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: Analyze the relationship between ∠7 and ∠6

∠7 and ∠6 are vertical angles. Vertical angles are equal. So, \(m\angle6 = m\angle7=55^{\circ}\).

Step2: Analyze the relationship between ∠7 and ∠5

∠7 and ∠5 are supplementary angles (they form a linear pair). So, \(m\angle5=180^{\circ}-m\angle7 = 180^{\circ}- 55^{\circ}=125^{\circ}\).

Step3: Analyze the relationship between ∠1 and ∠4

∠1 and ∠4 are vertical angles. So, \(m\angle1 = m\angle4\). Also, \(m\angle1=180^{\circ}-m\angle2\), and since ∠2 and ∠7 are corresponding angles (assuming the two horizontal lines are parallel), \(m\angle2 = m\angle7 = 55^{\circ}\), then \(m\angle1=125^{\circ}\), \(m\angle4 = 125^{\circ}\), and \(m\angle1 + m\angle4=125^{\circ}+125^{\circ}=250^{\circ}\).

Step4: Analyze the relationship for option D

We know \(m\angle1 = 125^{\circ}\), \(m\angle6 = 55^{\circ}\), \(m\angle7 = 55^{\circ}\), \(m\angle4 = 125^{\circ}\). Then \(m\angle1 + m\angle6=125^{\circ}+55^{\circ}=180^{\circ}\) and \(m\angle7 + m\angle4=55^{\circ}+125^{\circ}=180^{\circ}\)

Answer:

A. \(m\angle6 = 55^{\circ}\), C. \(m\angle1 + m\angle4 = 250^{\circ}\), D. \(m\angle1 + m\angle6 = m\angle7 + m\angle4\)