Let’s solve the problem step-by-step:
### Step 1: Understand the Concept
1) Vertical angles are equal. This means that if two angles are vertical angles, they must have the same measure.
### Step 2: Set Up the Equation
2) We are given two expressions for the vertical angles:
- First angle: [tex]\(4t + 16\)[/tex]
- Second angle: [tex]\(5t - 21\)[/tex]
Since vertical angles are equal, we can set these two expressions equal to each other:
[tex]\[
4t + 16 = 5t - 21
\][/tex]
### Step 3: Solve for [tex]\( t \)[/tex]
3) To find the value of [tex]\( t \)[/tex], we solve the equation:
[tex]\[
4t + 16 = 5t - 21
\][/tex]
First, isolate the term with [tex]\( t \)[/tex] on one side of the equation. Subtract [tex]\( 4t \)[/tex] from both sides:
[tex]\[
16 = t - 21
\][/tex]
Next, add 21 to both sides to solve for [tex]\( t \)[/tex]:
[tex]\[
16 + 21 = t
\][/tex]
Thus:
[tex]\[
t = 37
\][/tex]
### Step 4: Find the Angle Measurements
4) Now substitute [tex]\( t = 37 \)[/tex] back into the expressions for the angles to find their measures.
First angle:
[tex]\[
4t + 16
\][/tex]
Substitute [tex]\( t = 37 \)[/tex]:
[tex]\[
4(37) + 16 = 148 + 16 = 164 \degree
\][/tex]
Second angle:
[tex]\[
5t - 21
\][/tex]
Substitute [tex]\( t = 37 \)[/tex]:
[tex]\[
5(37) - 21 = 185 - 21 = 164 \degree
\][/tex]
### Final Answer:
- The value of [tex]\( t \)[/tex] is [tex]\( 37 \)[/tex].
- The two angle measurements are [tex]\( 164 \degree \)[/tex] and [tex]\( 164 \degree \)[/tex].
