Question

$\mathit{m}\mathbf{\angle }\mathbf{1}\mathbf{=}\mathbf{3}\mathit{x}\mathbf{-}\mathbf{5}$and $\mathit{m}\mathbf{\angle }\mathbf{2}\mathbf{=}\mathit{x}\mathbf{+}\mathbf{25}$.

a.role="math" localid="1646463936005" $\mathit{x}\mathbf{=}\underset{\mathbf{¯}}{\mathbf{?}}$

b. $\mathit{m}\mathbf{\angle }\mathbf{1}\mathbf{=}\underset{\mathbf{¯}}{\mathbf{?}}$(numerical value)

Short answer

a. The value of is 15.

b. The value of the measure of angle ∠1 is 40°.

Step-by-step solution

Part a. Step 1. Observe the given diagram.

The given diagram is:

Part a. Step 2. Write the theorem of vertical angles.

The theorem of vertical angles states that the vertical angles are congruent.

Part a. Step 3. Find the value of x.

From the given diagram it can be noticed that the angles∠1 and∠2are the vertical angles.

Therefore, by using the theorem of vertical angles it can be said that the angles∠1 and∠2 are the congruent angles.

That implies the angles∠1 and∠2 have same measure.

$$\begin{gathered} m\angle 1=m\angle 2 \\ 3x-5=x+25 \\ 3x-x=25+5 \\ 2x=30 \\ x=\frac{30}{2} \\ x=15 \end{gathered}$$

Therefore, the value of x is 15.

Part b. Step 1. Observe the given diagram. 

The given diagram is:

Part b. Step 2. Write the value of x.

The value of x is 15.

Part b. Step 3. Find the measure of the angle ∠1.

Substitute 15 for x into the equation $m\angle 1=3x-5$.

$$\begin{gathered} m\angle 1=3\left(15\right)-5 \\ =45-5 \\ =40 \end{gathered}$$

Therefore, the measure of angle ∠1 is 40°.