Question

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

Explanation:

Step1: Determine the relationship between angles

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

Step2: Solve the equation for \( x \)

Simplify the left - hand side of the equation: \( 7x+2+x + 8=90 \).
Combine like terms: \( 8x+10 = 90 \).
Subtract 10 from both sides: \( 8x=90 - 10=80 \).
Divide both sides by 8: \( x=\frac{80}{8}=10 \).

Step3: Calculate the measure of \( \angle DCE \)

Substitute \( x = 10 \) into the expression for \( \angle DCE \), which is \( (7x + 2)^\circ \).
\( 7\times10+2=70 + 2=72^\circ \).

Answer:

The measure of \( \angle DCE \) is \( 72^\circ \).