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NAME DATE COPYRIGHT © 2011 JUMP MATH: TO BE COPIED Protractors Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2 O-89 DATE NAME 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. base line 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° 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. O-90 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 8.2 COPYRIGHT © 2011 JUMP MATH: TO BE COPIED Drawing a triangle DATE NAME Drawing Perpendicular Lines and Bisectors Drawing a line segment perpendicular to AB through point P Using a set square P A P B A P B P A A B Here point P is on AB. B Here point P is outside AB. Using a protractor P A B P A P B P B A Here point P is on AB. A B Here point P is outside AB. Drawing the perpendicular bisector of line segment AB Using a set square or protractor M A A B A Step 1: Use a ruler to determine the midpoint of the line segment. Label it M. M B A B M B M Step 2: Use a set square or a protractor to draw a line perpendicular to AB that passes through M. The line you have drawn is the perpendicular bisector of AB. Using a compass and a straightedge COPYRIGHT © 2011 JUMP MATH: TO BE COPIED ℓ A Step 1: Put your compass point at one end of the line segment. Construct an arc as shown. B A B Step 2: With your compass at the same radius, construct a second arc centred at B as shown. Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2 A B Step 3: Construct a line ℓ through the intersection points of the two arcs. This is the perpendicular bisector of AB. O-91 DATE NAME Drawing Parallel Lines Drawing a line parallel to AB through point P Using a set square P A P A B B A Step 2: Use the set square and a straightedge to draw a perpendicular to AB. Step 1: Line up one of the short sides of the set square with AB. P P A B Step 3: Draw a line perpendicular to the new line that passes through P. B Step 4: Erase the line you no longer need. Using a protractor P A P P B A B 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 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. Using a compass and a straightedge ℓ ℓ m A B A Step 1: Mark any two points A, B on the line. Construct the perpendicular bisector of AB. O-92 D B Step 2: Mark any two points C, D on the line you drew. Construct the perpendicular bisector of CD. Label it m. Line m is parallel to AB. Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2 COPYRIGHT © 2011 JUMP MATH: TO BE COPIED C NAME DATE Distance Between Parallel Lines 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. 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. COPYRIGHT © 2011 JUMP MATH: TO BE COPIED 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 draw the perpendicular? Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2 O-93 DATE NAME Sum of the Angles in a Triangle (1) What is the sum of the angles in a triangle? INVESTIGATION 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° 50° 90° 70° 100° 70° 110° 70° 120° 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° Circle the combinations of a 50° angle and another angle that will make a triangle. 80° 70° 90° 80° 90° 90° 90° 100° ake a prediction: M To make a triangle, the total measures of any two angles must be less than 90° 110° °. B. List the sum of the measures of the angles in each triangle. 57° 70° 90° 69° ° + ° + ° = ° ° + ° + ° ° = ° 24° 41° 116° ° + 23° 24° ° + ° = ° ° + 132° ° + 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 °. O-94 ° = Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2 COPYRIGHT © 2011 JUMP MATH: TO BE COPIED 54° 20° DATE NAME Sum of the Angles in a Triangle (2) C. Calculate the sum of the angles. 70° 90° 20° 57° ° + ° + ° = ° 41° 54° ° + ° + ° = ° 116° 24° 24° 23° 69° 132° ° + ° + ° = ° ° + ° + ° = ° What do you notice about the sums of the angles? D. Cut out a paper triangle and fold it as follows: C A Step 1: Find the midpoints of the sides adjacent to the largest angle (measure or fold). Draw a line between the midpoints. COPYRIGHT © 2011 JUMP MATH: TO BE COPIED B 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. The three vertices folded together add up to a straight angle. What is the sum of the angles in a straight angle? So ∠A + ∠B + ∠C = A C B Step 3: Fold the other two vertices of the triangle so that they meet the top vertex. ° ° E.Could you fold the vertices of any triangle as you did in part D and get a straight angle? Do the results of the paper folding 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 °. Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2 O-95 NAME DATE Quadrilaterals (1) 2 3 4 5 6 COPYRIGHT © 2011 JUMP MATH: TO BE COPIED 1 O-96 Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2 NAME DATE Quadrilaterals (2) 8 9 10 11 12 COPYRIGHT © 2011 JUMP MATH: TO BE COPIED 7 Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2 O-97 DATE NAME Sorting Quadrilaterals Properties of sides Shapes with the property Properties of angles No equal sides No equal angles 1 pair of equal sides 1 pair of equal angles 2 pairs of equal sides 2 pairs of equal angles Equal sides are adjacent Equal angles are adjacent Equal sides are opposite Equal angles are opposite 4 equal sides 4 equal angles 1 pair of parallel sides No pairs of angles add to 180° 2 pairs of parallel sides 2 pairs of angles add to 180° Perpendicular diagonals 4 pairs of angles add to 180° Equal diagonals 6 pairs of angles add to 180° Properties of sides Shapes with the property Properties of angles Equal diagonals One of the diagonals is also an angle bisector Exactly one diagonal bisects the other Both diagonals are angle bisectors Both diagonals bisect each other Shapes with the property COPYRIGHT © 2011 JUMP MATH: TO BE COPIED Perpendicular diagonals Shapes with the property O-98 Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2 NAME DATE Triangles Set 1 Set 2 COPYRIGHT © 2011 JUMP MATH: TO BE COPIED Set 3 Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2 O-99 DATE NAME Triangles for Sorting B A E D C I G H F L J M O-100 N O Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2 COPYRIGHT © 2011 JUMP MATH: TO BE COPIED K DATE NAME Two Pentagons O N A Q R M B K L P S T a) How many right angles does each pentagon have? b)Cut the pentagons out. Fold the pentagons so that you can see that the remaining angles are all equal. c) Does each side on pentagon A have a side of the same length on pentagon B? d)Match each side from pentagon A with a side of the same length from pentagon B. Do not use the same side twice. KL = LM = MN = NO = OK = COPYRIGHT © 2011 JUMP MATH: TO BE COPIED e) If you place one pentagon on top of the other, do they match? f) Are they of the same shape? g) Can we say that these pentagons have the same sides and angles? h) Are the pentagons congruent? i) Explain why these pentagons have the same sides and angles but aren’t congruent. Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2 O-101 DATE NAME Investigating Congruence INVESTIGATION Do you have to check that all 3 pairs of sides and all 3 pairs of angles are equal to determine if two triangles are congruent? 3 sides Conjecture: If two triangles have 3 pairs of equal sides, then the triangles are congruent. Test the conjecture: Take 3 straws of different lengths. Make a triangle with them and trace it. See if you can make a different triangle with the same three straws. Try with a different set of three straws. Are your triangles congruent? Do you think this result will be true for all triangles? 2 sides and the angle between Conjecture: If 2 sides and the angle between them in one triangle are equal to 2 sides and the angle between them in another triangle, the triangles are congruent. Test the conjecture: Draw 3 triangles, each with one side 3 cm long, one side 5 cm long, and a 45° angle between these sides. Are the triangles congruent? Do you think this result will be true for all triangles? Pick three measurements: a = cm, b = cm, C = ° Draw three triangles, each with one side a cm long, one side b cm long, and a C° angle between these sides. Are your triangles congruent? 2 angles and the side between them Test this conjecture: Draw 3 triangles, each with a 60° angle, a 45° angle, and a side 5 cm long between these angles. Are the triangles congruent? Do you think this result will be true for all triangles? Pick three measurements: a = cm, B = °, C = ° Draw three triangles, each with a B° angle, a C° angle, and a side a cm long between these angles. Are your triangles congruent? O-102 Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2 COPYRIGHT © 2011 JUMP MATH: TO BE COPIED Conjecture: If 2 angles and the side between them in one triangle are equal to 2 angles and the side between them in another triangle, the triangles are congruent. DATE NAME Congruence Rules on Geometer’s Sketchpad (1) 1.Using the polygon tool, construct a triangle, ABC. Measure the sides and the angles of your triangle. 2. a) Construct a point D. Using a command for constructing circles and the length of AB as the radius, construct a circle with centre D. Construct line segment DE = AB. Hide the circle. b) Using a command for constructing circles and the length of BC as the radius, construct a circle with centre E. c) Using a command for constructing circles and the length of AC as the radius, construct a circle with centre D. d) Construct a point that is on both circles. Label it F. Use the polygon tool to construct triangle DEF. Hide the circles. 3. a) Which sides are equal in ∆ABC and ∆DEF? = , = , = b) Measure the angles of ∆DEF. What can you say about ∆ABC and ∆DEF? 4. Try to move the vertices of ∆DEF around. a) How does your triangle change? Which transformations can you make: rotations, reflections, translations? b) Can you move ∆DEF onto ∆ABC to check whether they are congruent? Do you need to reflect the triangle to do that? COPYRIGHT © 2011 JUMP MATH: TO BE COPIED 5. Move ∆DEF away from ∆ABC. Try to move the vertices of ∆ABC around. a) Are the changes you can make in this triangle different from the changes you could make to ∆DEF? Why? b) What happens to ∆DEF when you modify ∆ABC? What can you say about the triangles ∆ABC and ∆DEF? 6. Are three side lengths enough to determine a unique triangle? Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2 O-103 DATE NAME Congruence Rules on Geometer’s Sketchpad (2) 1.Using the polygon tool, construct a triangle, ABC. Measure the sides and the angles of your triangle. 2. a) Construct a point D. Using a command for constructing circles and the length of AB as the radius, construct a circle with centre D. Construct line segment DE = AB. Hide the circle. b) Using a command for constructing circles and the length of BC as the radius, construct a circle with centre E. c) Select E as a centre of rotation. Use a command in the Transformation menu to mark ∠ABC as the angle of rotation. Rotate point D around E by the angle chosen. Construct a ray from E through the image of D. d) Construct a point that is on the ray and the circle. Label it F. Use the polygon tool to construct triangle DEF. Hide the circle and the ray EF. 3. a) Which sides and angles are equal in ∆ABC and ∆DEF? = , = , = b) Measure the rest of the sides and angles of ∆DEF. What can you say about ∆ABC and ∆DEF? 4. Try to move the vertices of ∆DEF around. a) How does your triangle change? Which transformations can you make: rotations, reflections, translations? b) Can you move ∆DEF onto ∆ABC to check whether they are congruent? Do you need to reflect the triangle to do that? a) Are the changes you can make in this triangle different from the changes you could make to ∆DEF? Why? b) What happens to ∆DEF when you modify ∆ABC? What can you say about the triangles ∆ABC and ∆DEF? 6. Are two sides and the angle between them enough to determine a unique triangle? O-104 Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2 COPYRIGHT © 2011 JUMP MATH: TO BE COPIED 5. Move ∆DEF away from ∆ABC. Try to move the vertices of ∆ABC around. DATE NAME Congruence Rules on Geometer’s Sketchpad (3) 1. Using the polygon tool, construct a triangle, ABC. Measure the sides and the angles of your triangle. 2. a) Construct a point D. Using a command for constructing circles and the length of AB as the radius, construct a circle with centre D. Construct line segment DE = AB. Hide the circle. b) Select E as a centre of rotation. Use a command in the Transformation menu to mark ∠ABC as the angle of rotation. Rotate point D around E by the angle chosen. Construct a ray from E through the image of D. c) Select D as a centre of rotation. Mark ∠BAC as the angle of rotation. Rotate point E around D by the angle chosen. Construct a ray from D through the image of E. d) Construct a point F that is the intersection point of both rays. Use the polygon tool to construct triangle DEF. Hide the rays. 3. a) Which sides and angles are equal in ∆ABC and ∆DEF? = , = , = b) Measure the rest of the sides and angles of ∆DEF. What can you say about ∆ABC and ∆DEF? 4. Try to move the vertices of ∆DEF around. a) How does your triangle change? Which transformations can you make: rotations, reflections, translations? b) Can you move ∆DEF onto ∆ABC to check whether they are congruent? Do you need to reflect the triangle to do that? COPYRIGHT © 2011 JUMP MATH: TO BE COPIED 5. Move ∆DEF away from ∆ABC. Try to move the vertices of ∆ABC around. a) Are the changes you can make in this triangle different from the changes you could make to ∆DEF? Why? b) What happens to ∆DEF when you modify ∆ABC? What can you say about the triangles ∆ABC and ∆DEF? 6. Are two angles and a side between them enough to determine a unique triangle? Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2 O-105 DATE NAME Proving the Pythagorean Theorem (1) REMINDER Area of triangle = base × height ÷ 2 1. PQRS is a rectangle. a) Find the areas of the shapes: i) PQRS R 3 cm Q ii) ∆PRS T 5 cm iii) ∆PTS P b) Describe how the three areas in a) are related. 7 cm S 2. CDEA, AFGB, and CBHJ are squares. a) The base of the two shapes is given. Find the heights of the shapes. i) ∆CBD ii) ∆CBG CDEA base: DC base: DC height: AC height: AC AFGB base: BG base: BG height: height: E D F A A G iii) ∆CAJ B iv) ∆HBA LKJC base: JC base: JC height: height: BHKL base: BH base: BH height: height: A C J A K C B L H J L K B H b) Write the area of each triangle in terms of another shape. 1 1 i) Area of ∆CBD = of area of CDEA ii) Area of ∆CBG = of area of 2 2 O-106 iii) Area of ∆CAJ = 1 of area of 2 iv) Area of ∆HBA = 1 of area of 2 Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2 COPYRIGHT © 2011 JUMP MATH: TO BE COPIED C B C DATE NAME Proving the Pythagorean Theorem (2) 3.Fill in the blanks to finish proving the Pythagorean Theorem the same way Euclid did it 2300 years ago. In a triangle ABC with ∠A = 90°, AB2 + AC2 = BC2. a) The picture at right shows the Pythagorean Theorem. What special quadrilateral is each of these quadrilaterals? CDEA square AFGB CBHJ LKJC BHKL b) Area of ∆CBD = of area of square CDEA, because e) ∆CBD ≅ ∆CAJ by congruence rule, because DC = , CB = ∠DCB = 90° + ∠ = ∠ J K H E g)Area of ∆CBG = of area of square , because they both have base and height . h)Area of ∆HBA = of area of rectangle , because COPYRIGHT © 2011 JUMP MATH: TO BE COPIED B L f) Area of CDEA = area of LKJC because D F A they both have base and height . i)Mark the equal line segments in the picture. j) ∆CBG ≅ by congruence rule, because BG = , ∠CBG = 90° + ∠ = ∠ G C they both have base CJ and height . d) Mark the equal line segments in the picture. F A they both have base and height . D c)Area of ∆CAJ = of area of rectangle , because E G C L B CB = J H K k) Area of AFGB = area of BHKL because j) AB2 + AC2 = BC2, because Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2 O-107 NAME DATE COPYRIGHT © 2011 JUMP MATH: TO BE COPIED Circles O-108 Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2 DATE NAME Angle Properties Supplementary angles are angles that add to 180°. 1 2 A 60° B 3 120° 4 Corresponding angles make a pattern similar to the letter F. l l 1 l 2 m n 5 m m 3 n 6 l m 4 n n 8 7 Corresponding Angle Theorem: When lines are parallel, corresponding angles are equal. The reverse statement is true as well: When corresponding angles are equal, the lines are parallel. Alternate angles make a pattern similar to the letter Z. l l 3 m 6 4 p 5 n r Alternate Angle Theorem: When lines are parallel, alternate angles are equal. The reverse statement is true as well: When alternate angles are equal, the lines are parallel. Opposite angles are created when two lines intersect. 1 COPYRIGHT © 2011 JUMP MATH: TO BE COPIED Opposite Angle Theorem: Opposite angles are equal: ∠1 = ∠2 and ∠3 = ∠4 4 3 2 Complementary angles are angles whose sum is 90°. B A x 60° 30° y 1 2 Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2 O-109 DATE NAME Properties of Angles in a Triangle Sum of Angles in a Triangle Theorem The sum of the angles in a triangle is 180°. ∠A + ∠B + ∠C = 180° B A C Exterior Angle Theorem The measure of an exterior angle in a triangle is equal to the sum of the opposite angles in the triangle. ∠x = ∠a + ∠b b a c x Isosceles Triangle Theorem The base angles in an isosceles triangle are equal. The reverse statement is true as well: COPYRIGHT © 2011 JUMP MATH: TO BE COPIED If two angles in a triangle are equal, then the triangle is isosceles, with the equal sides adjacent to the third angle in the triangle. O-110 Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2 