Two figures that have the exact same shape, but not necessarily the exact same size, are called similar

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1 Similar and Congruent Figures Lesson 2.3 Two figures that have the exact same shape, but not necessarily the exact same size, are called similar figures. The parts of similar figures that match are called corresponding parts. Look at the two triangles below. C A T D O 8 G Corresponding Angles C and D A and O T and G Corresponding Sides CT and DG TA and GO AC and OD Corresponding angles in similar figures are congruent or equal in measure. For example, in the triangles above, both C and D measure 35. You can show that two parts of a figure are congruent using the congruence symbol: C D. What do you notice about the corresponding sides CA and DO? AT and OG? CT and DG? They are not equal which means they are not congruent. However, they do all have something in common. Look at the ratio of each of these pairs of corresponding sides. CA DO = _ 3 6 = _ 1 2 AT OG = _ 8 = 1 _ 2 CT DG = 6 12 = _ 1 2 Each pair of corresponding sides can be reduced to the ratio of 1 : 2. The reduced ratio 1 : 2 is the scale factor for the triangles. This means the sides are proportional. The ratio 1 : 2 means triangle DOG is twice as big as triangle CAT. The two triangles above are similar because the corresponding angles are congruent and the corresponding sides are proportional. You can show that two figures are similar using the similar symbol: CAT ~ DOG. Similar Not Similar 8 Lesson 2.3 ~ Similar and Congruent Figures

2 EXPLORE! Step 1: Draw a right triangle with sides cm and 20 cm on a piece of graph paper. easure the third side and record its value to the nearest tenth of a centimeter. Label the triangle AT. Step 2: Draw a right triangle with sides 7.5 cm and cm. easure the third side to the nearest tenth of a centimeter. Record its value. Label the triangle SIP. similar triangles Step 3: Compare the ratios of the corresponding sides. Write each ratio without decimals in the fraction. cm A S 7.5 cm I 20 cm T cm P A SI = 7.5 = 0 75 = _ 2 1 AT IP = T SP = Step : Use a protractor to measure the angles in each triangle. Round your answer to the nearest degree. (m is read measure of angle.) m = m S = m A = m I = m T = m P = Step 5: Are AT and SIP similar? Explain your answer. If so, write the scale factor for AT to SIP. Step 6: Draw a third triangle similar to AT. Explain how you know your triangle is similar to AT. U Example 1 DOT is similar to SUN. Find the scale factor from DOT to SUN. O in in 6 in 6 in D 3 in T S N 5 in Solution Write a ratio of a side length in DOT to its corresponding side length in SUN. Simplify the ratio. The scale factor from DOT to SUN is 3_ 5 or 3 : = 3 _ 5 Lesson 2.3 ~ Similar and Congruent Figures 9

3 Example 2 Determine whether rectangle CARS is similar to rectangle BIKE. If so, find the scale factor. A 2 cm C cm R S I 3 cm B 6 cm K E Solution Determine if corresponding angles are congruent. C B A I R K S E Determine the ratio of corresponding heights. CA BI = _ 2 3 Determine the ratio of corresponding lengths. SC EB = _ 6 = _ 2 3 CARS ~ BIKE because the corresponding angles are congruent and the corresponding sides have equal ratios. The rectangles have a scale factor of 2 : 3. Example 3 Consider the two squares below. Are the two squares similar? Explain your reasoning. 5 m 5 m Solution Since every angle is 90, all corresponding angles are equal. Since every side is 5 m, the ratio of any pair of corresponding sides is _ 5 5 = _ 1 1. This means all corresponding sides have the same ratio. Yes, they are similar. The corresponding angles are congruent and the corresponding sides have equal ratios. They have a scale factor of 1 : 1. Although the two squares in Example 3 are similar, they have a more specific relationship. When two shapes are the exact same shape and the exact same size, they are congruent figures. Congruent Not Congruent 50 Lesson 2.3 ~ Similar and Congruent Figures

