Chapter 4.4, Problem 26WE

a.

To determine

To draw an isosceles $△ABC$ such that $\overrightarrow{AB}\cong \overrightarrow{AC}$ & $m\angle A={80}^{\circ }$ . Then, draw $AX$ the bisector of an exterior angle at A.

Answer to Problem 26WE

Yes, $AX\parallel BC$

Explanation of Solution

The Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles opposite to those sides are congruent.

  

Now, see the figure carefully

1.

  $\angle CAX=\angle XAY$

Reason: Definition of angle bisector

2.

  $\angle BAC+\angle CAX+\angle XAY=180$

Reason: Angle addition postulate

3. This implies;

  $\begin{array}{l}2\angle CAX=180-80\\ \angle CAX=\frac{100}{2}=50\end{array}$

4. Now consider $△ABC$

  $m\angle A=80;\overrightarrow{AB}\cong \overrightarrow{AB}$ ;

Reason: Given

5.

  $m\angle B=m\angle C$

Reason: Isosceles theorem

6.

  $m\angle BAC+m\angle B+m\angle C=180$

Reason: The sum of angles of a triangle is 180.

7. This implies;

  $\begin{array}{l}2m\angle C=180-80\\ m\angle C=50\end{array}$

8.

  $\angle CAX\cong \angle C$

Reason: Proved in step 3 & 7 and make alternate interior angles

9.

  $AX\parallel BC$

Reason: If two lines are cut by a transversal and alternate interior angles are, then the lines are parallel.

Yes, $AX\parallel BC$

b.

To determine

To find out whether the answer will be changed if $m\angle A$ changed

Answer to Problem 26WE

No.

Explanation of Solution

No, answer will remain same. If $m\angle A$ changed, it doesn’t affect our answer.