Question

2 multiple choice 4 points
\\( \overleftrightarrow { a b } \\) and \\( \overleftrightarrow { c d } \\) intersect at point \\( e, m \angle a e c = 6 x + 20 \\), and \\( m \angle d e b = 10 x \\).
what is the value of \\( x \\)?
10
\\( 21 \frac { 1 } { 4 } \\)
\\( 4 \frac { 3 } { 8 } \\)
5
3 multiple choice 4 points
which pair of angles \\( x \\) and \\( y \\) are supplementary?
\\( m \angle x = 76 \\)
\\( m \angle y = 14 \\)
\\( m \angle x = 113 \\)
\\( m \angle y = 67 \\)
\\( m \angle x = 140 \\)
\\( m \angle y = 190 \\)
\\( m \angle x = 180 \\)
\\( m \angle y = 180 \\)

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. Since \(\angle AEC\) and \(\angle DEB\) are vertical angles, we have \(6x + 20=10x\).

Step2: Solve the equation for \(x\)

Subtract \(6x\) from both sides: \(20 = 10x-6x\), which simplifies to \(20 = 4x\). Then divide both sides by \(4\): \(x=\frac{20}{4}=5\).

for question 3:
Supplementary angles sum to \(180^{\circ}\).

• For \(m\angle x = 76\) and \(m\angle y = 14\), \(76 + 14=90

eq180\).

• For \(m\angle x = 113\) and \(m\angle y = 67\), \(113+67 = 180\).
• For \(m\angle x = 140\) and \(m\angle y = 190\), \(140 + 190=330

eq180\).

• For \(m\angle x = 180\) and \(m\angle y = 180\), \(180+180 = 360

eq180\).

Answer:

\(5\)