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.

$\overline{\mathbf{AB}}\mathbf{\perp }\overline{\mathbf{CD}}$

Short answer

a. From the given diagram it can be concluded that angles∠8and∠9are adjacent angles.

The theorem of perpendicular lines states that if two lines are perpendicular then the lines form adjacent congruent angles.

Like the lines, ABand CDare perpendicular lines and the angles∠8 and∠9are adjacent angles, therefore the angles∠8 and∠9will be congruent angles.

That implies, $\angle 8\cong \angle 9$.

b. The name of the theorem that justifies the conclusion is thetheorem of perpendicular lines.

Step-by-step solution

Part a. Step 1. Observe the given diagram.

The given diagram is:

Part a. Step 2. Definition of congruent angles.

The congruent angles are the angles which have the same measure.

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

Given that $\overline{AB}\perp \overline{CD}$.

That implies the lines AB and CD are perpendicular lines.

From the given diagram it can be concluded that the angles∠8 and∠9are adjacent angles.

The theorem of perpendicular lines states that if two lines are perpendicular then the lines form adjacent congruent angles.

As the lines,AB and CD are perpendicular lines and the angles∠8 and∠9 are adjacent angles, therefore the angles∠8 and∠9 will be congruent angles.

That implies, $\angle 8\cong \angle 9$.

Part b. Step 1. Observe the given diagram.

The given diagram is:

Part b. Step 2. Write the theorem of perpendicular lines.

The theorem of perpendicular lines states that if two lines are perpendicular then the lines form the adjacent congruent angles.

Part b. Step 3. Write the name of the theorem that justifies the conclusion. 

The name of theorem that justifies the conclusion is theorem of perpendicular lines.