Question

1. select all true statements if n || m.

a. ( mangle2 = 60 )
b. ( mangle3 = 100 )
c. ( mangle2 + mangle4 = 80 )
d. ( mangle2 + mangle3 = 80 )
e. ( mangle2 = 20 )

Explanation:

Step1: Find \(m\angle1\)

Since \(\angle1\) and the \(60^{\circ}\) angle are supplementary (linear - pair), \(m\angle1 = 180^{\circ}-60^{\circ}=120^{\circ}\).

Step2: Use the property of parallel lines (\(n\parallel m\)) and the triangle - angle - sum theorem

Because \(n\parallel m\), the alternate - interior angle property gives that the sum of the angles in the triangle formed (with angles \(20^{\circ}\), \(\angle2\), and \(\angle3\)) and the relationship with \(\angle1\).
We know that \(\angle2 = 20^{\circ}\) (alternate - interior angles for the transversal cutting \(n\) and \(m\)).
\(\angle4=60^{\circ}\) (alternate - interior angles for the transversal cutting \(n\) and \(m\)).
Using the angle - sum property of a triangle (\(m\angle2 + m\angle3+m\angle1=180^{\circ}\)), substituting \(\angle2 = 20^{\circ}\) and \(m\angle1 = 120^{\circ}\), we get \(20^{\circ}+m\angle3 + 120^{\circ}=180^{\circ}\), so \(m\angle3=40^{\circ}\).
Now check each option:

• Option A: \(m\angle2 = 20^{\circ}

eq60^{\circ}\), so A is false.

• Option B: \(m\angle3 = 40^{\circ}

eq100^{\circ}\), so B is false.

• Option C: \(m\angle2 + m\angle4=20^{\circ}+60^{\circ}=80^{\circ}\), so C is true.
• Option D: \(m\angle2 + m\angle3=20^{\circ}+40^{\circ}=60^{\circ}

eq80^{\circ}\), so D is false.

• Option E: \(m\angle2 = 20^{\circ}\), so E is true.

Answer:

C. \(m\angle2 + m\angle4 = 80\), E. \(m\angle2 = 20\)