Chapter 5.6, Problem 19E

### Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Chapter
Section

### Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085
Textbook Problem

# Exercises 18 and 19 are based on a theorem ( not stated) that is the converse of Theorem 5.6.3. See the figure on page 267. Given: N P = 6 , M N = 9 , P Q = 4 , and M Q = 6 ; m ∠ P = 62 ∘ and m ∠ M = 36 ∘ Find: m ∠ Q N M (HINT: N P M N = P Q M Q . )

To determine

To find:

mQNM.

Explanation

Given:

NQâ†’ bisects âˆ MNP, NP=6,MN=9,PQ=4,MQ=6,mâˆ P=62âˆ˜,mâˆ M=36âˆ˜.

Theorem used:

Angle bisector theorem:

If a ray bisects one angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the two sides that form the bisected angle.

Calculation:

Given: NQâ†’ bisects âˆ MNP.

Then by angle-bisector theorem, the ray NQâ†’ divides the opposite side into segments whose lengths are proportional to the lengths of the two sides that form the bisected angle.

â‡’NPMN=PQMQ

In a triangle the sum of the angles is 180âˆ˜.

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