   Chapter 3.5, Problem 26E ### Elementary Geometry for College St...

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

#### Solutions

Chapter
Section ### Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698
Textbook Problem
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# In Exercises 25 and 26, complete each proof shown on page 160. Given: Equilateral Δ A B C and D - B - C Prove: D A > A C PROOF Statements Reasons 1. ? 1. Given 2. Δ A B C is equiangular, so m ∠ A B C = m ∠ C 2. ? 3. m ∠ A B C > m ∠ D ( ∠ D of Δ A B D ) 3. The measure of an exterior ∠ of a Δ is greater than the measure of either nonadjacent interior ∠ 4. ? 4. Substitution 5. ? 5. ?

To determine

To prove:

The side DA is greater than the side AC.

Explanation

Given:

Consider the figure,

Figure (1)

From figure (1),

Equilateral ΔABC and D-B-C

If the measure of one angle of a triangle is greater than the measure of a second angle, than the side opposite to the larger angle is longer than the side opposite to the smaller angle.

The measure of an exterior angle of a triangle is greater than the measure of either nonadjacent interior angle of a triangle.

Definition:

The triangle whose all the sides are equal and the measures of all the angle are same, that triangle is called equilateral or equiangular triangle.

Approach:

Consider an equilateral ΔABC,

Since, in equilateral triangle the measures of all the angle are same, then

mABC=mC(1)

Consider the triangle ΔABD,

Since, the measure of an exterior angle of a triangle is greater than the measure of either nonadjacent interior angle of a triangle. Then,

mABC>mD(2)

Substitute equation (1) in equation (2).

mC>mD(3)

Since, the measure of one angle of a triangle is greater than the measure of a second angle, than the side opposite to the larger angle is longer than the side opposite to the smaller angle. Then, from equation (3) the side DA is greater than the side AC. So,

DA>AC

 PROOF Statements Reasons 1. Equilateral ΔABC and D-B-C 1

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