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G7-3 M easuring and Drawing Angles and Triangles Pages 102–104 Goals Curriculum Expectations Students will measure and draw angles. Ontario: 5m51, 5m52, 5m54, 6m48, 6m49, 7m3, 7m4, 7m46 WNCP: 6SS1, review, [T, R, V] PRIOR KNOWLEDGE REQUIRED Materials Knows what an angle is Can name an angle and identify a named angle dice geoboards and elastics grid paper Dynamic geometry software (optional) Vocabulary angle vertex arms acute obtuse Review the concept of an angle’s size. Draw two angles: ASK: Which angle is smaller? Which corner is sharper? The diagram on the left is larger, but its corner is sharper, and mathematicians say that this angle is smaller. The distance between the ends of the arms in both diagrams is the same, but this does not matter; angles are made of rays and these can be extended. What matters is the “sharpness.” The sharper the corner on the “outside” of the angle, the narrower the space between the angle’s arms. Explain that the size of an angle is the amount of rotation between the angle’s arms. The smallest angle is closed; both arms are together. Draw the following picture to illustrate what you mean by smaller and larger angles. Smaller Larger You might also illustrate what the size of an angle means by opening a book to different angles. Draw some angles and ask your students to order them from smallest to largest. A E-6 B C D E Teacher’s Guide for Workbook 7.1 COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED You can show how much an angle’s arm rotates with a piece of chalk. Draw a line on the board then rest the chalk along the line’s length. Fix the chalk to one of the line’s endpoints and rotate the free end around the endpoint to any desired position. Define acute and obtuse angles in relation to right angles. Obtuse angles are larger than a right angle; acute angles are smaller than a right angle. ExTRa PRaCTICE: 1. Copy the shapes onto grid paper and mark any right, acute, and obtuse angles. Which shape has one internal right angle? What did you use to check? 2. Which figures at left have a) all acute angles? b) all obtuse angles? c) some acute and some obtuse angles? Introduce protractors. On the board, draw two angles that are close to each other—say, 50° and 55°—without writing the measurements and in a way that makes it impossible to compare the angles visually. aSK: How can you tell which angle is larger? Invite volunteers to try different strategies they suggest (such as copying one of the angles onto tracing paper and comparing the tracing to the other angle, or creating a copy of the angle by folding paper). Lead students to the idea of using a measurement tool. ExaMPLE: Explain that to measure an angle, people use a protractor. A protractor has 180 subdivisions around its curved edge. These subdivisions are called degrees ( °). Degrees are a unit of measurement, so just as we write cm or m when writing a measurement for length, it is important to write the ° symbol for angles. origin base line Using protractors. Show your students how to use a protractor on the board or on the overhead projector. Identify the origin (the point at which all the degree lines meet) and the base line (the line that goes through the origin and is parallel to the straight edge). When using a protractor, students must • place the vertex of the angle at the origin; • position the base line along one of the arms of the angle. COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED You could draw pictures (see below) to illustrate incorrect protractor use, or demonstrate it using an overhead projector and a transparent protractor. Geometry 7-3 E-7 Introduce the degree measures for right angles (90°), acute angles (less than 90°), and obtuse angles (between 90° and 180°). Point out that there are two scales on a protractor because the amount of rotation can be measured clockwise or counter-clockwise. Students should practise choosing the correct scale by deciding whether the angle is acute or obtuse, then saying whether the measurement should be more or less than 90°. (You may want to do some examples as a class first.) Then have students practise measuring angles with protractors. Include some cases where the arms of the angles have to be extended first. Introduce angles in polygons, then have students measure the angles in several polygons. Students can draw polygons (with both obtuse and acute angles) and have partners measure the angles in the polygons. Drawing angles. Model drawing angles step by step (see Workbook p. 104 or BLM Measuring and Drawing Angles and Triangles, p E-42). Emphasize the correct position of the protractor. Have students practise drawing angles. You could also use Activity 2 for that purpose. Have students practise drawing lines that intersect at a given angle. Another way to practise drawing angles is to construct triangles with given angle measures. ACTIVITIES 1–4 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 different numbers of given angles. EXAMPLES: d) e) 12° 9° 4° process Expectation Selecting tools and strategies, Technology E-8 2. Students will need a die, a protractor, and a sheet of paper. Draw a starting line on the paper. Roll the die and draw an angle of the measure given by the die; use the starting line as your base line and draw the angle counter-clockwise. Label your angle with its degree measure. For each next roll, draw an angle in the counter-clockwise direction so that the base line of your angle is the arm drawn at the previous roll. The measure of the new angle is the sum of the result of the die and the measure of the angle in the previous roll. Stop when there is no room to draw an angle of the size given by the roll. For example, if the first three rolls are 4, 5, and 3, the picture will be as shown. 3. Teach students to draw and measure angles using Geometer’s Sketchpad. Then ask them to try moving different points (on the arms of the angle, or its vertex) so that the size of the angle becomes, say, 50°. Is it easy or hard to do? When you move the line segments, does Teacher’s Guide for Workbook 7.1 COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED 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 Sample ANSWERS: the angle change? (yes) When you move the vertex or other point on the arms, does the angle change? (yes) Show students how to draw an angle of fixed measure (using menu options). Will moving the endpoints change the size of the angle now? (no) Show students how to draw angles equal to a given angle. 4. Have students draw polygons in Geometer’s Sketchpad and measure the size of the angles and the length of the sides of these polygons. Have students check that the angle measures they obtain make sense. For example, clicking on three vertices of a quadrilateral and then using menu options to measure the angle might produce different angles, depending on the order in which the vertices were selected. Also, the software sometimes measures angles in the wrong direction, producing an answer more than 180°. process Expectation Technology, Reflecting on the reasonableness of the answer Extensions COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED 1. Angles on an analogue clock. What is the angle between the hands at 12:24? 13:36? 15:48? (Draw the hands first!) To guide students to the answers, draw an analogue clock that shows 3:00 on the board. Ask your students what angle the hands create. What is the measure of that angle? If the time is 1:00, what is the measure of the angle between the hands? Do you need a protractor to tell? Have students write the angle measures for each hour from 1:00 to 6:00. Which number do they skip count by? An hour is 60 minutes and a whole circle is 360°. What angle does the minute hand cover every minute? (6°) How long does it take the hour hand to cover that many degrees? How do you know? (12 minutes, because the hour hand covers only one twelfth of the full circle in an hour, moving 12 times slower than the minute hand) If the time is 12:12, where do the hour hand and the minute hand point? What angle does each hand make with a vertical line? What is the angle between the hands? (ANSWER: The hour hand points at one minute and the angle that it makes with the vertical line is 6°. The minute hand points at 12 minutes and the angle that it makes with the vertical line is 12 × 6 = 72°. The angle between the hands is 72° − 6° = 64°.) 2. Some scientists think that moths travel at a 30° angle to the sun when they leave home at sunrise. Note that the sun is far away, so all the rays it sends to us seem parallel. sun’s rays flower N 30° evening Geometry 7-3 moth home W morning E S E-9 a) What angle do the moths need to travel at to find their way back at sunset? Hint: Where is the sun in the evening? b) A moth sees the light from the candle flame and thinks it’s the sun. The candle is very near to us, and the rays it sends to us go out in all directions. Where does the moth end up? Draw the moth’s path. 30° 30° COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED E-10 Teacher’s Guide for Workbook 7.1 G7-4 Perpendicular Lines Pages 105–106 Goals Curriculum Expectations Students will identify and draw perpendicular lines. Ontario: 7m4, 7m5, 7m46 WNCP: 6SS1, 7SS3, [CN, V, T] PRIOR KNOWLEDGE REQUIRED Materials Knows what a right angle is Can name an angle and identify a named angle Dynamic geometry software (optional) protractors set squares Vocabulary angle perpendicular slant line right angle arms Introduce perpendicular lines (lines that meet at 90°) and show how to mark perpendicular lines with a square corner. Draw several pairs of intersecting lines on the board and have students identify the perpendicular lines. Include pairs of lines that are not horizontal and line segments that intersect in different places and at different angles (see examples below). Invite volunteers to check whether the lines are perpendicular using a corner of a page, a protractor, and/or a set square. Ask students where they see perpendicular lines or line segments—also called perpendiculars—in the environment (sides of windows and desks, intersections of streets, etc.). Extra practice: COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED Which lines look like they are perpendicular? K C P B F a) b) c) J d) O L G Q D A H M I R N E A perpendicular through a point. Explain that sometimes we are interested in a line that is perpendicular to a given line, but we need an additional condition—the perpendicular should pass through a given point. In each diagram below, have students identify first the lines that are perpendicular to the segment AB, then the lines that pass through point P, and finally the one line that satisfies both conditions. G D K a) b) E G B P C F P A C B F A D H E H Constructing a perpendicular through a point. Model using a set square (and then a protractor) to construct a perpendicular through a point that is not on the line. Emphasize the correct position of the set square (one Geometry 7-4 E-11 side coinciding with the given line, the other touching the given point) and the protractor (the given line should pass through the origin and through the 90° mark). Have students practise the construction. Circulate among the students to ensure that they are using the tools correctly. Then invite volunteers to model the construction of a perpendicular through a point that is on the line. (Emphasize the difference in the position of the set square: the square corner is now at the point, though one arm still coincides with the given line.) Then have students practise this construction as well. Extra practice: Draw a pair of perpendicular slant lines (i.e., lines that 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? (rectangle) Bonus Draw a slant line and a point not on the line. Using a protractor and a ruler or set square, 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 point to the given line. Measure the distance from the point to the line along the perpendicular. Then mark a point on the given line that is the same distance from the intersection as the given point. Now draw a perpendicular to the given line through this point as well. Finally, draw a perpendicular to the last line through the given point.) connection Science Why perpendiculars are important. Discuss with students why perpendiculars are important and where are they used in real life. For example, you can explain that it is easy to determine a vertical line (using gravity—just hang a stone on a rope and trace the rope), but you need a right angle to make sure that the floor of a room is horizontal. process Expectation 1. Have students use Geometer’s Sketchpad to: Technology a) Draw a line. Label it m. b) Mark a point A on the line m. c) Draw another line through point A. d) Measure the angle between the two lines. e) Try to make the angle a right angle by moving the points around. Is it easy or hard to do? f) Check whether the lines stay perpendicular when you move any of the points in the picture. g) Repeat parts a) through f) with a point not on the line. Note that when point A is not on the line, the second line might be modified so that it does not intersect the line m, and the angle you measure disappears. Explain that you need a method to draw perpendicular lines that will keep them perpendicular even if the points are moved around. E-12 Teacher’s Guide for Workbook 7.1 COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED ACTIVITIES 1–2 Teach students to draw perpendicular lines using the perpendicular line command from the menu. Do these lines stay perpendicular to the given line even if points are moved around? (yes) 2. Have students draw a triangle using the polygon tool in Geometer’s Sketchpad. Ask them to move the points around to make it look like a right triangle. Then ask them to measure the angles of the triangle and to check whether it is indeed a right triangle. 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) Ask students to think about how they draw a right triangle on paper. What tools do they use and why? (a protractor or a set square) What tools could we use instead of protractors and set square in Geometer’s Sketchpad? (perpendicular lines) Then have students draw a right triangle in Geometer’s Sketchpad. Ask them to check that the triangle remains a right triangle even if points are moved around. process Expectation Technology process Expectation Selecting tools and strategies Students can also add the measures of the acute angles in the right triangle they created, and check that the sum remains 90° even when the triangle is modified. Extension process Expectation Visualizing Discuss with your 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. P P B A B COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED A Geometry 7-4 E-13 G7-5 Perpendicular Bisectors Pages 107–108 Goals Curriculum Expectations Students will identify and draw perpendicular bisectors. Ontario: 7m1, 7m3, 7m4, 7m48 WNCP: 6SS1, 7SS3, [C, R, V, T] PRIOR KNOWLEDGE REQUIRED Can identify and construct a perpendicular using a set square or a protractor Can identify and mark right angles Can name line segments and identify named line segments Can draw and measure with a ruler Vocabulary line segment midpoint right angle perpendicular bisector Example: Materials paper circles or BLM Circles (p E-51) Dynamic geometry software (optional) Introduce the notation for equal line segments. Explain that when we want to show that line segments are