Lines and Angles
88. The Crossing Lines · vertically opposite angles are always equal
The vertically opposite angles ∠BOC and ∠AOD are always equal.
When two lines intersect, they form two pairs of vertically-opposite angles. Vertically-opposite angles are always equal because each pair shares supplements along the same straight line. ∠AOC = ∠BOD and ∠AOD = ∠BOC.
What this lesson covers
Try to break it
Drag point C around the circle. See how the angles ∠BOC and ∠AOD change, but they always stay exactly equal to each other!
How you build it
Draw two intersecting lines.
- Draw a straight line and label its ends A and B.
- Draw another straight line crossing the first one. Mark the intersection as O and label the ends C and D.
The proof, step by step
Prove that the vertically opposite angles ∠BOC and ∠AOD are equal.
- AOB and COD are straight lines intersecting at O.
- ∠AOC + ∠BOC = 180° (Linear pair on line AOB)
- ∠AOC + ∠AOD = 180° (Linear pair on line COD)
- Therefore, ∠BOC = ∠AOD (Both equal to 180° - ∠AOC)
- Similarly, ∠AOC = ∠BOD.
Worked example
Two straight lines intersect at O. If ∠AOC = 65°, what is the measure of ∠BOD?
Vertically opposite angles are equal. Since ∠AOC and ∠BOD are vertically opposite, ∠BOD = ∠AOC = 65°.
- 25°
- 65° — correct
- 105°
- 115°