Chapter 2.6, Problem 5CE

a.

To determine

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

Answer to Problem 5CE

  $\angle 1\cong \angle 3$ , $\angle 8\cong \angle 9$ , $\angle 6\cong \angle 7$ , and $\angle 4\cong \angle 2$

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:

  $\angle 6=\angle 7$ given.

Now,

  $\angle 4+\angle 6={180}^0$ (given)….(i).

  $\angle 2+\angle 7={180}^0$ (given)…..(ii).

  $\angle 6\cong \angle 7$ (given)…..(iii).

Now, subtracting equation (i) from equation (ii),

  $\angle 4\cong \angle 2$ ( $\because \angle 6\cong \angle 7$ )

Conclusion:

So, the conclusion is $\angle 1\cong \angle 3$ , $\angle 8\cong \angle 9$ , $\angle 6\cong \angle 7$ , and $\angle 4\cong \angle 2$ .

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 which are opposite each other when two lines cross are equal.

In Figure 4:

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

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

In Figure 5:

  $\angle 6=\angle 7$ (Given)

Now,

  $\angle 4+\angle 6={180}^0$ (Given)….(i).

  $\angle 2+\angle 7={180}^0$ (Given)…..(ii).

  $\angle 6\cong \angle 7$ (Given)…..(iii).

Now, subtracting equation (i) from equation (ii),

  $\angle 4\cong \angle 2$ ( $\because \angle 6\cong \angle 7$ )

Conclusion:

Therefore, the concepts used are Vertically Opposite Angles, Linear Pair and using the given information to solve.