Showing Triangles are Similar: AA, SSS and SAS Sections 7.3 &7.4

Objectives Identify similar triangles using the AA Similarity Postulate and the SSS and SAS Similarity Theorems (similarity shortcuts). Use similar triangles to solve problems.

Key Vocabulary None

Postulates 15 Angle – Angle (AA) Similarity Postulate

Theorems 7.2 Side – Side – Side (SSS) Similarity Theorem 7.3 Side – Angle – Side (SAS) Similarity Theorem

Do you really have to check all the sides and angles? Similar Triangles Are the triangles below similar? Do you really have to check all the sides and angles? NO, there are Shortcuts for determining Similarity.

Triangle Similarity Shortcuts

Postulate 15 Angle-Angle (AA) Similarity Angle – Angle (AA) Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. Example: If ∠K≅∠Y and ∠J≅∠X, then ∆JKL∼∆XYZ.

Example 1a A. Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.

Example 1a Since mB = mD, B ≅ D By the Triangle Sum Theorem, 42 + 58 + mA = 180, so mA = 80. Since mE = 80, A ≅ E. Answer: So, ΔABC ~ ΔEDF by the AA Similarity.

Example 1b B. Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.

Example 1b QXP ≅ NXM by the Vertical Angles Theorem. Since QP || MN, Q ≅ N. Answer: So, ΔQXP ~ ΔNXM by the AA Similarity.

Your Turn: A. Determine whether the triangles are similar. If so, write a similarity statement. A. Yes; ΔABC ~ ΔFGH B. Yes; ΔABC ~ ΔGFH C. Yes; ΔABC ~ ΔHFG D. No; the triangles are not similar.

Your Turn: B. Determine whether the triangles are similar. If so, write a similarity statement. A. Yes; ΔWVZ ~ ΔYVX B. Yes; ΔWVZ ~ ΔXVY C. Yes; ΔWVZ ~ ΔXYV D. No; the triangles are not similar.

Thales The Greek mathematician Thales was the first to measure the height of a pyramid by using geometry. He showed that the ratio of a pyramid to a staff was equal to the ratio of one shadow to another.

Example 2 If the shadow of the pyramid is 576 feet, the shadow of the staff is 6 feet, and the height of the staff is 5 feet. Explain why Thales’ method worked to find the height of the pyramid? Find the height of the pyramid?

Example 2 Explain why Thales’ method worked to find the height of the pyramid? The Triangles formed by the pyramid and its shadow, and the staff and its shadow are similar triangles by the AA Postulate. Find the height of the pyramid? Solution: The pyramid is 480 ft tall. Similarity proportion Substitution Cross product Simplify Divide by 6

If two pairs of angles are congruent, then the triangles are similar. Example 1 Use the AA Similarity Postulate Determine whether the triangles are similar. If they are similar, write a similarity statement. Explain your reasoning. SOLUTION If two pairs of angles are congruent, then the triangles are similar. 1. G  L because they are both marked as right angles.

Find mF to determine whether F is congruent to J. Example 1 Use the AA Similarity Postulate 2. Find mF to determine whether F is congruent to J. mF + 90° + 61° = 180° Triangle Sum Theorem mF + 151° = 180° Add. mF = 29° Subtract 151° from each side. Both F and J measure 29°, so F  J. ANSWER By the AA Similarity Postulate, FGH ~ JLK

Are you given enough information to show that RST is similar to Example 2 Use the AA Similarity Postulate Are you given enough information to show that RST is similar to RUV? Explain your reasoning. SOLUTION Redraw the diagram as two triangles: RUV and RST. From the diagram, you know that both RST and RUV measure 48°, so RST  RUV. Also, R  R by the Reflexive Property of Congruence. By the AA Similarity Postulate, RST ~ RUV.

