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One angle in a pair of vertical angles is [tex]\(4t + 16\)[/tex] and the other angle is [tex]\(5t - 21^\circ\)[/tex].

1. Define vertical angles.
2. Write the equation to find the angle measurements.
3. Solve for [tex]\(t\)[/tex].
4. Determine the two angle measurements.

Sagot :

GinnyAnsweridn
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].
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