Question

1. in the figure shown, lines f and g are parallel. select all statements that must be true.

a. angle 4 is congruent to angle 6.
b. m∠1 = m∠6
c. angle 1 and angle 7 are supplementary.
d. angle 1 is congruent to angle 5.
e. m∠3 = m∠6
f. m∠7 + m∠8 = 180°

Explanation:

Step1: Recall parallel - line angle relationships

When two parallel lines are cut by a transversal, alternate - interior angles are congruent, corresponding angles are congruent, and same - side interior angles are supplementary.

Step2: Analyze option a

Angle 4 and angle 6 are alternate - interior angles. By the alternate - interior angles theorem, when lines \(f\) and \(g\) are parallel, \(\angle4\cong\angle6\). So, option a is true.

Step3: Analyze option b

\(\angle1\) and \(\angle6\) are neither corresponding, alternate - interior, nor vertical angles. There is no relationship that makes \(m\angle1 = m\angle6\) when \(f\) and \(g\) are parallel. So, option b is false.

Step4: Analyze option c

\(\angle1\) and \(\angle7\) are same - side interior angles. By the same - side interior angles theorem, when \(f\) and \(g\) are parallel, \(\angle1\) and \(\angle7\) are supplementary, i.e., \(m\angle1 + m\angle7=180^{\circ}\). So, option c is true.

Step5: Analyze option d

\(\angle1\) and \(\angle5\) are corresponding angles. By the corresponding angles postulate, when \(f\) and \(g\) are parallel, \(\angle1\cong\angle5\). So, option d is true.

Step6: Analyze option e

\(\angle3\) and \(\angle6\) are neither corresponding, alternate - interior, nor vertical angles. There is no relationship that makes \(m\angle3 = m\angle6\) when \(f\) and \(g\) are parallel. So, option e is false.

Step7: Analyze option f

\(\angle7\) and \(\angle8\) are a linear pair. By the linear - pair postulate, for any two adjacent angles that form a straight line, \(m\angle7 + m\angle8 = 180^{\circ}\), regardless of whether the lines \(f\) and \(g\) are parallel or not. So, option f is true.

Answer:

A. Angle 4 is congruent to angle 6.
C. Angle 1 and angle 7 are supplementary.
D. Angle 1 is congruent to angle 5.
F. \(m\angle7 + m\angle8 = 180^{\circ}\)