Your Turn Use the AA Similarity Postulate Determine whether the triangles are similar. If they are similar, write a similarity statement. 1. ANSWER yes; RST ~ MNL 2. ANSWER yes; GLH ~ GKJ

Use Similar Triangles Your Turn Write a similarity statement for the triangles. Then find the value of the variable. 3. ANSWER ABC ~ DEF; 9 4. ABD ~ EBC; 3 ANSWER

Assignment 7.3 Pg 375 – 378: #1 – 45 odd

Theorem 7.2 Side-Side-Side (SSS) Similarity Side – Side – Side (SSS) Similarity Theorem: If the corresponding side lengths of two triangles are proportional, then the two triangles are similar. Example:

Theorem 7.3 Side-Angle-Side (SAS) Similarity Side – Angle Side (SAS) Similarity Theorem: If two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, then the two triangles are similar. Example:

Review - Similar Triangles Previously, we learned how to determine if two triangles were congruent (SSS, SAS, ASA, AAS). There are also several tests to prove triangles are similar. Postulate 15 – AA Similarity 2 s of a Δ are  to 2 s of another Δ Theorem 7.2 – SSS Similarity 3 corresponding sides of 2 Δs are proportional Theorem 7.3 – SAS Similarity 2 corresponding sides of 2 Δs are proportional and the included s are 

Example 3a A. Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning. Answer: So, ΔABC ~ ΔDEC by the SSS Similarity Theorem.

Example 3b B. Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning. By the Reflexive Property, M  M. Answer: Since the lengths of the sides that include M are proportional, ΔMNP ~ ΔMRS by the SAS Similarity Theorem.

Your Turn: A. Determine whether the triangles are similar. If so, choose the correct similarity statement to match the given data. A. ΔPQR ~ ΔSTR by SSS Similarity Theorem B. ΔPQR ~ ΔSTR by SAS Similarity Theorem C. ΔPQR ~ ΔSTR by AAA Similarity Theorem D. The triangles are not similar.

Example 3b B. Determine whether the triangles are similar. If so, choose the correct similarity statement to match the given data. A. The triangles are not similar. B. ΔAFE ~ ΔACB by SSS Similarity Theorem C. ΔAFE ~ ΔAFC by SSS Similarity Theorem D. ΔAFE ~ ΔBCA by SSS Similarity Theorem

Example 4: In the figure, and Determine which triangles in the figure are similar.

Example 4: by the Alternate Interior Angles Theorem. Vertical angles are congruent, Answer: Therefore, by the AA Similarity Theorem,

Your Turn: In the figure, OW = 7, BW = 9, WT = 17.5, and WI = 22.5. Determine which triangles in the figure are similar. I Answer:

Example 5: ALGEBRA Given QT 2x 10, UT 10, find RQ and QT.

Example 5: Since because they are alternate interior angles. By AA Similarity, Using the definition of similar polygons, Substitution Cross products

Example 5: Distributive Property Subtract 8x and 30 from each side. Divide each side by 2. Now find RQ and QT. Answer:

Your Turn: ALGEBRA Given and CE x + 2, find AC and CE. Answer:

Example 6 If ΔRST and ΔXYZ are two triangles such that = which of the following would be sufficient to prove that the triangles are similar? A B C R  S D __ 2 3 ___ RS XY

Example 6 __ 2 3 If = = , then you know that all the sides are proportional by the same scale factor, . This is sufficient information by the SSS Similarity Theorem to determine that the triangles are similar. ___ RS XY RT XZ Answer: B

Your Turn: A. = B. mA = 2mD C. = D. = AC DC 4 3 BC 5 EC Given ΔABC and ΔDEC, which of the following would be sufficient information to prove the triangles are similar? A. = B. mA = 2mD C. = D. = ___ AC DC __ 4 3 BC 5 EC

Example 7: INDIRECT MEASUREMENT: Josh wanted to measure the height of the Sears Tower in Chicago. He used a 12-foot light pole and measured its shadow at 1 P.M. The length of the shadow was 2 feet. Then he measured the length of the Sears Tower’s shadow and it was 242 feet at that time. What is the height of the Sears Tower?

Example 7: Assuming that the sun’s rays form similar triangles, the following proportion can be written. Now substitute the known values and let x be the height of the Sears Tower. Substitution Cross products

Example 7: Simplify. Divide each side by 2. Answer: The Sears Tower is 1452 feet tall.

