Vertical Angles

Geometry · Axiom Academy

When two lines cross, the angles directly across from each other are always equal — see why, and put it to work. Two lines cross at a single point and carve the plane into four angles . Number them ∠1, ∠2, ∠3, ∠4 going around. The pairs that face each other across the crossing — ∠1 with ∠3, and ∠2 with ∠4 — are the vertical angles . ∠1 and ∠3 are one vertical pair (red) ∠2 and ∠4 are the other vertical pair (blue) The reason comes from the straight line . Two angles that sit side by side along a straight line are supplementary — they add to 180°. The sweep below shows the straight angle splitting, so you can see each adjacent pair total a half-turn. ∠1 + ∠2 = 180° and ∠2 + ∠3 = 180°, because each pair lies along a straight line. Both equal 180°, so ∠1 + ∠2 = ∠2 + ∠3. Cancel ∠2 and you are left with ∠1 = ∠3. The four angles complete one full rotation around the point: ∠1 + ∠2 + ∠3 + ∠4 = 360°. Vertical angles are congruent: ∠1 ≅ ∠3 and ∠2 ≅ ∠4 — always, for any crossing. If two angles are vertical angles, then they are congruent. Two facts do all the lifting: a vertical angle is equal to its partner, and an adjacent angle is 180° minus it. The animation tracks one moving angle and its 180° supplement together; the worked examples below use the same two moves. If ∠1 = 70°, find ∠3. Since ∠1 and ∠3 are vertical angles, they are congruent, so ∠3 = 70° . If ∠1 = 70°, find ∠2. Here ∠1 and ∠2 are adjacent, hence supplementary, so ∠2 = 180° − 70° = 110° .

This is the written version of the interactive lesson above. See the full Geometry course.