Question

a. In each exercise use the information given to conclude that two angles are congruent.

b. Name or state the definition or theorem that justifies your conclusion.

∠4 is supplementary to ∠6; ∠2 is supplementary to ∠7;$\angle 6\cong \angle 7$

Short answer

a.∠4is supplementary to∠6and∠2is supplementary to∠7.

Therefore, the sum of the angles and is and the sum of the angles∠2and∠7is also 180°.

That implies,

$\angle 4+\angle 6=180°\left(1\right)$and $\angle 2+\angle 7=180°\left(2\right)$.

Subtract equation (2) from the equation (1).

$$\begin{gathered} \angle 4+\angle 6-\left(\angle 2+\angle 7\right)=180°-180° \\ \angle 4+\angle 6-\angle 2-\angle 7=0\left(3\right) \end{gathered}$$

As, the angles∠6and∠7are the congruent angles, therefore $\angle 6=\angle 7$.

Substitute the above relation in equation (3).

$$\begin{gathered} \angle 4+\angle 6-\angle 2-\angle 7=0 \\ \angle 4+\angle 7-\angle 2-\angle 7=0 \\ \angle 4-\angle 2=0 \\ \angle 4=\angle 2 \end{gathered}$$

Therefore, .

b. The definition that justifies the conclusion is the definition of the supplementary angles which states that the supplementary angles are the angles whose sum is 180°.

Step-by-step solution

Part a. Step 1. Observe the given diagram.

The given diagram is:

Part a. Step 2. Definition of supplementary angles.

The supplementary angles are the angles whose sum is 180°.

Part a. Step 3. Use the given information to conclude that the two angles are congruent.

is supplementary to∠6and∠2is supplementary to∠7.

Therefore, the sum of angles ∠4 and ∠6 is 180° and the sum of angles ∠2 and ∠7 is also 180°.

That implies,

$\angle 4+\angle 6=180°\left(1\right)$and $\angle 2+\angle 7=180°\left(2\right)$.

Subtract equation (2) from equation (1).

$$\begin{gathered} \angle 4+\angle 6-\left(\angle 2+\angle 7\right)=180°-180° \\ \angle 4+\angle 6-\angle 2-\angle 7=0\left(3\right) \end{gathered}$$

As, angles ∠6 and ∠7 are congruent angles, therefore $\angle 6=\angle 7$.

Substitute the above relation in equation (3).

$$\begin{gathered} \angle 4+\angle 6-\angle 2-\angle 7=0 \\ \angle 4+\angle 7-\angle 2-\angle 7=0 \\ \angle 4-\angle 2=0 \\ \angle 4=\angle 2 \end{gathered}$$

Therefore, $\angle 4\cong \angle 2$.

Part b. Step 1. Observe the given diagram.

The given diagram is:

Part b. Step 2. Definition of the supplementary angles.

The supplementary angles are the angles whose sum is 180°.

Part b. Step 3. State the definition that justifies the conclusion. 

The definition that justifies the conclusion is the definition of the supplementary angles which states that the supplementary angles are the angles whose sum is 180°.