Fundamental Concepts
19. The Crossing Lines · vertically opposite angles are always equal
The vertically opposite angles (∠AOC and ∠BOD) are always equal.
When two lines intersect, they form two pairs of vertically-opposite angles. Let O be the point of intersection of lines AB and CD. Vertically-opposite angles are always equal because each pair shares two straight-line supplements. Drag A or C — ∠AOC always equals ∠BOD, and ∠AOD always equals ∠BOC.
What this lesson covers
Try to break it
What if you drag A and C until the lines cross at a perfect cross (90°)? Do the vertically opposite angles still match?
How you build it
Draw two intersecting lines at O.
- Place point A.
- Place point B. Points A and B set the first line.
- Draw the line through A and B.
- Place point C, off the line AB.
- Place point D. Points C and D set the second line.
- Draw the line through C and D. Where CD crosses AB is the point O — the two lines form vertically opposite angles.
The proof, step by step
Prove ∠AOC is equal to ∠BOD.
- ∠AOC and ∠AOD form a linear pair on line CD, so ∠AOC + ∠AOD = 180°.
- ∠BOD and ∠AOD form a linear pair on line AB, so ∠BOD + ∠AOD = 180°.
- Since both sums equal 180°, ∠AOC must equal ∠BOD.
Worked example
Two straight lines intersect at point O. If one of the vertically opposite angles measures 65°, what is the measure of its vertically opposite angle?
Vertically opposite angles are always equal. Therefore, the angle opposite to 65° is also 65°.
- 25°
- 65° — correct
- 115°
- 180°