4 Example a. Is Rectangle A similar to Rectangle B? b. Is Rectangle A similar to Rectangle C? Rectangle A Rectangle B Rectangle C 3 cm 7 cm 12 cm 6 cm cm Solutions a. All corresponding angles are equal (90 ). 6 cm Look at the ratio of corresponding sides in Rectangle A and Rectangle B. The ratio of short side to short side is 3_. The ratio of long side to long side is 6_ 7. 3_ 6_ 7. The sides are not proportional. Rectangle A is NOT similar to Rectangle B. b. All corresponding angles are equal (90 ). Look at the ratio of corresponding sides in Rectangle A and Rectangle C. The ratio of short side to short side is 3_ 6 = 1_ 2. The ratio of long side to long side is 6 12 = 1_ 2. The ratios are equal. The sides are proportional. Rectangle A is similar to Rectangle C. They have a scale factor of 1 : 2. exercises Find the corresponding sides and corresponding angles to complete each statement T T A 60 0 N O 60 0 P 18 9 A R 0 O 12 0 P _ _ N corresponds to _ TA corresponds to _ AN corresponds to _ N AR corresponds to _ A corresponds to _ A TR corresponds to _ T A R Lesson 2.3 ~ Similar and Congruent Figures 51

5 3. C 8 A. F 2 I 1 1 P L S I P 8 CP corresponds to _ C CA _ corresponds to _ A A corresponds to _ P corresponds to _ P E 2 R Y 6 A G 6 PL corresponds to _ LA corresponds to _ AY corresponds to _ PY corresponds to _ N P L A Y 5. Sketch two similar shapes to the right triangle below, one larger and one smaller. Include angle and side lengths in your drawing. m 6. Use this rectangle: 1 in 3 in a. Sketch a rectangle that is congruent to the given rectangle. Label the measurements of the sides. b. Sketch a rectangle that is similar to the given rectangle. Label the measurements of the sides. c. Sketch a rectangle that is not similar or congruent to the given rectangle. Label the measurements of the sides. 7. Are all squares similar? Explain your reasoning. 8. Are all rectangles similar? Explain your reasoning. 9. Two similar shapes have a scale factor of 2 : 5. Lourdes says they are squares with sides of 2 inches and 5 inches. Paul says they are squares with sides of 8 centimeters and 20 centimeters. Explain how both students can be correct.. Sketch two rectangles that are similar but not congruent. Explain how you know your answer is correct. Determine the scale factor for each pair of similar figures. 5 5 m 5.7 m Lesson 2.3 ~ Similar and Congruent Figures 30

6 Determine the scale factor for each pair of similar figures _ 8 inch 2 1_ inches 1_ 6 foot 3 1_ 3 feet 19. Determine which pairs of triangles are similar. Explain why they are similar and write their scale factor. Triangle A Triangle B Triangle C Triangle D Triangle E Review Solve each proportion. 20. x_ 6 = = a 22. _ 6 = _ y 23. x_ 8 = _ 5 = 18 y 25. _ 5 = _ x 9 8 Find the perimeter and area of each figure. 26. in m m 6 in 8 m Lesson 2.3 ~ Similar and Congruent Figures 53

7 Tic-Tac-Toe ~ Cook i e s Chocolate Chip Cookies makes 8 dozen 1 1_ 2 cups sugar 1_ 2 cups flour 1 1_ 2 cups brown sugar 2 teaspoons baking soda 2 cups butter 1_ 2 teaspoon salt eggs cups chocolate chips 2 teaspoons vanilla To find the amount of each ingredient needed to make 6 dozen cookies, use the ratio 6_ 8. Write a proportion to find the amount of sugar needed. 6_ 8 = x 8x = 9 x = or 1 1_ 1 1_ 2 8 cup Use proportions to complete the chart for all nine ingredients. Show all work neatly on a separate piece of paper. Write all decimals as fractions or mixed numbers. Ingredient dozen 6 dozen dozen Sugar Brown sugar Butter Tic-Tac-Toe ~ P icture Size s ost picture packages give you the option of buying pictures in five different sizes: 3 in by 5 in in by 6 in 5 in by 7 in 8 in by in 11 in by 1 in 1. A customer picked up her pictures and noticed that the 3 in by 5 in photo cut off a portion of her son s head, but the 8 in by in photo did not. Explain how this happened using ideas about similar figures. 2. Are any of the picture sizes similar to one another? Explain why or why not using diagrams, proportions and the definition of similar figures. 3. Give two different-sized pictures that would be similar to a in by 6 in picture.. Find or create a photo or drawing of yourself in two of the sizes listed above. What differences do you notice? Explain the differences using ideas of similar figures. Include the photos or drawings in your work. 5 Lesson 2.3 ~ Similar and Congruent Figures