Question

1. an angle measures seventeen more than three times a number. its supplement is three more than seven times the number. what is the measure of each angle in degrees?

Explanation:

Step1: Define the variable

Let the number be \( x \). Then the measure of the first angle is \( 3x + 17 \) degrees. The supplement of an angle is \( 180^\circ \) minus the angle, so the supplement of the first angle is \( 180-(3x + 17) \) degrees. And we know the supplement is also \( 7x+3 \) degrees.

Step2: Set up the equation

We set the two expressions for the supplement equal to each other:
\[
180-(3x + 17)=7x + 3
\]

Step3: Simplify the left - hand side

Simplify \( 180-(3x + 17) \):
\[
180 - 3x-17=163 - 3x
\]
So our equation becomes \( 163-3x = 7x+3 \)

Step4: Solve for \( x \)

Add \( 3x \) to both sides:
\[
163=7x + 3x+3
\]
\[
163 = 10x+3
\]
Subtract 3 from both sides:
\[
163 - 3=10x
\]
\[
160 = 10x
\]
Divide both sides by 10:
\[
x=\frac{160}{10}=16
\]

Step5: Find the measure of each angle

First angle: Substitute \( x = 16 \) into \( 3x + 17 \)
\[
3\times16+17=48 + 17=65
\]
Second angle (supplement): Substitute \( x = 16 \) into \( 7x+3 \)
\[
7\times16+3=112 + 3=115
\]
We can also check using the supplement formula: \( 180 - 65=115 \), which matches.

Answer:

The measure of the first angle is \( 65^\circ \) and the measure of its supplement is \( 115^\circ \)