Question

if ( mangle7 = 55^{circ} ), which of the following statements are true? select all that apply
a ( mangle6 = 55^{circ} )
b ( mangle5 = 135^{circ} )
c ( mangle1 + mangle4 = 250^{circ} )
d ( mangle1 + mangle6 = mangle7 + mangle4 )

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. Since \(\angle7\) and \(\angle5\) are vertical angles, \(m\angle5 = m\angle7=55^{\circ}\). Also, \(\angle7\) and \(\angle6\) are supplementary (\(\angle7+\angle6 = 180^{\circ}\)), so \(m\angle6=180 - 55=125^{\circ}\).

Step2: Use corresponding - angle and vertical - angle properties

\(\angle1\) and \(\angle5\) are corresponding angles. Since \(m\angle5 = 55^{\circ}\), \(m\angle1 = 55^{\circ}\). \(\angle4\) and \(\angle1\) are supplementary (\(\angle1+\angle4 = 180^{\circ}\)), so \(m\angle4=180 - 55 = 125^{\circ}\). Then \(m\angle1+m\angle4=55 + 125=180^{\circ}
eq250^{\circ}\).

Step3: Calculate \(m\angle1 + m\angle6\) and \(m\angle7 + m\angle4\)

\(m\angle1 + m\angle6=55+125 = 180^{\circ}\). \(m\angle7 + m\angle4=55 + 125=180^{\circ}\).

Answer:

D. \(m\angle1 + m\angle6=m\angle7 + m\angle4\)