Question

1. in the given diagram, two lines meet at a point that is also the endpoint of a ray.

a. in a complete sentence, describe an angle relationship that would help you solve for x.
b. determine the measures of $\angle afb$ and $\angle bfc$.

Explanation:

Part (a)

To solve for \( x \), we can use the angle relationship that the sum of \( \angle AFB \), \( \angle BFC \), and the right angle (since there is a square symbol indicating \( 90^\circ \)) is \( 180^\circ \) (they are supplementary as they form a straight line or a linear pair with the right angle considered). So, \( (3x + 4)^\circ+(5x - 2)^\circ + 90^\circ=180^\circ \) (the sum of angles on a straight line is \( 180^\circ \), and here we have three angles: \( \angle AFB \), \( \angle BFC \), and the right angle, which together make a straight line).

Step 1: Set up the equation

We know that the sum of \( \angle AFB=(3x + 4)^\circ \), \( \angle BFC=(5x - 2)^\circ \), and the right angle (\( 90^\circ \)) is \( 180^\circ \) (since they lie on a straight line). So,

$$ (3x + 4)+(5x - 2)+90 = 180 $$

Step 2: Simplify the left - hand side

Combine like terms:

$$ 3x+5x + 4-2 + 90=180\\ 8x+92 = 180 $$

Step 3: Solve for \( x \)

Subtract 92 from both sides:

$$ 8x=180 - 92\\ 8x = 88 $$

Divide both sides by 8:

$$ x=\frac{88}{8}=11 $$

Step 4: Find the measure of \( \angle AFB \)

Substitute \( x = 11 \) into \( (3x + 4)^\circ \):

$$ 3(11)+4=33 + 4=37^\circ $$

Step 5: Find the measure of \( \angle BFC \)

Substitute \( x = 11 \) into \( (5x - 2)^\circ \):

$$ 5(11)-2=55 - 2 = 53^\circ $$

Answer:

The sum of \( \angle AFB \), \( \angle BFC \), and the right angle is \( 180^\circ \) (they are supplementary as they lie on a straight line), so \( (3x + 4)+(5x - 2)+ 90 = 180 \) can be used to solve for \( x \).

Part (b)