NAME DATE COPYRIGHT © 2011 JUMP MATH: TO BE COPIED Rectangles Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2 O-111 DATE NAME Similarity Rules Using Geometer’s Sketchpad (1) INVESTIGATION are similar? Which information about a pair of triangles will ensure they A. SSS (side-side-side) a) Draw a triangle, ABC. Measure the sides. b) Create a new parameter, s, to be the scale factor. Set s = 2. c) Construct a triangle A′B′C′ with sides A′B′ = sAB, A′C′ = sAC, B′C′ = sBC. d) Measure the angles of both triangles. What do you notice? Are the triangles similar? e) Modify ∆ABC by moving the vertices. Does ∆A′B′C′ stay similar to ∆ABC? f) Conjecture: SSS is a similarity rule: A ' B ' A 'C ' B 'C ' If = = , then ∆ABC is similar to ∆A′B′C′. AB AC BC Change the scale factor (s) to three different values, including two decimals, one of them more than 1 and the other less than 1. Does ∆A′B′C′ stay similar to ∆ABC? Is the conjecture true? B. SAS (side-angle-side ) a) Draw a triangle, ABC. Measure the sides and the angles. b) Create a new parameter, s, to be the scale factor. Set s = 3. c) Construct a triangle A′B′C′ with sides A′B′ = sAB, A′C′ = sAC and ∠A′ = ∠A. d)Measure the side B′C′ and the angles ∠B′, ∠C′. Find the ratio B′C′ : BC. What do you notice? Are the triangles similar? e)Modify ∆ABC by moving the vertices. Does the triangle ∆A′B′C′ stay similar to ∆ABC? f) Conjecture: SAS is a similarity rule: A ' B ' A 'C ' If ∠A = ∠A′ and = , then ∆ABC is similar to ∆A′B′C′. AB AC Change the scale factor (s) to three different values, including two decimals, one of them more than 1 and the other less than 1. Does ∆A′B′C′ stay similar to ∆ABC? Is the conjecture true? O-112 Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2 COPYRIGHT © 2011 JUMP MATH: TO BE COPIED DATE NAME Similarity Rules Using Geometer’s Sketchpad (2) C. ASA (angle-side-angle) a) Draw a triangle, ABC. Measure the sides and the angles. b) Create a new parameter, s, to be the scale factor. Set s = 4. c) Construct a triangle A′B′C′ with sides A′B′ = sAB, ∠A′ = ∠A and ∠B′ = ∠B. d)Measure the sides B′C′, A′C′ and the angle ∠C′. Find the ratios B′C′ : BC and A′C′ : AC. What do you notice? Are the triangles similar? e) Modify ∆ABC by moving the vertices. Does the ∆A′B′C′ stay similar to ∆ABC? D. The ratios between corresponding sides of similar triangles are the same. In which parts—A, B or C—did you construct the triangles so that at least two pairs of corresponding sides had the same ratios? How was the construction in the other part different? a) Draw a triangle, ABC. Measure the sides and the angles. b)Construct a triangle A′B′C′ with ∠A′ = ∠A and ∠B′ = ∠B. What do you know about the angles C and C′? Explain. A ' B ' C ' B ' A 'C ' c)Measure the sides of ∆A′B′C′ . Find the ratios , , . What do AB CB AC you notice? Are the triangles similar? d) Modify ∆ABC by moving the vertices. COPYRIGHT © 2011 JUMP MATH: TO BE COPIED Does ∆A′B′C′ stay similar to ∆ABC? e) Conjecture: AA (angle-angle) is a similarity rule: If ∠A = ∠A′ and ∠B = ∠AB′, then ∆ABC is similar to ∆A′B′C′. Modify ∆A′B′C′ by moving the vertices. What do you notice about the ratios between the corresponding sides? Does ∆A′B′C′ stay similar to ∆ABC? Is the conjecture true? Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2 O-113 DATE NAME Area of Parallelogram and Triangle Parallelogram height bases A parallelogram is a quadrilateral with two pairs of parallel sides. Any pair of parallel sides can be chosen to be the bases. The distance between these two parallel sides is the height. The height is measured along a line perpendicular to the bases. This line can be drawn anywhere. In these pictures, the thick black line is one of the bases and the dashed line is the height. Area of a parallelogram = base × height Triangle height base A triangle is a polygon with three sides. Any side of a triangle can be the base. Draw a perpendicular from the vertex opposite the base to the base. The distance from the vertex to the base along that perpendicular is the height. Any triangle is half of a parallelogram with the same base and height. Area of a triangle = base × height ÷ 2 O-114 Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2 COPYRIGHT © 2011 JUMP MATH: TO BE COPIED Sometimes the height is outside the triangle. In these pictures, the thick black line is the base and the dashed line is the height.