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Transcript
8-3 PROVING TRIANGLES SIMILAR (p. 432-438) There is no need to do the Investigation. When we prove triangles congruent, we no longer use the definition of congruent triangles since we have learned our shorter methods such as SSS, ASA, SAS, AAS, and HL. Likewise, we do not have to always satisfy the definition of similar triangles in order to prove triangles similar. We are about to learn three shorter methods of proving triangles similar. Postulate 8-1 Angle-Angle Similarity (AA ~) Postulate If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Example: Sketch two triangles, ABC and XYZ. If A X and C Z, are the two triangles similar by AA ~? Example: In the following diagram, BC AD. Why are the triangles similar? Write a similarity statement. B E 42 42 A C D Theorem 8-1 Side-Angle-Side Similarity (SAS ~) Theorem If an angle of one triangle is congruent to an angle of a second triangle, and the sides including the two angles are proportional, then the triangles are similar. We will not be concerned with the proof of this theorem. Example: In the following diagram, m B 65 and m K 65. Are these triangles similar by SAS ~? If so, write a similarity statement. B 12 65 K 6 65 C A 20 10 J L Theorem 8-2 Side-Side-Side Similarity (SSS ~) Theorem If the corresponding sides of two triangles are proportional, then the triangles are similar. We will not be concerned with the proof of this theorem. Example: Are the following two triangles similar by SSS ~? If so, write a similarity statement. B 15 A K 21 24 7 5 J C 8 L Example: Explain why the following two triangles must be similar. Write a similarity statement. Z 18 V 24 12 Y 36 W X Do 2 on p. 434. You can apply these three methods (AA ~, SAS ~, and SSS ~) to find the lengths of unknown sides in similar triangles. Once you know that two triangles are similar, you then know that corresponding sides of these two triangles are proportional. This is sometimes abbreviated as CSSTP. So, you will then set up and solve a proportion to find an unknown side length. Example: ABCD is a parallelogram. Why is AWX ~ YWZ? Set up and solve a proportion to find WY. A X 5 4W 10 D Z B Y C Do 3 on p. 434. Indirect measurement is a process where you use similar triangles and known lengths to find distances that are difficult to measure directly. Therefore, you find these new lengths indirectly. You can use mirrors or shadows along with similar triangles to obtain lengths or distances indirectly. Example: Amber places a mirror 28 feet from the base of a flagpole. When she stands 3 feet from the mirror, she can see the top of the flagpole reflected in the mirror. If her eyes are 5 ft above the ground, how tall is the flagpole? Make a good sketch. Which angles are congruent in the two triangles? By what method are the triangles similar? Set up and solve a proportion to find the height of the flagpole. Do 4 on p. 435. Homework p. 435-438: 4,5,7,11,12,15,17,18,23b,24,26,28,38,45,46,53,56,57 15. Solve 2 x 3 x 7.5