + DBF) = EBF + EBF So EBF = 180 ÷ 2 = 90. D-110 Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry