Section 02
The relation between angle pairs when the two lines are parallel
When a straight line intersects two parallel straight lines, any pair of angles formed by the intersection is either equal in measure or supplementary. That single sentence powers every calculation in this lesson.
The four relations
Rule 1
Corresponding angles are equal
- m(∠1) = m(∠5)
- m(∠2) = m(∠6)
- m(∠3) = m(∠7)
- m(∠4) = m(∠8)
Rule 2
Alternating interior angles are equal
- m(∠3) = m(∠6)
- m(∠4) = m(∠5)
Rule 3
Alternating exterior angles are equal
- m(∠1) = m(∠8)
- m(∠2) = m(∠7)
Rule 4
Interior angles on the same side are supplementary
- m(∠3) + m(∠5) = 180°
- m(∠4) + m(∠6) = 180°
Example 1 — find the angle marked ?
In each figure the two horizontal lines are parallel. Find the measure of the marked angle and state the reason.
Figure 1 — given 70°
Answer: 70°
AB ∥ CD and GF is a transversal, so m(∠GEB) = m(∠EFD) = 70° — corresponding angles are equal in measure.
Figure 2 — given 52°
Answer: 52°
AB ∥ CD and EF is a transversal, so m(∠BEF) = m(∠EFC) = 52° — alternating interior angles are equal in measure.
Figure 3 — given 115°
Answer: 115°
IH ∥ ON and JK is a transversal, so m(∠JLI) = m(∠NMK) = 115° — alternating exterior angles are equal in measure.
Figure 4 — given 101°
Answer: 79°
BA ∥ CD and BC is a transversal, so m(∠B) + m(∠C) = 180° — interior angles on the same side are supplementary. Hence m(∠C) = 180° − 101° = 79°.
Try it yourself
In each of the following, two parallel lines are cut by a transversal. Name the angle pair first, then decide whether the angles are equal or supplementary, and only then compute. Practising the naming step is what makes exam questions fast.
- Given 63° at an exterior position, find the corresponding interior angle.
- Given 118° interior, find the co-interior angle on the same side.
- Given 47° interior, find its alternating interior partner.
Answers: 63° · 62° · 47°