Vertical Angles: Definition, Theorem & Examples

Definitionnoun phrase

What are vertical angles in math?

Vertical angles are the pair of opposite angles formed when two lines intersect. They share a vertex but no sides, and they are always equal in measure. Two crossing lines make two pairs of vertical angles: if one angle is 120°, the angle across from it is also 120°.

In Spanish: ángulos opuestos por el vértice
Also called: vertically opposite angles, opposite angles
Grade level: 6-8

Grade 7Grade 8GeometryCCSS 7.G.B.5CCSS HSG-CO.C.9GEDSATACTHiSETTABEASVABACCUPLACER

Vertical Angles in a picture

Look across the crossing point, not beside it: the angle directly opposite is the vertical angle, and it always has the same measure.

Formal definition and notation

When two lines intersect at a point O, they form four angles. Two of these angles are vertical angles when their sides are opposite rays: the sides of one angle extend through O to form the sides of the other. The Vertical Angles Theorem states that vertical angles are congruent.

Symbol	Read it as	Meaning
∠ 1 ≅ ∠ 3	angle 1 is congruent to angle 3	the two vertical angles have equal measures

Key facts

Vertical here means sharing a vertex, not standing up straight. Vertical angles can lie side to side across the page.
Two crossing lines make 4 angles: two pairs of equal vertical angles, and each angle is supplementary to its two neighbors.
If one of the four angles is 90°, all four are 90°, and the lines are perpendicular.
The four angles around the crossing point add to 360°.
Vertical angles are a classic first proof in high school geometry, because the argument uses only linear pairs.

Vertical Angles Theorem

m∠ 1 = m∠ 3 and m∠ 2 = m∠ 4

∠ 1, ∠ 3: one pair of opposite angles at the crossing
∠ 2, ∠ 4: the other opposite pair

Why it works: ∠ 1 + ∠ 2 = 180° and ∠ 3 + ∠ 2 = 180° (each pair forms a straight line), so ∠ 1 = ∠ 3.

Vertical Angles examples, step by step

Example 1: Read the opposite angle

Warm-up

Two lines cross. One of the angles measures 47°. Find the other three angles.

The angle opposite 47° is its vertical angle, so it is 47°.
Each neighbor forms a straight line with 47°: 180° − 47° = 133°.
The two neighbors are opposite each other, so both are 133°.

Answer: The angles are 47°, 133°, 47° and 133°.

Example 2: Solve for x

Core

Vertical angles measure (6x − 15)° and (4x + 21)°. Find x and the angle.

Vertical angles are equal: 6x − 15 = 4x + 21.
Subtract 4x: 2x − 15 = 21.
Add 15: 2x = 36, so x = 18.
Angle: 6(18) − 15 = 93°. Check: 4(18) + 21 = 93.

Answer: x = 18, and each vertical angle measures 93°.

Example 3: Mixed pair

Test-style

Two lines intersect. One angle is (2y + 10)°, and the angle next to it is (3y − 5)°. What is the measure of the angle vertical to the first one?

Neighboring angles form a linear pair: 2y + 10 + 3y − 5 = 180.
5y + 5 = 180, so y = 35.
First angle: 2(35) + 10 = 80°.
Its vertical angle is equal, so it is also 80°.

Answer: The vertical angle measures 80°.

Common mistakes

✗ MistakeSetting neighboring angles equal: 2x = 3x − 40 for angles that sit side by side.

✓ InsteadNeighbors on a line are supplementary: 2x + 3x − 40 = 180, so x = 44.

Only angles directly across from each other are equal. Side-by-side angles at a crossing add to 180.

✗ MistakeThinking vertical angles must point up and down.

✓ InsteadAngles opposite each other across any crossing point are vertical, even if they open left and right.

Vertical comes from vertex, the shared corner point, not from the up-down direction.

✗ MistakeCalling the two angles of a single bent line vertical angles.

✓ InsteadVertical angles need two full lines that cross, giving four angles.

With only one line and one ray you get a linear pair, which has no opposite angle.

Not to be confused with

Vertical Anglesvs.Adjacent angles

Adjacent angles share a vertex and a side and sit next to each other. At a crossing, adjacent angles add to 180°; vertical angles are equal.

Vertical Anglesvs.Supplementary angles

Supplementary angles add to 180°. Vertical angles are equal. At a crossing of two lines with 70° and 110° angles, 70° and 70° are vertical; 70° and 110° are supplementary.

Quick check: 3 questions

Question 1 of 3

Two lines cross. One angle is 64°. What is the angle directly opposite it?

Show answer and explanation

Answer: C. The opposite angle is a vertical angle, so it equals 64°. 116° is the neighboring angle (the supplement), 26° is the complement, and 296° subtracts from 360.

Question 2 of 3

Vertical angles measure (5x + 4)° and (7x − 20)°. What is x?

Show answer and explanation

Answer: B. Set them equal: 5x + 4 = 7x − 20 gives 24 = 2x, so x = 12 (each angle is 64°). 1613 comes from adding the angles to 180 instead of setting them equal, 8 subtracts 4 from 20 instead of adding, and 2 adds 5x and 7x instead of subtracting.

Question 3 of 3

At a crossing of two lines, one angle is 90°. What are the other three?

Show answer and explanation

Answer: A. The opposite angle is 90° (vertical), and each neighbor is 180° − 90° = 90°. So all four are right angles and the lines are perpendicular. The other choices ignore that neighbors must total 180 and opposites must match.

Vertical Angles: frequently asked questions

Are vertical angles always congruent?

Yes. Whenever two straight lines cross, each angle and the angle across from it have the same measure. The reason: both angles form a straight line with the same neighboring angle, so each equals 180° minus that neighbor. This result is called the Vertical Angles Theorem, and you may cite it as a reason in proofs.

Why are they called vertical angles?

The name comes from vertex, the shared corner point, not from the direction up and down. Vertical angles share a vertex and point away from each other across it. That is why many textbooks also call them vertically opposite angles. A pair that opens left and right is still a pair of vertical angles.

Can vertical angles be supplementary?

Only when both are 90°. Vertical angles are equal, so if they also add to 180°, each must be 90°. That happens exactly when the two lines are perpendicular. In every other crossing, the vertical angles are both acute or both obtuse, and their sum is not 180°.

Practice and tools

Effortless Math practicePractice on Effortless Math

Sources

Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 7.G.B.5, CCSS HSG-CO.C.9

Cite this entry

Effortless Math Team. “Vertical Angles.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/vertical-angles/

Flashcard

TermVertical Anglesacross from each other at a crossing

DefinitionThe opposite angles formed by two intersecting lines. They are always equal: if one is 70°, the angle across is 70°.

Study the full deck

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Vertical Angles

What are vertical angles in math?

Vertical Angles in a picture

Formal definition and notation

Key facts

Vertical Angles Theorem

Vertical Angles examples, step by step

Example 1: Read the opposite angle

Example 2: Solve for x

Example 3: Mixed pair

Common mistakes

Not to be confused with

Quick check: 3 questions

Vertical Angles: frequently asked questions

Are vertical angles always congruent?

Why are they called vertical angles?

Can vertical angles be supplementary?

Related terms

Practice and tools

Sources

Flashcard

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