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6.6. Theorems Involving Similarity www.ck12.org 6.6 Theorems Involving Similarity Here you will use similar triangles to prove new theorems about triangles. Can you find any similar triangles in the picture below? Watch This MEDIA Click image to the left for more content. http://www.youtube.com/watch?v=YLtQmuTY5rI James Sousa: Triangle Proportionality Theorem Guidance If two triangles are similar, then their corresponding angles are congruent and their corresponding sides are proportional. There are many theorems about triangles that you can prove using similar triangles. 1. Triangle Proportionality Theorem: A line parallel to one side of a triangle divides the other two sides of the triangle proportionally. This theorem and its converse will be explored and proved in Example A, Example B, and the practice exercises. 2. Triangle Angle Bisector Theorem: The angle bisector of one angle of a triangle divides the opposite side of the triangle into segments proportional to the lengths of the other two sides of the triangle. This theorem will be explored and proved in Example C. 3. Pythagorean Theorem: For a right triangle with legs a and b and hypotenuse c, a2 + b2 = c2 . This theorem will be explored and proved in the Guided Practice problems. Example A Prove that ∆ADE ∼ ∆ABC. 398 www.ck12.org Chapter 6. Similarity Solution: The two triangles share 6 A. Because DEkBC, corresponding angles are congruent. Therefore, 6 ADE ∼ = 6 ABC. The two triangles have two pairs of congruent angles. Therefore, ∆ADE ∼ ∆ABC by AA ∼. Example B Use your result from Example A to prove that AB AD = AC AE . EC AE . AB Therefore, AD Then, use algebra to show that DB AD = AC Solution: ∆ADE ∼ ∆ABC which means that corresponding sides are proportional. = AE . Now, you can use algebra to show that the second proportion must be true. Remember that AB = AD + DB and AC = AE + EC. AB AD AD + DB → AD DB → 1+ AD DB → AD AC AE AE + EC = AE EC = 1+ AE EC = AE = You have now proved the triangle proportionality theorem: a line parallel to one side of a triangle divides the other two sides of the triangle proportionally. Example C Consider ∆ABC with AE the angle bisector of 6 BAC and point D constructed so that DCkAE. Prove that EB BA = EC CA . 399 6.6. Theorems Involving Similarity www.ck12.org Solution: By the triangle proportionality theorem, EC BA EB EC = BA AD . Multiply both sides of this proportion by EC BA . EB BA EC · = · EC AD BA EB EC → = BA AD Now all you need to show is that AD = CA in order to prove the desired result. • • • • Because AE is the angle bisector of 6 BAC, 6 BAE ∼ = 6 EAC. ∼ Because DCkAE, 6 BAE = 6 BDC (corresponding angles). Because DCkAE, 6 EAC ∼ = 6 DCA (alternate interior angles). ∼ Thus, 6 BDC = 6 DCA by the transitive property. Therefore, ∆ADC is isosceles because its base angles are congruent and it must be true that AD ∼ = CA. This means that AD = CA. Therefore: EB BA = EC CA This proves the triangle angle bisector theorem: the angle bisector of one angle of a triangle divides the opposite side of the triangle into segments proportional to the lengths of the other two sides of the triangle. Concept Problem Revisited 400 www.ck12.org Chapter 6. Similarity There are three triangles in this picture: ∆BAC, ∆BCD, ∆CAD. All three triangles are right triangles so they have one set of congruent angles (the right angle). ∆BAC and ∆BCD share 6 B, so ∆BAC ∼ ∆BCD by AA ∼. Similarly, ∆BAC and ∆CAD share 6 C, so ∆BAC ∼ ∆CAD by AA ∼. By the transitive property, all three triangles must be similar to one another. Vocabulary A similarity transformation is one or more rigid transformations followed by a dilation. Two figures are similar if a similarity transformation will carry one figure to the other. Similar figures will always have corresponding angles congruent and corresponding sides proportional. The Pythagorean Theorem states that for a right triangle with legs a and b and hypotenuse c, a2 + b2 = c2 . An angle bisector divides an angle into two congruent angles. Guided Practice The large triangle above has sides a, b, and c. Side c has been divided into two parts: y and c − y. In the Concept Problem Revisited you showed that the three triangles in this picture are similar. 1. Explain why a c = 2. Explain why b c = c−y a . y b. 3. Use the results from #1 and #2 to show that a2 + b2 = c2 . Answers: 1. When triangles are similar, corresponding sides are proportional. Carefully match corresponding sides and you see that ac = c−y a . 2. When triangles are similar, corresponding sides are proportional. Carefully match corresponding sides and you see that bc = by . 401 6.6. Theorems Involving Similarity www.ck12.org 3. Cross multiply to rewrite each equation. Then, add the two equations together. a c−y = → a2 = c2 − cy c a b y = → b2 = cy c b → a2 + b2 = c2 − cy + cy → a2 + b2 = c2 You have just proved the Pythagorean Theorem using similar triangles. Practice Solve for x in each problem. 1. 2. 3. 402 www.ck12.org Chapter 6. Similarity 4. 5. 6. 403 6.6. Theorems Involving Similarity 7. Use the picture below for #8-#10. 8. Solve for x. 9. Solve for z. 10. Solve for y. Use the picture below for #11-#13. 404 www.ck12.org www.ck12.org 11. Assume that Chapter 6. Similarity b a = dc . Use algebra to show that b+a a = d+c c . 12. Prove that ∆Y ST ∼ ∆Y XZ 13. Prove that ST kXZ 14. Prove that a segment that connects the midpoints of two sides of a triangle will be parallel to the third side of the triangle. 15. Prove the Pythagorean Theorem using the picture below. 405 6.7. Applications of Similar Triangles www.ck12.org 6.7 Applications of Similar Triangles Here you will solve problems involving similar triangles and learn about special right triangles. Michael is 6 feet tall and is standing outside next to his younger sister. He notices that he can see both of their shadows and decides to measure each shadow. His shadow is 8 feet long and his sister’s shadow is 5 feet long. How tall is Michael’s sister? Watch This MEDIA Click image to the left for more content. http://www.youtube.com/watch?v=LhEe0kB4QIs James Sousa: Indirect Measurement Using Similar Triangles Guidance If two triangles are similar, then their corresponding angles are congruent and their corresponding sides are proportional. There are three criteria for proving that triangles are similar: 1. AA: If two triangles have two pairs of congruent angles, then the triangles are similar. 2. SAS: If two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar. 3. SSS: If three sides of one triangle are proportional to three sides of another triangle, then the triangles are similar. Once you know that two triangles are similar, you can use the fact that their corresponding sides are proportional and their corresponding angles are congruent to solve problems. In the Examples and practice, you will consider many different applications of similar triangles. Example A a) Prove that the two triangles below are similar. 406