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Math
GeometryElementary Geometry for College Students6th Edition

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Chapter 3.2, Problem 31E
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Elementary Geometry for College St...

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

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Chapter
Section
Problem 1E
Problem 2E
Problem 3E
Problem 4E
Problem 5E
Problem 6E
Problem 7E
Problem 8E
Problem 9E
Problem 10E
Problem 11E
Problem 12E
Problem 13E
Problem 14E
Problem 15E
Problem 16E
Problem 17E
Problem 18E
Problem 19E
Problem 20E
Problem 21E
Problem 22E
Problem 23E
Problem 24E
Problem 25E
Problem 26E
Problem 27E
Problem 28E
Problem 29E
Problem 30E
Problem 31E
Problem 32E
Problem 33E
Problem 34E
Problem 35E
Problem 36E
Problem 37E
Problem 38E
Problem 39E
Problem 40E
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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.

Chapter 3.2, Problem 31E, In Exercises 30 to 32, draw the triangles that are to be shown congruent separately. Then complete

Given: ∠ 1 ≅ ∠ 2 and M N ¯ ≅ Q P ¯
Prove: M Q ¯ ∥ N P ¯
(HINT: Show Δ N M P ≅ Δ Q P M .)

To determine

To prove:

The given statement ∠1≅∠2 and MN¯≅QP¯.

Explanation

Given:

The given statement is,

∠1≅∠2 and MN¯≅QP¯

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 corresponding sides and the included angle between them is congruent, then the triangles are congruent.

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

Approach:

The required triangles are,

Figure (2)

The given statements are,

∠1≅∠2 and MN¯≅QP¯

The side MP is common in both triangles ΔNMP and ΔQMP.

So,

MP¯≅MP¯

Use SAS criteria of triangle congruency.

ΔNMP≅ΔQMP

Use transitive property of congruence.

∠2≅∠4

∠2 and ∠4 are congruent corresponding angles.

Thus, MQ¯∥NP¯.

PROOF
Statements Reasons
1. MN¯≅QP¯ 1. Given
2.MP¯≅MP¯ 2

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