Introduction Congruent triangles have corresponding parts with angle measures that are the same and side lengths that are the same. If two triangles are congruent, they are also similar. Similar triangles have the same shape, but may be different in size. It is possible for two triangles to be similar but not congruent. Just like with determining congruency, it is possible to determine similarity based on the angle measures and lengths of the sides of the triangles : DEFINING SIMILARITY

Key Concepts To determine whether two triangles are similar, observe the angle measures and the side lengths of the triangles. When a triangle is transformed by a similarity transformation (a rigid motion [reflection, translation, or rotation] followed by a dilation), the result is a triangle with a different position and size, but the same shape. If two triangles are similar, then their corresponding angles are congruent and the measures of their corresponding sides are proportional, or have a constant ratio : DEFINING SIMILARITY

Key Concepts, continued The ratio of corresponding sides is known as the ratio of similitude. The scale factor of the dilation is equal to the ratio of similitude. Like with congruent triangles, corresponding angles and sides can be determined by the order of the letters. If is similar to, the vertices of the two triangles correspond in the same order as they are named : DEFINING SIMILARITY

Key Concepts, continued The symbol shows that parts are corresponding. ; they are equivalent. The corresponding angles are used to name the corresponding sides : DEFINING SIMILARITY

Key Concepts, continued Observe the diagrams of and. The symbol for similarity ( ) is used to show that figures are similar : DEFINING SIMILARITY

You can also use SSS and SAS to show similarity. SSS: If the three sides of one triangle are proportional to the three sides of the other triangle then the two triangles are similar. Example SSS: Show they are similar : DEFINING SIMILARITY 20 F G H 16 X 24 S 32 R

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Example: SAS If two sides of one triangle are proportional to two sides of another triangle and the included angles of these sides are congruent, then the two triangles are similar. Show they are similar : DEFINING SIMILARITY 12 R Q 8 S 6 9 U T

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Guided Practice Example Are these two triangles similar. How can you tell? : DEFINING SIMILARITY

Guided Practice Example 2 Are these two triangles similar? How can you tell? : DEFINING SIMILARITY

Key Concepts The Angle-Angle (AA) Similarity Statement is another statement that allows us to prove triangles are similar. The AA Similarity Statement allows that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar : APPLYING SIMILARITY USING THE ANGLE-ANGLE (AA) CRITERION

Key Concepts, continued Notice that it is not necessary to show that the third pair of angles is congruent because the sum of the angles must equal 180˚. Similar triangles have corresponding sides that are proportional. The Angle-Angle Similarity Statement can be used to solve various problems, including those that involve indirect measurement, such as using shadows to find the height of tall structures : APPLYING SIMILARITY USING THE ANGLE-ANGLE (AA) CRITERION

Guided Practice Example 3 Explain why, and then find the length of : APPLYING SIMILARITY USING THE ANGLE-ANGLE (AA) CRITERION

WORK: : APPLYING SIMILARITY USING THE ANGLE-ANGLE (AA) CRITERION

Guided Practice Example 4 Suppose a person 5 feet 10 inches tall casts a shadow that is 3 feet 6 inches long. At the same time of day, a flagpole casts a shadow that is 12 feet long. To the nearest foot, how tall is the flagpole? : APPLYING SIMILARITY USING THE ANGLE-ANGLE (AA) CRITERION

Guided Practice: Example 4, continued : APPLYING SIMILARITY USING THE ANGLE-ANGLE (AA) CRITERION

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Guided Practice Example 2 Determine whether the triangles are similar. Explain your reasoning : PROVING TRIANGLE SIMILARITY USING SAS AND SSS SIMILARITY

WORK: : PROVING TRIANGLE SIMILARITY USING SAS AND SSS SIMILARITY

Guided Practice Example 3 Determine whether the triangles are similar. Explain your reasoning : PROVING TRIANGLE SIMILARITY USING SAS AND SSS SIMILARITY

WORK: : PROVING TRIANGLE SIMILARITY USING SAS AND SSS SIMILARITY