Question

using algebra to solve for missing measures

1. if ( mangle pqt = 3x + 47 ) and ( mangle sqr = 6x - 25 ), find the measure of ( angle sqr ).
2. if ( ab perp cd ), ( mangle dce = 7x + 2 ) and ( mangle ecb = x + 8 ), find the measure of ( angle dce ).

Explanation:

Problem 10:

Step1: Identify vertical angles

Since \( \angle PQT \) and \( \angle SQR \) are vertical angles, they are equal. So we set up the equation:
\( 3x + 47 = 6x - 25 \)

Step2: Solve for \( x \)

Subtract \( 3x \) from both sides:
\( 47 = 3x - 25 \)

Add 25 to both sides:
\( 72 = 3x \)

Divide both sides by 3:
\( x = 24 \)

Step3: Find \( m\angle SQR \)

Substitute \( x = 24 \) into \( 6x - 25 \):
\( 6(24) - 25 = 144 - 25 = 119 \)

Step1: Use perpendicular lines property

Since \( AB \perp CD \), \( \angle DCB = 90^\circ \). And \( \angle DCE + \angle ECB = \angle DCB \), so:
\( (7x + 2) + (x + 8) = 90 \)

Step2: Solve for \( x \)

Combine like terms:
\( 8x + 10 = 90 \)

Subtract 10 from both sides:
\( 8x = 80 \)

Divide by 8:
\( x = 10 \)

Step3: Find \( m\angle DCE \)

Substitute \( x = 10 \) into \( 7x + 2 \):
\( 7(10) + 2 = 70 + 2 = 72 \)

Answer:

The measure of \( \angle SQR \) is \( 119^\circ \)

Problem 11: