Question

answer each question below. 14. an angle’s measure is 12 less than twice its (15) complement. find the measure of the angle. (c)15. the difference between twice an angle’s (16) measure and its supplement is 27. find the measure of the angle. 16. if a - p - b and if ap = 2x - 9, ab = 32, and (8) pb = 1.5x + 6, find pb.

Explanation:

Step1: Define the angle

Let the angle be $x$. Its complement is $90 - x$.

Step2: Set up the equation for question 14

The measure of the angle is 12 less than twice its complement, so $x=2(90 - x)-12$.

Step3: Solve the equation for question 14

Expand the right - hand side: $x = 180-2x - 12$.
Add $2x$ to both sides: $x + 2x=180 - 12$.
Combine like terms: $3x=168$.
Divide both sides by 3: $x = 56$.

Step4: Define the angle for question 15

Let the angle be $y$. Its supplement is $180 - y$.

Step5: Set up the equation for question 15

The difference between twice an angle's measure and its supplement is 27, so $2y-(180 - y)=27$.

Step6: Solve the equation for question 15

Expand the left - hand side: $2y-180 + y=27$.
Combine like terms: $3y=27 + 180$.
$3y=207$.
Divide both sides by 3: $y = 69$.

Step7: Use the segment addition postulate for question 16

Since $A - P - B$, then $AP+PB=AB$.
Substitute the given expressions: $(2x - 9)+(1.5x + 6)=32$.

Step8: Combine like terms for question 16

$2x+1.5x-9 + 6=32$.
$3.5x-3=32$.
Add 3 to both sides: $3.5x=32 + 3=35$.
Divide both sides by 3.5: $x = 10$.

Step9: Find $PB$ for question 16

Substitute $x = 10$ into the expression for $PB$: $PB=1.5\times10+6$.
$PB=15 + 6=21$.

Answer:

1. The measure of the angle is 56.
2. The measure of the angle is 69.
3. $PB = 21$.