Question

In the diagram, $\mathit{m}\angle \mathit{A}\mathit{X}\mathit{B}\mathbf{=}\mathbf{90}$. Name:

a. two congruent supplementary angles.

b. two supplementary angles that are not congruent.

c. two complementary angles.

d. a straight angle.

Short answer

a.The two congruent supplementary angles are$\angle \mathit{A}\mathit{X}\mathit{B}$and $\angle \mathit{B}\mathit{X}\mathit{D}$.

b. The two supplementary angles that are not congruent are$\angle \mathit{A}\mathit{X}\mathit{C}$and $\angle \mathit{C}\mathit{X}\mathit{D}$.

c. The two complementary angles are $\angle \mathit{B}\mathit{X}\mathit{C}$and $\angle \mathit{C}\mathit{X}\mathit{D}$.

d. The straight angle is $\angle \mathit{A}\mathit{X}\mathit{D}$.

Step-by-step solution

Part a. Step 1. Observe the given diagram.

The given diagram is:

Part a. Step 2. Description of step.

The congruent angles are the angles which have same measure.

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

Part a. Step 3. Name two congruent supplementary angles.

From the given diagram it can be noticed that the angles$\angle AXB$ and$\angle BXD$ forms the linear pair.

Therefore, the sum of the angles$\angle AXB$ and$\angle BXD$ is$180°$.

Therefore, it can be obtained that:

$$\begin{gathered} \angle AXB+\angle BXD=180° \\ 90°+\angle BXD=180° \\ \angle BXD=180°-90° \\ \angle BXD=90° \end{gathered}$$

Therefore,$\angle BXD=90°$

Therefore, it can be noticed that:

$\angle AXB=\angle BXD=90°$

It can also be noticed that $\angle AXB+\angle BXD=180°$.

Therefore, as$\angle AXB+\angle BXD=180°$ and $\angle AXB=\angle BXD=90°$.

Therefore, the angles$\angle AXB$ and$\angle BXD$ are congruent supplementary angles.

Therefore, the two congruent supplementary angles are$\angle AXB$ and $\angle BXD$.

Part b. Step 1. Observe the given diagram.

The given diagram is:

Part b. Step 2. Description of step.

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

Part b. Step 3. Description of step.

From the given diagram it can be noticed that the angles$\angle AXC$ and$\angle CXD$ forms the linear pair.

Therefore, the sum of the angles$\angle AXC$ and$\angle CXD$ is $180°$.

From the diagram it can also be noticed that the angles$\angle AXC$ and$\angle CXD$ do not have same measure.

Therefore, the angles$\angle AXC$ and$\angle CXD$ are not congruent.

As, the sum of the angles $\angle AXC$and $\angle CXD$is $180°$and the angles $\angle AXC$and $\angle CXD$are not congruent.

Therefore, the two supplementary angles that are not congruent are$\angle AXC$ and $\angle CXD$.

Part c. Step 1. Observe the given diagram.

The given diagram is:

Part c. Step 2. Description of step.

The complementary angles are the angles whose sum is $90°$.

Part c. Step 3. Description of step.

From the given diagram it can be noticed that the angles$\angle AXB$ and$\angle BXD$ forms the linear pair.

Therefore, the sum of the angles$\angle AXB$ and$\angle BXD$ is$180°$.

Therefore, it can be obtained that:

$$\begin{gathered} \angle AXB+\angle BXD=180° \\ 90°+\angle BXD=180° \\ \angle BXD=180°-90° \\ \angle BXD=90° \end{gathered}$$

Therefore,$\angle BXD=90°$

By using the angle addition postulate, it can be said that:

$\angle BXC+\angle CXD=\angle BXD$

Therefore, it can be obtained that:

$$\begin{gathered} \angle BXC+\angle CXD=\angle BXD \\ \angle BXC+\angle CXD=90° \end{gathered}$$

As,,$\angle BXC+\angle CXD=90°$ therefore the angles$\angle BXC$ and$\angle CXD$ are complementary angles.

The two complementary angles are$\angle BXC$and $\angle CXD$.

Part d. Step 1. Observe the given diagram.

The given diagram is:

Part d. Step 2. Description of step.

A straight angle is an angle which has measure of $\mathbf{180}\mathbf{°}$.

Part d. Step 3. Description of step.

From the given diagram it can be noticed that the angle$\angle AXD$ has measure of $180°$.

Therefore, the angle$\angle AXD$ is a straight angle.