equal, we add the same number of marks across each line segment. This is particularly useful for sketches, when you are not drawing everything exactly to scale. Introduce the midpoint. Model finding the midpoint of a segment using a ruler, and mark the halves of the line segment as equal, then have students practise this skill. Introduce bisectors. Explain that a bisector of a line segment is a line (or ray or line segment) that divides the line segment into two equal parts. There can be many bisectors of a line segment. Ask students to draw a line segment with several bisectors. ASK: Can a line have a bisector? What about a ray? (no, because lines and rays have no midpoints) Draw a scalene triangle. Choose a side and draw a bisector to that side that passes through the vertex of the triangle that is opposite that side. Repeat with the other sides. What do you notice? (ANSWER: All three bisectors, called medians, pass through the same point.) Introduce perpendicular bisectors. Of all the bisectors of a line segment only one is perpendicular to the line segment, and it is called the perpendicular bisector. The perpendicular bisector of a line segment • divides the line segment into two equal parts AND • intersects the line segment at right angles (90°). Point out that there are two parts in the definition, and both must be true. ASK: How can we draw a perpendicular bisector? How is that problem E-14 Teacher’s Guide for Workbook 7.1 COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED Extra practice: similar to constructing a bisector? Constructing a perpendicular? Lead students to the idea that they should first find the midpoint of the line segment, then construct a perpendicular through that point. Have students practise drawing perpendicular bisectors using set squares and protractors. process Expectation Changing into a known problem E D F J G C H B Present several diagrams that combine intersecting line segments, perpendicular lines, and bisectors, such as the one at left, and have students identify equal segments, then find perpendicular lines and bisectors. (Example: Find a bisector of EG. Is it a perpendicular bisector?) Ask other questions about the diagram, such as: • J is the midpoint of what segment? (CG) Why not AD? How do you know? (We do not have any information about the lengths of DJ and AJ. They look equal, but might have different lengths.) • Name three line segments GJ is perpendicular to. A Finally, have students identify all the perpendicular bisectors and the line segments they bisect. (ANSWERS: AD, JA, and JD bisect line segment CG; CJ, JG, and CG bisect line segment FH) process assessment Workbook Question 8f) – [C,], 7m3 Ask students to find examples of equal segments, midpoints, and perpendicular bisectors in the classroom or elsewhere, such as in letters of the alphabet, in pictures or photographs, and so on. ACTIVITIES 1–3 process Expectation Technology, Selecting tools and strategies 2. Paper folding and line segments COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED A C B 1. Show students how to construct the midpoint of a line segment in Geometer’s Sketchpad. Explain that there is no command that will construct a perpendicular bisector of a line segment. Challenge students to find a way to construct a perpendicular bisector of a line segment using the tools they are familiar with in Geometer’s Sketchpad. (Find the midpoint of the line segment, then construct a line through the midpoint and perpendicular to the given line segment) PROMPT: Think of the way you construct a perpendicular line segment on paper. Which tools can you use? (protractor or set square and ruler) Which tool replaces a protractor or a set square in Geometer’s Sketchpad? (“Perpendicular Line” in the “Construct” menu) Draw a line segment AB dark enough that you can see through the paper. Fold the paper so that A meets B. What line has your crease made? (ANSWER: a perpendicular bisector) Use a ruler and protractor to check your answer. 3. Paper folding and circles Give each student a circle (you can use BLM Circles) and ask students to draw and label a triangle on their circle. a) Fold the circle in half so that A meets B. Geometry 7-5 E-15 Look at the line that the crease in your fold makes. Is it a bisector of angle C? Is it a perpendicular bisector of line segment AB? b) Fold the circle in half again, this time making A meet C. What two properties will the crease fold have? c) Repeat, making B meet C. At what point in the circle will all three perpendicular bisectors meet? Extensions 1. Given a triangle ABC, how can you use perpendicular bisectors to help you draw the circle going through the points A, B, and C? Draw an acute scalene triangle. Draw the perpendicular bisectors by hand—do not cut and fold the triangle. 