Chapter 2.6, Problem 6CE

a.

To determine

To use the given information to conclude that two angles are congruent.

Answer to Problem 6CE

  $\angle 1\cong \angle 3$ , $\angle 8\cong \angle 9$ , and $\angle 6\cong \angle 7$

Explanation of Solution

Given:

  

In Figure 1:

  $\angle 1\cong \angle 3$ (Vertically opposite angles)

In Figure 2:

No pair of angles are congruent. Instead $\angle 4+\angle 5={180}^0$ (Linear Pair).

In Figure 3:

No pair of angles are congruent.

In Figure 4:

  $\angle 8\cong \angle 9={90}^0$ and $\angle 8+\angle 9={180}^0$ (Linear pair)

In Figure 5:

  $\angle 6=\angle 7$ because $XZ$ is the angle bisector of $\angle WXY$ .

Conclusion:

So, the conclusion is $\angle 1\cong \angle 3$ , $\angle 8\cong \angle 9$ , and $\angle 6\cong \angle 7$ .

b.

To determine

To name or state the definition or theorem that justifies the conclusions in part ( a ).

Explanation of Solution

Given:

  

In Figure 1:

  $\angle 1\cong \angle 3$ (Vertically opposite angles)

Vertically opposite angles: The angles formed opposite each other on intersection of two lines. They are always equal.

In Figure 4:

  $\angle 8\cong \angle 9={90}^0$ (by diagram)

But $\angle 8+\angle 9={180}^0$ (Linear Pair)

Linear Pair: Two angles are said to be linear pair if they are adjacent angles produced by two intersecting lines. The measure of a straight angle is ${180}^0$ , so a linear pair must add up to ${180}^0$ .

In Figure 5:

  $\angle 6=\angle 7$ because $XZ$ is the angle bisector of $\angle WXY$ .

Angle Bisector: It is a line segment that divides the given angle into two halves.

Conclusion:

Therefore, the concepts used are Vertically Opposite Angles, Linear Pair and Angle Bisector.