To find the measures of angles P and Q, let's go step-by-step through the given conditions and calculations.
1. Understand the relationship:
- Angles P and Q are supplementary, which means their measures add up to 180 degrees.
- Let the measure of angle Q be [tex]\( x \)[/tex] degrees.
- The measure of angle P is given to be 3 times the measure of angle Q minus 16 degrees, so we can write [tex]\( \text{mP} = 3x - 16 \)[/tex].
2. Set up the equation based on the supplementary condition:
- The sum of angles P and Q is 180 degrees:
[tex]\[
x + (3x - 16) = 180
\][/tex]
3. Solve the equation:
- Combine like terms:
[tex]\[
x + 3x - 16 = 180 \quad \Rightarrow \quad 4x - 16 = 180
\][/tex]
- Add 16 to both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
4x - 16 + 16 = 180 + 16 \quad \Rightarrow \quad 4x = 196
\][/tex]
- Divide both sides by 4 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{196}{4} \quad \Rightarrow \quad x = 49
\][/tex]
4. Find the measure of angle P:
- Using the value of [tex]\( x \)[/tex] (which is the measure of angle Q), substitute back to find the measure of angle P:
[tex]\[
\text{mP} = 3x - 16 \quad \Rightarrow \quad \text{mP} = 3 \times 49 - 16 = 147 - 16 = 131
\][/tex]
So, the measures of angles P and Q are:
- Angle Q: [tex]\( 49 \)[/tex] degrees.
- Angle P: [tex]\( 131 \)[/tex] degrees.