2. Start with a paper circle. Choose a point C on the circle and draw a right angle so that its arms intersect the circle. Label the points where the arms intersect the circle A and B and draw the line segment AB. Repeat with several circles to produce different right triangles. (You can use BLM Circles for this Extension.) A Fold the circle in two across the side AC so that A falls on C (creating a perpendicular bisector of AC). Mark the point where B falls on the circle. Repeat with the side BC, marking the point where A falls on the circle. What do you notice? (The image of A is the same as the image of B.) What type of special quadrilateral have you created? (a rectangle) Repeat the exercise starting with an obtuse or an acute angle C. Do the images of A and B coincide? B COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED C E-16 Teacher’s Guide for Workbook 7.1 G7-6 Parallel Lines Pages 109–111 Goals Curriculum Expectations Students will identify and draw parallel lines. Ontario: 7m1, 7m2, 7m46 WNCP: 5SS5, 7SS3, [C, R, PS] PRIOR KNOWLEDGE REQUIRED Can identify and construct a perpendicular using a set square or a protractor Can identify and mark right angles Can name a line segment and identify a named line segment Can draw and measure with a ruler Vocabulary line segment parallel right angle perpendicular Materials BLM Distance Between Parallel Lines (p E-46) BLM Drawing Parallel Lines (p E-44) Introduce parallel lines—straight lines that never intersect, no matter how much they are extended. Show how to mark parallel lines with the same number of arrows. Have students identify parallel lines in several diagrams. Then have students identify and mark parallel sides of polygons. Include polygons that have pairs of parallel sides that are neither horizontal nor vertical. Introduce the symbol || for parallel lines, label the vertices of the polygons used above, and have students state which sides are parallel using the new notation (EXAMPLE: AB || CD). Ask students to think about where they see parallel lines. Some examples of parallel lines in the real world are a double centerline on a highway and the edges of construction beams. p ? COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED m process assessment Workbook Question 8 [R, C], 7m2 n Determining if two lines are parallel. Have students draw a triangle on grid paper. Then ask them to draw line segments that are parallel to the sides of the triangle. For each pair of parallel lines (say, m and n) ask students to draw a perpendicular (say, p) to one of the lines (m) so that it intersects both lines. Ask students to predict what the angle between n and p is. Ask students to explain their prediction. Then have them measure the angle between the lines. Was the prediction correct? Have students check the prediction using the other two pairs of parallel lines they drew. Explain that this property—If one line in a pair of parallel lines interests a third line at a right angle, the other parallel line also makes a right angle with the same line—allows us to check whether two lines are parallel and to construct parallel lines. Drawing a line segment parallel to a given line segment. Have students problem-solve how to construct parallel lines using what they’ve just learned about a perpendicular to parallel lines. As a prompt, you could use the Geometry 7-6 E-17 process assessment [PS], 7m1 rectangle and square problems from G7-4 (Extra practice and Bonus, p E-12—students are required to draw a rectangle or a square using a point and a pair of parallel lines). ASK: What do you know about the sides of a rectangle? How does constructing a rectangle mean that you constructed a pair of parallel lines? Model drawing the line parallel to a given line through a point using a protractor (see p 111 in the Workbook or BLM Drawing Parallel Lines), and then model doing the same thing using a set square. Have students practise drawing parallel lines using both tools. ACTIVITIES 1–2 1. Paper folding. Draw a line dark enough so you can see it through the page. Fold the paper so that you can find a line perpendicular to AB that does not bisect AB. How can you use this crease to find a line parallel to AB? How can you use a ruler or any right angle to find a line parallel to AB? ANSWER: Fold the paper so that the perpendicular to AB falls onto itself. The crease is perpendicular to the perpendicular, so it is parallel to AB. 2. Students can investigate distances between parallel lines with BLM Distance Between Parallel Lines. Extensions process assessment 7m1, [PS] 2. Draw: a) a hexagon with three parallel sides b) an octagon with four parallel sides c) a heptagon 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: a) E-18 b) c) Teacher’s Guide for Workbook 7.1 COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED 1. A plane is a flat surface. It has length and width, but 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. Can you find a pair of lines not in a plane that never meet and do not intersect? d) e) f) 3. Lines are parallel if they point in the same direction—that’s why we use arrows to show parallel lines! We can regard direction as an angle with a horizontal line. For example, if two lines are both vertical, they both make a right angle with a horizontal line, and they are parallel. The choice of a horizontal line as a benchmark is arbitrary—it is just a convention; any line could be used for that purpose. Indeed, any two lines perpendicular to a third line are parallel. Students can complete the Investigation on BLM Properties of Parallel Lines (p E-45) to learn what happens when parallel lines meet a third line at different angles. 