Question

1. if ( overleftrightarrow{ab} perp overleftrightarrow{cd} ), ( mangle dce = 7x + 2 ) and ( mangle ecb = x + 8 ), find the measure of ( angle dce ).

Explanation:

Step1: Recognize right angle

Since \( \overrightarrow{AB} \perp \overrightarrow{CD} \), \( \angle DCB = 90^\circ \), so \( m\angle DCE + m\angle ECB = 90^\circ \).

Step2: Substitute angle expressions

Substitute \( m\angle DCE = 7x + 2 \) and \( m\angle ECB = x + 8 \) into the equation: \( (7x + 2) + (x + 8) = 90 \).

Step3: Solve for x

Simplify: \( 8x + 10 = 90 \). Subtract 10: \( 8x = 80 \). Divide by 8: \( x = 10 \).

Step4: Find \( m\angle DCE \)

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

Answer:

\( 72^\circ \)