Question

Can you help me write proofs for these, PLEASE!!!

Answer 1

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. $\overleftrightarrow{HM}$║$\overleftrightarrow{JK}$ by basic proportionality theorem

Detailed methods used to prove response

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

$\displaystyle 1. \hspace{0.3 cm} \frac{GJ}{HK} = \frac{GK}{GM}$                            ${}$          1.  Given

2. ∠1 ≅ ∠G                          ${}$                   2. Given

3. ΔKHG is an isosceles triangle${}$         3. Definition

4. HK = HG ${}$                                            4. Legs of an isosceles triangle

$\displaystyle 5. \hspace{0.3 cm} \frac{GJ}{HG} = \frac{GK}{GM}$              ${}$                         5. Substitution property

6. $\overleftrightarrow {HM}$ ║ $\overleftrightarrow{JK}$                      ${}$                    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:

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