4. Ask students if they can draw a parallelogram that’s not a rectangle using only a ruler and a set square. SOLUTION: Step 2: Draw two of the parallel sides using the triangle. Step 3: Use the ruler to complete the figure. COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED Step 1: Draw one side of the parallelogram. Geometry 7-6 E-19 G7-7 Angle Relationships Pages 112–113 Goals Curriculum Expectations Ontario: 7m3, 7m7, 7m46, 7m47 WNCP: 6SS2, 7SS1, [C, R, V] Students will find angles in triangles, discover the sum of the angles in a triangle, and use this sum to solve problems. PRIOR KNOWLEDGE REQUIRED Can use a protractor to measure angles Materials BLM Sum of the Angles in a Triangle (pp E-47–E-48) Vocabulary straight angle acute, obtuse, right angle adjacent angles intersecting lines linear pair A straight angle is formed when the arms of the angle point in exactly opposite directions and form a straight line through the vertex of the angle. Adjacent angles share an arm and a vertex. a c a b f d a b Two adjacent angles are a linear pair if, together, they form a straight angle. Have students name all the pairs of adjacent angles in the picture at left. b ASK: Which angles make a linear pair? Which two combinations of angles make a straight angle? 1 cup 3 = 1 cup 2 5 cup 6 Point out that the same procedure applies to capacities, volumes, and areas. 4 cm3 + 2 cm3 = 6 cm3 45° 30° E-20 What happens with angles? Their measures are also added: the measure of the large angle at left is 30° + 45° = 75°. Sum of the angles around a point. Ask students to draw a pair of intersecting lines, measure the angles, and write the measures on the picture. Then ask them to add up the measures. What is the sum of the angles around the point? Did they all get the same answer? Show the Teacher’s Guide for Workbook 7.1 COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED + Angle measures in adjacent angles add to the measure of the large angle. Draw a line segment divided into two smaller segments. Mark the lengths of the smaller line segments. ASK: What is the length of the whole line segment? How do you know? What do you do with the lengths of the smaller line segments to obtain the length of the whole line segment? 53° 127° 53° 128° picture at left (mention that these angles were made by two intersecting lines) and ask students how they can tell that whoever measured the angles made a mistake. (Any two adjacent angles in this picture make a linear pair, so their sums should all be 180°). Sum of the angles in a triangle. Have students complete the Investigation on BLM Sum of the Angles in a Triangle. What is the sum of the angles in a triangle? Then have students draw several triangles, measure their angles, and add the measures. Did they get the same sum every time? Explain that the sums might not add to 180° due to mistakes in measurement. The protractor, though a convenient tool, is imprecise. 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? (180° minus the sum of the other two angles) Have students find the measures of the angles in several problems of this sort, then proceed to more complicated questions, such as the following: process Expectation Reflecting on the reasonableness of the answer • All the angles in a triangle are equal. What is the size of each angle? •O ne angle of a triangle is 30°. The other two angles are equal. What is the size of these angles? • A triangle has two equal angles. One of the angles in this triangle is 90°. What are the sizes of the other two angles? With the last question, ask whether the equal angles can be 90° each. Have students explain why this is not possible. (The two angles would already add to 180°, leaving no room for the third angle.) Invite a volunteer to draw the correct triangle on the board and mark the measures of the angles. Then present a similar problem: • A triangle has two equal angles. One of the angles in this triangle is 50°. What are the sizes of the other two angles? ASK: How is this problem different from the previous problem? (The given process Expectation COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED Reflecting on other ways to solve the problem process assessment Workbook, Question 11 [C, V], 7m7 angle is an acute angle, not a right angle.) Can a triangle have two angles of 50°? What is the third angle then? (80°) Draw an acute isosceles triangle on the board and ask volunteers to mark the angles on the picture. Then draw another acute isosceles triangle, mark the base angles as equal, and mark the unequal angle as 50°. Can this situation happen? What are the measures of the other two angles? Extra practice: a) If half an angle is 20°, the whole b) If one-third of an angle is 30°, angle is ______°. the whole angle is ______°. 20° Geometry 7-7 30° E-21 c) What is half of 90°? ______° d) If one-quarter of an angle is 25°, the whole angle is ______°. 