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2 answers:
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Exterior angle theorem and properties of the isosceles triangles are used
prove the specified relations.
Correct responses:
9. ΔPAT ~ ΔPTB by AA similarity postulate
15. ║
by basic proportionality theorem
9. A two column proof is presented as follows;
Statement Reasons
1. ∠PTA = ∠B 1. Given
2. ∠PAT = ∠ATB + ∠B 2. Exterior angle of a triangle theorem
3. ∠BTP = ∠ATB + ∠PTA 3 Angle addition postulate
4. ∠BTP = ∠ATP + ∠B 4. Substitution property of equality
5. ∠PAT = ∠BTP 5. Substitution property
6. ΔPAT ~ ΔPTB 6. AA similarity postulate
The Angle-Angle, AA, similarity postulate states that two triangles are similar if two angles in one triangle are each equal to two angles in the other triangle.
15. A two column proof is presented as follows;
Statement Reasons
1. Given
2. ∠1 ≅ ∠G 2. Given
3. ΔKHG is an isosceles triangle 3. Definition
4. HK = HG 4. Legs of an isosceles triangle
5. Substitution property
6. ║
6. BPT, Basic Proportionality Theorem
Basic Proportionality Theorem, BPT, states that if a line parallel to one of the sides of a triangle, intersects the other two sides, then the line divides the other two sides in the same proportion.
Learn more about basic proportionality theorem, AA similarity postulate here: