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Sections 8-3/8-5: April 24, 2012

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Warm-up: (10 mins) Practice Book: Practice 8-2 # 1 – 23 (odd)

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Warm-up: (10 mins)

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Questions on Homework?

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Review Name the postulate you can use to prove the triangles are congruent in the following figures:

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Sections 8-3/8-5: Ratio/Proportions/Similar Figures Objective: Today you will learn to prove triangles similar and to use the Side- Splitter and Triangle-Angle-Bisector Theorems.

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Angle-Angle Similarity (AA ∼ ) Postulate Geogebra file: AASim.ggb

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Angle-Angle Similarity (AA ∼ ) Postulate

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Example 1: Using the AA ∼ Postulate, show why these triangles are similar ∠ BEA ≅∠ DEC because vertical angles are congruent ∠ B ≅∠ D because their measures are both 600 ΔBAE ∼ ΔDCE by AA ∼ Postulate.

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SAS ∼ Theorem ΔABC ∼ ΔDEF

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SAS ∼ Theorem Proof

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SSS ∼ Theorem

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SSS ∼ Theorem Proof

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Example 2: Explain why the triangles are similar and write a similarity statement.

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Example 3: Find DE

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Real World Example How high must a tennis ball must be hit to just pass over the net and land 6m on the other side?

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Use Similar Triangles to find Lengths

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Use Similar Triangles to Heights

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Section 8-5: Proportions in Triangles Open Geogebra file SideSplitter.ggb

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Side-Splitter Theorem

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Example 4: Use the Side-Splitter Theorem to find the value of x

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Example 5: Find the value of the missing variables

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Corollary to the Side-Splitter Theorem

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Example 6: Find the value of x and y

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Example 7: Find the value of x and y

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Sail Making using the Side-Splitter Theorem and its Corollary What is the value of x and y?

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Triangle-Angle-Bisector Theorem

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Triangle-Angle-Bisector Theorem Proof

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Example 8: Using the Triangle-Angle- Bisector Theorem, find the value of x

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Example 9: Fnd the value of x

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Theorems Angle-Angle Similarity (AA ∼ ) Postulate: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. 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. Side-Side-Side Similarity (SSS ∼ ) Theorem: If the corresponding sides of two triangles are proportional, then the triangles are similar. Side-Splitter Theorem: If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. Corollary to the Side-Splitter Theorem: If three parallel lines intersect two transversals, then the segments intercepted on the transversals are proportional. Triangle-Angle-Bisector Theorem: If a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle.

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Wrap-up Today you learned to prove triangles similar and to use the Side-Splitter and Triangle-Angle-Bisector Theorems. Tomorrow you’ll learn about Similarity in Right Triangles Homework (H) p. 436 # 4-19, 21, 24-28 p. 448 # 1-3, 9-15 (odd), 25, 27, 32, 33 Homework (R) p. 436 # 4-19, 24-28 p. 448 # 1-3, 9-15 (odd), 32, 33

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