25° ? Bonus What is one-third of 120°? ______° ACTIVITIES 1–2 1. Create right angles by paper folding Give your students a piece of paper that has no corners (e.g., an oval or a cloud shape). Ask students to create a right angle on the paper by folding. (POSSIBLE ANSWER: Fold once, unfold to see your straightline crease, then fold again across the crease so that the sides of the crease coincide.) ASK: When you folded the paper the first time, what is the size of the process assessment 7m7, [C] angle you created? (180°, a straight angle) What fraction of the straight angle is the right angle? (one half) Use your answers to these two questions to explain why your angle is a right angle. (ANSWER: A right angle is 90°, so two right angles are 180°, or a straight angle. A right angle is exactly half of a straight angle. In other words, when you fold the paper the second time and the halves of the first crease coincide, you know that the two angles you have created are equal. Together, the angles make a straight angle, so they have to be right angles.) 2. Start with a cloud- or oval-shaped piece of paper. How can you fold the paper to create the following? Students can work independently or in pairs to try and create the angle and shapes listed. Invite students to share strategies and solutions with the class, then, if necessary, give students the correct sequence of steps to follow. Do not unfold the paper unless it is mentioned as part of instructions. The instructions might be easier to understand if you actually perform the folding as you go along. ANSWERS: a) Make a 45° angle: Create a right angle by the method of Activity 1. Fold the right angle through the vertex so that its arms fall onto each other. This divides the angle into two equal parts, so the angle is a 45° angle. E-22 Teacher’s Guide for Workbook 7.1 COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED a) a 45° angle b) a square c) a rectangle b) Make a square: i) Fold the paper to make a line segment. Fold the line segment onto itself to create a right angle; the vertex of the right angle will be one of the vertices of the square. ii) Fold the right angle in two to make a 45° angle. Sides of square iii) Fold the corner of the 45° angle onto one of the arms, so that parts of that arm meet each other. Trace the fold you created with a pencil. iv) Unfold once and trace the crease that is revealed with a pencil. v) Unfold once again and flip the paper over. The lines you drew form two more sides of the square. c) Make a rectangle: COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED Geometry 7-7 i) Repeat step i) from part b). Sides of rectangle E-23 ii) Choose a point on one of the arms of your right angle to be the second vertex of the rectangle. Fold the paper through the chosen point so that the arm of the right angle you created folds onto itself. This will be the third side of the rectangle. Second corner of the rectangle Third side of the rectangle COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED iii) Flip the paper over. Fold the bottom part of the strip backwards so that both creases forming the sides of the strip fold onto themselves. E-24 Teacher’s Guide for Workbook 7.1 NAME DATE Measuring and Drawing Angles and Triangles Measuring an angle 30° arm origin If the arms are too short to reach the protractor scale, lengthen them. Step 1: Place the origin of the protractor over the vertex of the angle. Step 2: Rotate the protractor so the base line is exactly along one of the arms of the angle. 0° 0° 180° base line Step 3: Look at that arm of the angle and choose the scale that starts at 0°. Step 4: Use that scale to find the measurement. Drawing an angle angle mark angle mark 60° Step 1: Draw a line segment. Step 2: Place the protractor with the origin on one endpoint. This point will be the vertex of the angle. Step 3: Hold the protractor in place and mark a point at the angle measure you want. Step 4: Draw a line from the vertex through the angle mark. Drawing lines that intersect at an angle 45° P Step 1: Draw a line. Mark a point P on the line. 45° P Step 2: Draw an angle of the given measure using P as vertex. P Step 3: Extend the arms of your angle to form lines. 90° 30° 5 cm 90° 5 cm 5 cm Step 1: Sketch the Step 2: Use a ruler to draw triangle you want to draw. one side of the triangle. E-42 90° 30° 5 cm Step 3: Use a protractor to draw the angles at each end of this side. Extend the arms until they intersect. 90° 30° 5 cm Step 2: Erase any extra arm lengths. Blackline Master — Geometry — Teacher’s Guide for Workbook 7.1 COPYRIGHT © 2010 JUMP MATH: TO BE COPIED Drawing a triangle NAME DATE COPYRIGHT © 2010 JUMP MATH: TO BE COPIED Circles Blackline Master — Geometry — Teacher’s Guide for Workbook 7.1 E-51 NAME DATE Distance Between Parallel Lines InveStIgatIon Does the distance between parallel lines depend on where you measure it? a. Measure the line segments with endpoints on the two parallel lines with a ruler. Write the lengths of the line segments on the picture. B. Use a square corner to draw at least three perpendiculars from one parallel line to the other, as shown. Measure the distance between the two parallel lines along the perpendiculars. What do you notice? C. Explain why all the perpendiculars you drew in part B are parallel. e. Measure the line