Your Turn: On her trip along the East coast, Jennie stops to look at the tallest lighthouse in the U.S. located at Cape Hatteras, North Carolina. At that particular time of day, Jennie measures her shadow to be 1 feet 6 inches in length and the length of the shadow of the lighthouse to be 53 feet 6 inches. Jennie knows that her height is 5 feet 6 inches. What is the height of the Cape Hatteras lighthouse to the nearest foot? Answer: 196 ft

Find the ratios of the corresponding sides. SOLUTION Example 1 Use the SSS Similarity Theorem Determine whether the triangles are similar. If they are similar, write a similarity statement and find the scale factor of Triangle B to Triangle A. Find the ratios of the corresponding sides. SOLUTION All three ratios are equal. So, the corresponding sides of the triangles are proportional. PR SU 12 6 12 ÷ 6 6 ÷ 6 = 2 1 RQ UT 10 5 10 ÷ 5 5 ÷ 5 QP TS 8 4 8 ÷ 4 4 ÷ 4 45

The scale factor of Triangle B to Triangle A is . 2 1 Example 1 Use the SSS Similarity Theorem ANSWER The scale factor of Triangle B to Triangle A is . 2 1 By the SSS Similarity Theorem, PQR ~ STU. 46

Is either DEF or GHJ similar to ABC? Example 2 Use the SSS Similarity Theorem Is either DEF or GHJ similar to ABC? SOLUTION Look at the ratios of corresponding sides in ABC and DEF. 1. Shortest sides = 6 4 AB DE 3 2 Longest sides 12 8 CA FD Remaining sides 9 BC EF ANSWER Because all of the ratios are equal, ABC ~ DEF. 47

Look at the ratios of corresponding sides in ABC and GHJ. 2. Example 2 Use the SSS Similarity Theorem Look at the ratios of corresponding sides in ABC and GHJ. 2. Shortest sides = 6 AB GH 1 Longest sides 12 14 CA JG 7 Remaining sides BC HJ 9 10 ANSWER Because the ratios are not equal, ABC and GHJ are not similar. 48

Your Turn Use the SSS Similarity Theorem Determine whether the triangles are similar. If they are similar, write a similarity statement. 1. ANSWER yes; ABC ~ DFE 2. ANSWER no

C and F both measure 61°, so C  F. Example 3 Use the SAS Similarity Theorem Determine whether the triangles are similar. If they are similar, write a similarity statement. SOLUTION C and F both measure 61°, so C  F. Compare the ratios of the side lengths that include C and F. = AC DF 3 5 Shorter sides 6 10 CB FE Longer sides The lengths of the sides that include C and F are proportional. ANSWER By the SAS Similarity Theorem, ABC ~ DEF. 50

Separate the triangles, VYZ and VWX, and label the side lengths. Example 4 Similarity in Overlapping Triangles Show that VYZ ~ VWX. Separate the triangles, VYZ and VWX, and label the side lengths. SOLUTION V  V by the Reflexive Property of Congruence. Shorter sides 4 + 8 4 = VY VW 12 3 1 Longer sides 5 + 10 5 ZV XV 15 51

The lengths of the sides that include V are proportional. Example 4 Similarity in Overlapping Triangles The lengths of the sides that include V are proportional. ANSWER By the SAS Similarity Theorem, VYZ ~ VWX. 52

Your Turn Use the SAS Similarity Theorem Determine whether the triangles are similar. If they are similar, write a similarity statement. Explain your reasoning. 3. ANSWER No; H  M but 6 8 12 ≠ .

lengths of the sides that include P are proportional, Your Turn Use the SAS Similarity Theorem Determine whether the triangles are similar. If they are similar, write a similarity statement. Explain your reasoning. 4. ANSWER Yes; P  P, and the lengths of the sides that include P are proportional, so PQR ~ PST by the SAS Similarity Theorem. , 6 3 PS PQ = 2 1 10 5 PT PR ;

Summary of Similarity Shortcuts

Assignment 7.4 Pg. 382 – 385: # 1 – 29 odd, 33 – 37 odd

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