Question

reason: corresponding angles

4.
(14x + 7)°
103
14x + 7 = 77
100° 77
x = 5
reason: corresponding angles

8.
4y - 13 = 5x - 68
(4y - 13)° (3x)°
(5x - 68)°

x = ______
reason: ______

y = ______
reason: ______

Explanation:

Step1: Solve for \( x \)

We know that \( 3x \) and \( 5x - 68 \) are alternate interior angles (since the lines are parallel, alternate interior angles are equal). So we set up the equation:
\( 3x = 5x - 68 \)
Subtract \( 3x \) from both sides:
\( 0 = 2x - 68 \)
Add 68 to both sides:
\( 2x = 68 \)
Divide both sides by 2:
\( x = 34 \)

Step2: Solve for \( y \)

We are given the equation \( 4y - 13 = 5x - 68 \). We already found that \( x = 34 \), so substitute \( x = 34 \) into the equation:
\( 4y - 13 = 5(34) - 68 \)
Calculate the right side:
\( 5(34)=170 \), so \( 170 - 68 = 102 \)
Now we have:
\( 4y - 13 = 102 \)
Add 13 to both sides:
\( 4y = 115 \)
Wait, that can't be right. Wait, maybe the angles \( (4y - 13)^\circ \) and \( 3x^\circ \) are supplementary? Wait, no, looking at the diagram, maybe \( (4y - 13) \) and \( 3x \) are supplementary? Wait, no, the original equation written is \( 4y - 13 = 5x - 68 \). Wait, let's check again.

Wait, maybe I made a mistake in the angle relationship. Let's re - examine. If the two horizontal lines are parallel and the transversal cuts them, then \( 3x \) and \( 5x - 68 \) are alternate interior angles, so \( 3x=5x - 68 \), which gives \( x = 34 \). Then, if \( (4y - 13) \) and \( 3x \) are supplementary (since they form a linear pair), then \( (4y - 13)+3x = 180 \).

Substitute \( x = 34 \) into \( 3x \): \( 3\times34 = 102 \)
So \( 4y-13 + 102=180 \)
\( 4y + 89 = 180 \)
\( 4y=180 - 89=91 \)
\( y=\frac{91}{4}=22.75 \). But the written equation is \( 4y - 13 = 5x - 68 \). Let's use the written equation.

Substitute \( x = 34 \) into \( 5x-68 \): \( 5\times34 - 68=170 - 68 = 102 \)
So \( 4y-13 = 102 \)
\( 4y=102 + 13=115 \)
\( y=\frac{115}{4}=28.75 \). Wait, there must be a mistake in the angle relationship.

Wait, maybe the angles \( (4y - 13) \) and \( 5x - 68 \) are corresponding angles? Wait, the diagram shows two parallel lines cut by a transversal. Let's assume that \( 3x \) and \( 5x - 68 \) are alternate interior angles, so \( 3x=5x - 68 \), solving:

\( 3x=5x - 68 \)

\( 68 = 5x-3x \)

\( 2x=68 \)

\( x = 34 \)

Then, if \( (4y - 13) \) and \( 3x \) are equal (corresponding angles), then \( 4y-13 = 3x \)

Substitute \( x = 34 \), \( 3x = 102 \)

\( 4y-13 = 102 \)

\( 4y=102 + 13=115 \)

\( y=\frac{115}{4}=28.75 \)

But let's go back to the given equation in the diagram: \( 4y - 13 = 5x - 68 \)

Substitute \( x = 34 \) into \( 5x - 68 \): \( 5\times34-68 = 170 - 68=102 \)

So \( 4y-13 = 102 \)

\( 4y=102 + 13 = 115 \)

\( y=\frac{115}{4}=28.75 \)

Answer:

\( x = 34 \) (Reason: Alternate Interior Angles are equal)

\( y = 28.75 \) (Reason: Using the given equation \( 4y - 13 = 5x - 68 \) and substituting \( x = 34 \))