Introduction When a series of similarity transformations are performed on a triangle, the result is a similar triangle. When triangles are similar, the.

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Presentation transcript:

Introduction When a series of similarity transformations are performed on a triangle, the result is a similar triangle. When triangles are similar, the corresponding angles are congruent and the corresponding sides are of the same proportion. It is possible to determine if triangles are similar by measuring and comparing each angle and side, but this can take time. There exists a set of similarity statements, similar to the congruence statements, that let us determine with less information whether triangles are similar. 1.6.2: Applying Similarity Using the Angle-Angle (AA) Criterion

Key Concepts The Angle-Angle (AA) Similarity Statement is one 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. 1.6.2: 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. 1.6.2: Applying Similarity Using the Angle-Angle (AA) Criterion

Common Errors/Misconceptions misidentifying congruent parts because of the orientation of the triangles misreading similarity statements as congruent statements incorrectly creating proportions between corresponding sides 1.6.2: Applying Similarity Using the Angle-Angle (AA) Criterion

Guided Practice Example 2 Explain why , and then find the length of 1.6.2: Applying Similarity Using the Angle-Angle (AA) Criterion

Guided Practice: Example 2, continued Show that the triangles are similar. According to the diagram, and . by the Angle-Angle (AA) Similarity Statement. 1.6.2: Applying Similarity Using the Angle-Angle (AA) Criterion

Guided Practice: Example 2, continued Find the length of . Corresponding sides of similar triangles are proportional. Create and solve a proportion to find the length of 1.6.2: Applying Similarity Using the Angle-Angle (AA) Criterion

Guided Practice: Example 2, continued The length of is 5.52 units. Corresponding sides are proportional. Substitute known values. Let x represent the length of (2.72)(6.9) = (3.4)(x) Solve for x. 18.768 = 3.4x x = 5.52 ✔ 1.6.2: Applying Similarity Using the Angle-Angle (AA) Criterion

Guided Practice: Example 2, continued http://www.walch.com/ei/00155 1.6.2: 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? 1.6.2: Applying Similarity Using the Angle-Angle (AA) Criterion

Guided Practice: Example 4, continued Identify the known information. The height of a person and the length of the shadow cast create a right angle. The height of the flagpole and the length of the shadow cast create a second right angle. You can use this information to create two triangles. Draw a picture to help understand the information. 1.6.2: Applying Similarity Using the Angle-Angle (AA) Criterion

Guided Practice: Example 4, continued 1.6.2: Applying Similarity Using the Angle-Angle (AA) Criterion

Guided Practice: Example 4, continued Determine if the triangles are similar. Two pairs of angles are congruent. According to the Angle-Angle (AA) Similarity Statement, the triangles are similar. Corresponding sides of similar triangles are proportional. 1.6.2: Applying Similarity Using the Angle-Angle (AA) Criterion

Guided Practice: Example 4, continued Find the height of the flagpole. Create and solve a proportion to find the height of the flagpole. 1.6.2: Applying Similarity Using the Angle-Angle (AA) Criterion

Guided Practice: Example 4, continued The flagpole is 20 feet tall. Corresponding sides are proportional. Let x represent the height of the flagpole. Simplify. Solve for x. x = 20 ✔ 1.6.2: Applying Similarity Using the Angle-Angle (AA) Criterion

Guided Practice: Example 4, continued http://www.walch.com/ei/00156 1.6.2: Applying Similarity Using the Angle-Angle (AA) Criterion