Chapter 3.CR, Problem 14CR

### 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
1 views

# Given: A C â†’ bisects âˆ  B A D Prove: A D > C D

To determine

To prove:

The side AD¯ is greater than the side CD¯.

Explanation

Given:

The given figure is,

Figure (1)

From figure (1).

Approach:

An angle bisector divides the angle in two congruent angles.

Opposite side of larger angle has the larger length in a triangle.

Approach:

Consider the triangle Î”ABD,

From the given data,

The angle bisector ACÂ¯ bisects âˆ BAD in two congruent angles mâˆ 1 and mâˆ 2. So,

mâˆ 1=mâˆ 2

Consider the triangle Î”ABC,

Since, the measure of the exterior angle mâˆ ACD of the triangle is the sum of measures of the two nonadjacent angles mâˆ 1 and mâˆ B. Then,

mâˆ ACD=mâˆ 1+mâˆ B

Then, the exterior angle mâˆ ACD is greater than the angle mâˆ 1. So,

mâˆ ACD>mâˆ 1

Since, the angle mâˆ 1 is equal to the angle mâˆ 2 then,

mâˆ ACD>mâˆ 2

Since, the side opposite to larger angle in a triangle has the larger length

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