Chapter 3.2, Problem 32E

### Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698

Chapter
Section

### Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698
Textbook Problem
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# In Exercises 30 to 32, draw the triangles that are to be shown congruent separately. Then complete the proof. Given: M N Â¯ âˆ¥ Q P Â¯ and M Q Â¯ âˆ¥ N P Â¯ Prove: M Q Â¯ â‰… N P Â¯ (HINT: Show Î” M Q P â‰… Î” P N M .)

To determine

To prove:

The given statement MN¯QP¯ and MQ¯NP¯.

Explanation

Given:

The given statement are,

MNÂ¯âˆ¥QPÂ¯ and MQÂ¯âˆ¥NPÂ¯

Figure (1)

Properties used:

(1) If two lines are cut by a transversal so that two corresponding angles are congruent, then these lines are parallel.

(2) In two triangles, if two angles and one non included side are congruent, then the triangles are congruent.

(3) If two triangles are congruent to each other then all sides of the triangle are congruent to each other.

Approach:

The required figure is,

Figure (2)

The given statements are,

MNÂ¯âˆ¥QPÂ¯ and MQÂ¯âˆ¥NPÂ¯

Line MN and line QP are parallel lines and âˆ 1 and âˆ 2 are corresponding angles.

So,

âˆ 1â‰…âˆ 2

Line MQ and line NP are parallel lines and âˆ 3 and âˆ 4 are corresponding angles.

So,

âˆ 3â‰…âˆ 4

The side PM is common in both triangles Î”MQP and Î”PNM.

So,

PMÂ¯â‰…PMÂ¯

Use AAS criteria of triangle congruency.

Î”MQPâ‰…Î”PNM

Use transitive property of congruence.

MQÂ¯â‰…NPÂ¯

Thus, MQÂ¯â‰…NPÂ¯.

 PROOF Statements Reasons 1

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