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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 Similar Polygons Two polygons are similar polygons iff the corresponding angles are congruent and the corresponding sides are proportional. N Similarity Statement: C M CORN ~ MAIZ Corresponding Angles: O C M O N R C A M A Z R I N Z StatementOof Proportionality: R CO OR RN NC A MA AI IZ ZM I Example 1 Triangles ABC and ADE are D B similar. Find the A value of x. 9 cm 6 cm 8 cm C x E Example 2 Are the triangles below similar? 8 4 6 3 37 53 10 5 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 50 40 B 50 40 A D E Investigation 1 Step 1: Draw ΔABC where m<A and m<B equal sensible values of your choosing. C 50 40 B A 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 50 40 B 50 40 A D E Investigation 1 Now, are your triangles similar? What would you have to check to determine if they are similar? F C 50 40 B 50 40 A D E 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. 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? Investigation 2 What if you decide to indirectly measure a height on a day when there are no shadows? The following GSP Animation will help you discover an alternate method of indirect measurement using a mirror. Example 7 Your eye is 168 centimeters from the ground and you are 114 centimeters from the mirror. The mirror is 570 centimeters from the flagpole. How tall is the flagpole? Investigation 3 Each group will be given one of the three candidates for similarity shortcuts. Each group member should start with a different triangle and complete the steps outlined for the investigation. Share your results and make a conjecture based on your findings. 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 9 Use your new conjectures to find the missing measure. 28 24 24 x 18 y Example 10 Find the value of x that makes ΔABC ~ ΔDEF. Assignment • P. 384-387: 1-4, 7, 8, 10, 12, 14-17, 20, 30, 31, 32, 36, 41, 42 • P. 391-395: 4, 6-8, 10-14, 33, 39, 40 • Challenge Problems