Question

10.
(6y - 15)° (5x)°
(4y + 25)°
x = ________
reason: ___________________________
y = ________
reason: ___________________________

Explanation:

Step1: Solve for \( x \)

The angles \( (6y - 15)^\circ \) and \( (5x)^\circ \) are supplementary (they form a linear pair), so their sum is \( 180^\circ \). But also, \( (6y - 15)^\circ \) and \( (4y + 25)^\circ \) are equal because they are alternate exterior angles (since the lines are parallel and cut by a transversal). Wait, first, let's find \( y \) using the alternate exterior angles.

Wait, actually, \( (6y - 15)^\circ \) and \( (4y + 25)^\circ \) are equal (alternate exterior angles). So:
\( 6y - 15 = 4y + 25 \)
Subtract \( 4y \) from both sides: \( 2y - 15 = 25 \)
Add 15 to both sides: \( 2y = 40 \)
Divide by 2: \( y = 20 \)

Now, \( (6y - 15)^\circ \) and \( (5x)^\circ \) are supplementary (linear pair), so:
\( (6y - 15) + 5x = 180 \)
Substitute \( y = 20 \):
\( 6(20) - 15 + 5x = 180 \)
\( 120 - 15 + 5x = 180 \)
\( 105 + 5x = 180 \)
Subtract 105: \( 5x = 75 \)
Divide by 5: \( x = 15 \)

Step2: Verify \( x \) and \( y \)

For \( y \): Alternate exterior angles are equal, so \( 6y - 15 = 4y + 25 \) gives \( y = 20 \). For \( x \): Linear pair angles sum to \( 180^\circ \), so \( (6(20) - 15) + 5x = 180 \) gives \( x = 15 \).

Answer:

\( x = 15 \) (Reason: Linear pair angles sum to \( 180^\circ \), and using \( y = 20 \) from alternate exterior angles equality)
\( y = 20 \) (Reason: Alternate exterior angles are equal, so \( 6y - 15 = 4y + 25 \))