Geometry 8.2 Proving Triangle Similarity by AA 8.2 Warmup Find the value of x. 1. 2. 3. 4. November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA 8.2 Warmup Find the indicated measure in JKLM. 1. ML 2. MJ 3. JN 4. MK 5. m∠MJK 6. m∠LMJ 7. m∠MKL 8. m∠LJM November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

8.2 Proving Triangle Similarity by AA Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA 8.2 Essential Question What can you conclude about two triangles when you know that two pairs of corresponding angles are congruent? November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA Goals Identify similar triangles. Prove triangles similar. Solve problems using similar triangles. ~ November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA Consider these two triangles… November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA These two angles are congruent… November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA These two angles are congruent… November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA And these two angles are congruent… November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA Two triangles, three congruent angles. November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA These triangles are similar. November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA Do we really need all three angles? NO! November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA What must the last angle in each triangle be? ? ? 70 30 70 30 November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA The sum of the angles in each triangle must be 180. 80 80 70 30 70 30 November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA We only need to know two angles are congruent. The remaining angles must also be congruent. Third Angles Thm 70 30 70 30 November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

AA~ Postulate (Angle-Angle) If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA Example 1 B T A M N Is BAT ~ MAN? November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA Example 1 Solution M Is BAT ~ MAN? B T A A N November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA Example 1 Solution M Is BAT ~ MAN? A N B T A Yes. Why? AA~ Postulate November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA Example 1 Solution B T A M N BAT ~ MAN November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA Abbreviation Instead of writing AA Postulate or AA Similarity Postulate we will abbreviate this as AA~ November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA Your Turn 1 Are these triangles similar? If so, why? Then write a statement of similarity. F T A N M November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA Your Turn 1 Solution FAT  MAN Why? Vertical angles. F T A N M November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA Your Turn 1 Solution T  N Why? Alternate Interior Angles F T A N M November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA Your Turn 1 Solution Are the triangles similar? Yes. Why? AA~ (Notice, F  M, but we only need two congruent angles.) F T A N M November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA Your Turn 1 Solution Statement of Similarity: FAT ~MAN F T Be sure to match up corresponding vertices correctly! A N M November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA Example 2 a. Are these triangles similar? If so, write a similarity statement. Y Yes, they are similar by AA~. J 85 85 65 30 30 65 X Z K H XYZ ~ HJK November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA Example 2 b. List all the pairs of congruent angles and write the statement of proportionality. Y X  H Y  J Z  K J 85 85 65 30 30 65 X Z K H 𝑋𝑌 𝐻𝐽 = 𝑌𝑍 𝐽𝐾 = 𝑋𝑍 𝐻𝐾 November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA Your Turn 2 a. Are these triangles similar? If so, write a similarity statement. ∠𝑁≅∠𝑁 (reflexive) ∠𝑁𝐵𝐴≅∠𝐹 (rt. Angles congruent) N Yes, they are similar by AA~. NBA ~ NFL A B F L November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA Your Turn 2 b. List all the pairs of congruent angles and write the statement of proportionality. N N  N NBA  NFL NAB  NLF A B 𝑁𝐵 𝑁𝐹 = 𝑁𝐴 𝑁𝐿 = 𝐵𝐴 𝐹𝐿 F L November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

sides are proportional. If two triangles are similar, then corresponding sides are proportional. November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA Example 3 Show ∆𝐴𝐵𝐶~∆𝐴𝐷𝐸. Find AD. Since 𝐵𝐶 ∥ 𝐷𝐸 , ∠𝐴𝐵𝐶≅∠𝐴𝐷𝐸. ∠𝐴≅∠𝐴, by the reflexive property. So, ∆𝐴𝐵𝐶~∆𝐴𝐷𝐸 by AA~ D B A E 16 C 10 30 November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA Example 3 𝐴𝐵 𝐴𝐷 = 𝐴𝐶 𝐴𝐸 16 𝐴𝐷 = 20 30 20𝐴𝐷=480 𝐴𝐷=24 D B A E 16 C 10 30 ? 24 20 November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA Your Turn 3 Solve for x. 20 45 24 x November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA Example 4 The triangles are similar. Find the value of the variable. 24 x + 4 10 16 November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA Example 5 Jimmy, who is 36 inches tall, casts a shadow that is 48 inches long. A nearby windmill has a shadow that is 64 feet long. How tall is the windmill? November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA Example 5 Solution Jimmy, who is 36 inches tall, casts a shadow that is 48 inches long. A nearby windmill has a shadow that is 64 feet long. How tall is the windmill? First, get uniform units! November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA Example 5 Solution Jimmy, who is 3 feet tall, casts a shadow that is 4 feet long. A nearby windmill has a shadow that is 64 feet long. How tall is the windmill? x ft 64 ft Now label the drawing. 3 ft 4 ft November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA Example 5 Solution 3 64 x Set up the proportion. 4 November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA Example 5 Solution 3 64 x Solve it. 4 November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA Example 5 Solution 3 64 48 The windmill is 48 feet tall. Answer it. 4 November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.4 Similar Triangles Example 6 Find the height of the tree to the nearest foot. 4𝑓𝑡 10 𝑖𝑛 5.5 𝑓𝑡 = 𝑥 𝑓𝑡 125 𝑓𝑡 𝑥= 1318.1818 12 58 𝑖𝑛 66 𝑖𝑛 = 𝑥 𝑖𝑛 1500 𝑖𝑛 𝑥=109.8484 𝑓𝑡 66𝑥=87000 About 110 𝑓𝑡 𝑥=1318.1818 𝑖𝑛 What is the height to the nearest inch? 𝑥=.8484 𝑥12=10.18 𝑥=109 𝑓𝑡 10 𝑖𝑛 November 16, 2018 Geometry 8.4 Similar Triangles

Geometry 8.2 Proving Triangle Similarity by AA Your Turn 6 The Washington Monument is 169.3 m tall, and the base is 16.8 m on each side. For a class project, you are going to make a scale model that is going to be 2 m tall. What will the length of each side of the base be? November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA Your Turn 6 169.3 2 ? 16.8 November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA Summary Two triangles are similar if two angles of one triangle are congruent to two angles of the other triangle. (AA~ postulate) If triangles are similar, sides are proportional. Also, medians, altitudes, and other corresponding lengths are proportional. November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA

Geometry 8.2 Proving Triangle Similarity by AA Homework November 16, 2018 Geometry 8.2 Proving Triangle Similarity by AA