segments you drew between the two given parallel lines in part D. What do you notice? F. To measure the distance between two parallel lines, draw a line segment perpendicular to both lines and measure it. Does the distance between parallel lines depend on where you measure it? E-46 Blackline Master — Geometry — Teacher’s Guide for Workbook 7.1 COPYRIGHT © 2010 JUMP MATH: TO BE COPIED D. A parallelogram is a 4-sided polygon with opposite sides parallel. You can draw parallelograms by using anything with parallel sides, like a ruler. Place a ruler across both of the parallel lines and draw a line segment along each side of the ruler. Use this method to draw at least 3 parallelograms with different angles. NAME DATE Drawing Parallel Lines Drawing a line parallel to AB through point P Using a set square P A P P A B Step 1: Line up one of the short sides of the set square with AB. P A B A Step 2: Use the set square and a straight-edge to draw a perpendicular to AB. B B Step 3: Draw a line perpendicular to the new line that passes through P. Step 4: Erase the line you no longer need. Using a protractor P A P P B A B B A B Step 2: Line up the 90° line on the protractor with the line segment drawn in step 1, and the straight side of the protractor with point P. Draw a line parallel to AB. Erase the first perpendicular you drew. COPYRIGHT © 2010 JUMP MATH: TO BE COPIED Step 1: Line up the 90° line on the protractor with AB. Use the straight side of the protractor to draw a line segment perpendicular to AB. A P E-44 Blackline Master — Geometry — Teacher’s Guide for Workbook 7.1 NAME DATE Properties of Parallel Lines Investigation What happens if two lines meet a third line at the same angle, but it is not a right angle? A. Draw a pair of parallel lines and a third line intersecting both at an angle that is not a right angle. B. ∠ABD = ∠ACE = 70°. Draw a perpendicular to BD through point A. Extend it to meet CE. Is the line you drew perpendicular to CE? Check using a protractor. What can you say about the lines BD and CE? E D C B A D C. ∠ABD = ∠ACE = 70°. Are the lines BD and CE parallel? E B A C D. Draw a pair of lines that intersect at a 40° angle. Draw a third line that meets one of the lines at the same angle. Try to make the third line parallel to one of the lines you started with. Check by drawing a perpendicular. E. Compare the pattern between the equal angles ∠ABD and ∠ACE in parts B and D. Which one looks more like the angles marked in the letter C and which one is more like angles in the letter F? COPYRIGHT © 2010 JUMP MATH: TO BE COPIED Mathematicians have proved that if two lines meet with a third line at the same angles creating a pattern like in the letter F, the lines are parallel. When the lines meet at a right angle, you do not have to worry about the pattern of equal angles—they are all right angles. Blackline Master — Geometry — Teacher’s Guide for Workbook 7.1 C E B D A C E A B D E-45 NAME DATE Sum of the Angles in a Triangle Investigation What is the sum of the angles in a triangle? A. Circle the combinations of a 70° angle and another angle that will make a triangle. (Hint: Imagine the sides of the triangle extended—will they ever intersect?) 70° 80° 70° 90° 70° 100° 70° 110° 70° 120° Circle the combinations of a 50° angle and another angle that will make a triangle. 50° 100° 50° 110° 50° 120° 50° 130° 50° 140° Circle the combinations of a 90° angle and another angle that will make a triangle. 90° 70° 90° 80° 90° 90° 90° 100° 90° Make a prediction: To make a triangle, the total measures of any two angles must be less than °. 110° B. List the sum of the measures of the angles in each triangle. 90° ° + ° + ° = ° COPYRIGHT © 2010 JUMP MATH: TO BE COPIED 118° 55° 68° 42° 57° 20° 70° ° + ° + ° = ° 25° 20° 25° 130° ° + ° + ° = ° ° + ° + ° = ° What do you notice about the sums of the angles? Do you think this result will be true for all triangles? Make a conjecture: The sum of the three angles in any triangle will always be °. Blackline Master — Geometry — Teacher’s Guide for Workbook 7.1 E-47 NAME DATE C. Calculate the sum of the angles. 70° °+ 90° °+ 42° 57° 20° °= ° °+ 68° 55° °+ °= 25° 25° 118° ° 130° 20° °+ °+ °= ° °+ °+ °= ° What do you notice about the sums of the angles? D. Cut out a paper triangle and fold it as follows: C A B Step 1: Find the midpoints of the sides adjacent to the largest angle (measure or fold). Draw a line between the midpoints. A B C Step 2: Fold the triangle along the new line so that the top vertex meets the base of the triangle. You will get a trapezoid. A C B Step 3: Fold the other two vertices of the triangle so that they meet the top vertex. The three vertices folded together add up to a straight angle. What is the sum of the angles in a straight angle? ° e. Could you fold the vertices of any triangle along a line and get a straight angle as you did in part D? Do the results of the paper folding activity support your conjecture in part B? Explain. F. In fact, it has been mathematically proven that… the sum of the angles in a triangle is E-48 °. Blackline Master — Geometry — Teacher’s Guide for Workbook 7.1 COPYRIGHT © 2010 JUMP MATH: TO BE COPIED So ∠A + ∠B + ∠C = °