Question

1. diagram with angles ( y^circ ), ( x^circ ), ( 72^circ ), ( z^circ ) and a right angle ( angle a ) and ( angle b ) are complementary angles. if the...

Explanation:

To solve for the angles \( x^\circ \), \( y^\circ \), and \( z^\circ \), we analyze the relationships between the angles (supplementary, complementary, vertical angles, etc.).

Step 1: Find \( x^\circ \)

The angle \( x^\circ \) and the \( 72^\circ \) angle are supplementary (they form a linear pair, so their sum is \( 180^\circ \)):
\[
x + 72 = 180
\]
Subtract \( 72 \) from both sides:
\[
x = 180 - 72 = 108
\]

Step 2: Find \( z^\circ \)

The \( z^\circ \) angle and the \( 72^\circ \) angle are complementary (they form a right angle, so their sum is \( 90^\circ \)):
\[
z + 72 = 90
\]
Subtract \( 72 \) from both sides:
\[
z = 90 - 72 = 18
\]

Step 3: Find \( y^\circ \)

The \( y^\circ \) angle, the right angle (\( 90^\circ \)), and \( z^\circ \) are supplementary (they form a linear pair, so their sum is \( 180^\circ \)):
\[
y + 90 + z = 180
\]
We already found \( z = 18 \), so substitute:
\[
y + 90 + 18 = 180
\]
Simplify:
\[
y + 108 = 180
\]
Subtract \( 108 \) from both sides:
\[
y = 180 - 108 = 72
\]

Final Answers

• \( x = \boldsymbol{108^\circ} \)
• \( y = \boldsymbol{72^\circ} \)
• \( z = \boldsymbol{18^\circ} \)

Answer:

To solve for the angles \( x^\circ \), \( y^\circ \), and \( z^\circ \), we analyze the relationships between the angles (supplementary, complementary, vertical angles, etc.).

Step 1: Find \( x^\circ \)

The angle \( x^\circ \) and the \( 72^\circ \) angle are supplementary (they form a linear pair, so their sum is \( 180^\circ \)):
\[
x + 72 = 180
\]
Subtract \( 72 \) from both sides:
\[
x = 180 - 72 = 108
\]

Step 2: Find \( z^\circ \)

The \( z^\circ \) angle and the \( 72^\circ \) angle are complementary (they form a right angle, so their sum is \( 90^\circ \)):
\[
z + 72 = 90
\]
Subtract \( 72 \) from both sides:
\[
z = 90 - 72 = 18
\]

Step 3: Find \( y^\circ \)

The \( y^\circ \) angle, the right angle (\( 90^\circ \)), and \( z^\circ \) are supplementary (they form a linear pair, so their sum is \( 180^\circ \)):
\[
y + 90 + z = 180
\]
We already found \( z = 18 \), so substitute:
\[
y + 90 + 18 = 180
\]
Simplify:
\[
y + 108 = 180
\]
Subtract \( 108 \) from both sides:
\[
y = 180 - 108 = 72
\]

Final Answers

• \( x = \boldsymbol{108^\circ} \)
• \( y = \boldsymbol{72^\circ} \)
• \( z = \boldsymbol{18^\circ} \)