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Geometry

Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen

$Math Studyset Vaia Explanations$ Math

Student Edition · 227 pages

ISBN: 9780395977279

Chapter 4: Q18WE. (page 126)

Write proofs in two-column form.
Given: $\bar{T M} ≅ \bar{P R}$ ; $\bar{T M} ∥ \bar{R P}$
Prove: $△ T E M ≅ △ P E R$

Short Answer

Expert verified

The two-column proof is:

Statements	Reasons
$\bar{T M} ∥ \bar{R P}$	Given
$∠ E T M = ∠ E P R, ∠ T M E = ∠ P R E$	Alternate interior angles
$\bar{T M} ≅ \bar{P R}$	Given
$△ T E M ≅ △ P E R$	ASA Postulate

Step by step solution

01

Step 1. Observe the figure.

The figure is:

02

Step 2. Description of step.

Given that $\bar{T M} ≅ \bar{P R}$ and $\bar{T M} ∥ \bar{R P}$ .

As, $\bar{T M} ∥ \bar{R P}$ and from the given diagram it can be noticed that the angles $∠ E T M$ and $∠ E P R$ are the alternate interior angles.

Therefore, $∠ E T M = ∠ E P R$ .

That implies, $∠ E T M ≅ ∠ E P R$ .

As, $\bar{T M} ∥ \bar{R P}$ and from the given diagram it can be noticed that the angles $∠ T M E$ and $∠ E R P$ are the alternate interior angles.

Therefore, $∠ T M E = ∠ P R E$ .

That implies, $∠ T M E ≅ ∠ P R E$ .

Therefore, it can be seen that $∠ E T M = ∠ E P R$ , $\bar{T M} = \bar{P R}$ and $∠ T M E = ∠ P R E$ .

Therefore, the triangles $△ T E M$ and $△ P E R$ are the congruent triangles by using the ASA postulate.

03

Step 3. Write the proof in two-column form.

The proof in two-column form is:

Statements	Reasons
$\bar{T M} ∥ \bar{R P}$	Given
$∠ E T M = ∠ E P R$ , $∠ T M E = ∠ P R E$	Alternate interior angles
$\bar{T M} ≅ \bar{P R}$	Given
$△ T E M ≅ △ P E R$	ASA Postulate

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Most popular questions from this chapter

Decide whether you can deduce by the SSS, SAS, or ASA postulate that another triangle is congruent to $Δ A B C$ If so, write the congruence and name the postulate used. If not, write no congruence can be deduced.

Decide whether you can deduce by the SSS, SAS, or ASA postulate that another triangle is congruent to $Δ A B C$ . If so, write the congruence and name the postulate used. If not, write no congruence can be deduced.

For the following figure, do the SAS postulates justify that the two triangles are congruent?

Decide whether you can deduce by the SSS, SAS, or ASA postulate that another triangle is congruent to $Δ A B C$ . If so, write the congruence and name the postulate used. If not, write no congruence can be deduced.

For the following figure, can the triangle be proved congruent. If so, what postulate can be used?

See all solutions

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