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Similar Polygons Two polygons are similar polygons iff the corresponding angles are congruent and the corresponding sides are proportional. Similarity Statement: N C M CORN ~ MAIZ Corresponding Angles: C M C R I O NA N Z O R A M StatementOof Proportionality: R CO OR RN NC MA AI IZ ZM A I Z Example 1 Triangles ABC and ADE are similar. Find the value of x. D B A 6 cm 9 cm 8 cm C x E Example 1 Triangles ABC and ADE are similar. Find the value of x. 8 𝑥+8 = 6 9 D B A 6 cm 9 cm 8 cm 6𝑥 + 48 = 72 C 6𝑥 = 24 𝑥=4 x E Example 2 Are the triangles below similar? 8 4 6 3 37 53 5 10 Do you REALLY have to check all the sides and angles? Investigation 1 In this Investigation we will check the first similarity shortcut. If the angles in two triangles are congruent, are the triangles necessarily similar? F C A 50 40 B D 50 40 E Investigation 1 Step 1: Draw ΔABC where m<A and m<B equal sensible values of your choosing. C A 50 40 B Investigation 1 Step 1: Draw ΔABC where m<A and m<B equal sensible values of your choosing. Step 2: Draw ΔDEF where m<D = m<A and m<E = m<B and AB ≠ DE. F C A 50 40 B D 50 40 E Investigation 1 Now, are your triangles similar? What would you have to check to determine if they are similar? F C A 50 40 B D 50 40 E 6.4-6.5: Similarity Shortcuts Objectives: 1. To find missing measures in similar polygons 2. To discover shortcuts for determining that two triangles are similar Angle-Angle Similarity AA Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. Example 3 Determine whether the triangles are similar. Write a similarity statement for each set of similar figures. Example 3 Yes, from AA. ∆𝐶𝐷𝐸~∆𝐾𝐺𝐻 Yes, from AA. (They share angle A) ∆𝐴𝐵𝐸~∆𝐴𝐶𝐷 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 4 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, find the height of the pyramid. Example 5 Explain why Thales’ method worked to find the height of the pyramid? Example 6 If a person 5 feet tall casts a 6-foot shadow at the same time that a lamppost casts an 18-foot shadow, what is the height of the lamppost? Example 6 5 𝑥 = 6 18 6𝑥 = 90 𝑥 = 15′ Side-Side-Side Similarity SSS Similarity Theorem: If the corresponding side lengths of two triangles are proportional, then the two triangles are similar. Side-Angle-Side Similarity 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 8 Are the triangles below similar? Why or why not? Example 8 Are the triangles below similar? Why or why not? Yes, from SSS since all the corresponding sides have the same ratio 2/3. Yes, from SAS since the two pairs of corresponding sides have the ratio 6/7 and they have a congruent angle between them. Example 9 Use your new conjectures to find the missing measure. 28 24 24 x 18 y Example 9 Use your new conjectures to find the missing measure. 24 𝑥 = 18 24 𝑥 = 32 24 28 = 18 𝑦 𝑦 = 21 28 24 24 x 18 y Example 10 Find the value of x that makes ΔABC ~ ΔDEF. Example 10 Find the value of x that makes ΔABC ~ ΔDEF. 4 𝑥−1 = 12 18 12𝑥 − 12 = 72 12𝑥 = 84 𝑥=7