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: Vertical angles

Vertical angles are equal. Since \(\angle7\) and \(\angle5\) are vertical angles, \(m\angle5 = m\angle7=55^{\circ}\). \(\angle5\) and \(\angle6\) are supplementary (\(\angle5+\angle6 = 180^{\circ}\)), so \(m\angle6=180 - 55=125^{\circ}\). \(\angle7\) and \(\angle8\) are supplementary (\(\angle7+\angle8 = 180^{\circ}\)), so \(m\angle8 = 125^{\circ}\).

Step2: Corresponding angles and linear - pair

\(\angle1\) and \(\angle5\) are corresponding angles (assuming the two horizontal lines are parallel). So \(m\angle1=m\angle5 = 55^{\circ}\). \(\angle1\) and \(\angle2\) are supplementary (\(\angle1+\angle2=180^{\circ}\)), so \(m\angle2 = 125^{\circ}\). \(\angle3\) and \(\angle1\) are vertical angles, so \(m\angle3=m\angle1 = 55^{\circ}\). \(\angle4\) and \(\angle2\) are vertical angles, so \(m\angle4=m\angle2=125^{\circ}\).

Step3: Check each option

• Option A:

Since \(\angle7\) and \(\angle5\) are vertical angles (\(m\angle5 = 55^{\circ}\)), and \(\angle5+\angle6=180^{\circ}\) (linear - pair), \(m\angle6=180 - 55=125^{\circ}\). So \(m\angle6
eq55^{\circ}\).

• Option B:

Since \(\angle7\) and \(\angle5\) are vertical angles, \(m\angle5 = m\angle7 = 55^{\circ}
eq135^{\circ}\).

• Option C:

\(m\angle1 = 55^{\circ}\), \(m\angle4=125^{\circ}\). Then \(m\angle1 + m\angle4=55 + 125=180^{\circ}
eq250^{\circ}\).

• Option D:

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

Answer:

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