Question

Angles are created when lines a and b are cut by a transversal, t. These angles are labeled in the diagram.

The lines a and b are horizontal and angle 1 and angle 2 are on the either sides of the transversal and above line a, angles 3 and 4 are on the either sides of the transversal and below line a, angles 5 and 6 are on the either sides of the transversal and above line b, angles 7 and 8 are on the either sides of the transversal and below line b.

Select the claim that is true about lines a and b for all cases.

A.
If lines a and b are cut by a transversal, t, such that
m
∠
1
=
m
∠
5
,
m
∠
2
=
m
∠
6
,
m
∠
3
=
m
∠
7
, and
m
∠
4
=
m
∠
8
, then the lines must be parallel.

B.
If lines a and b are cut by a transversal, t, such that
m
∠
1
+
m
∠
5
=
180
°
,
m
∠
2
+
m
∠
6
=
180
°
,
m
∠
3
+
m
∠
7
=
180
°
, and
m
∠
4
+
m
∠
8
=
180
°
, then the lines must be parallel.

C.
If lines a and b are cut by a transversal, t, such that
m
∠
1
=
m
∠
5
,
m
∠
2
=
m
∠
6
,
m
∠
3
=
m
∠
7
, and
m
∠
4
=
m
∠
8
, then the lines must be perpendicular.

D.
If lines a and b are cut by a transversal, t, such that
m
∠
1
+
m
∠
5
=
90
°
,
m
∠
2
+
m
∠
6
=
90
°
,
m
∠
3
+
m
∠
7
=
90
°
, and
m
∠
4
+
m
∠
8
=
90
°
, then the lines must be perpendicular.

Answer 1

Answer: A.

Reason: A is the converse of the Corresponding Angles Postulate — if a transversal makes each pair of corresponding angles equal, the two lines are parallel.

Why the others fail:
- B is false: corresponding angles summing to 180° does not force the lines to be parallel (they would be parallel only if each corresponding angle were equal, which would require each to be 90°).
- C is false: equality of corresponding angles does not imply the angles are 90°, so the lines need not be perpendicular.
- D is false: corresponding angles being complementary (sum 90°) does not force the two lines to be perpendicular.