Vertically Opposite Angles

Learn that vertically opposite angles are always equal

Understanding Vertically Opposite Angles

When two straight lines intersect (cross each other), they form 4 angles. The angles that are opposite each other are called vertically opposite angles.

Key Rule:

Vertically opposite angles are EQUAL

Two Lines Intersecting

When two lines cross: ∠a = ∠c = 115° and ∠b = ∠d = 65°

Vertically opposite pairs:

$\angle a = \angle c = 115°$ and $\angle b = \angle d = 65°$

Identifying Vertically Opposite Pairs

When two lines intersect, there are 2 pairs of vertically opposite angles:

Angles that are opposite each other (not next to each other)
The two angles in each pair are equal

Example: Finding the pairs

∠w = ∠y (first pair) and ∠x = ∠z (second pair)

Pair 1: $\angle w = \angle y$ and Pair 2: $\angle x = \angle z$

Finding Unknown Angles

You can use two properties together to find unknown angles:

Vertically opposite angles are equal
Angles on a straight line sum to 180°

Worked Example: Find angles b, c and d

Two straight lines cross. One angle is 53°. Find ∠b, ∠c and ∠d.

Solution:

Step 1: Find ∠c (vertically opposite to 53°)

$\angle c = 53°$ (vertically opposite angles are equal)

Step 2: Find ∠b (on a straight line with 53°)

$\angle b = 180° - 53° = 127°$

Step 3: Find ∠d (vertically opposite to ∠b)

$\angle d = 127°$

Answer: $\angle b = 127°$ , $\angle c = 53°$ , $\angle d = 127°$

Practice Problems

Practice 1: Find angle x

Two straight lines cross. Find ∠x.

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∠x is NOT vertically opposite to 58°. It is next to it.

∠x and 58° are angles on a straight line:

$\angle x = 180° - 58° = 122°$

Practice 2: Find angles b, c and d

Two straight lines cross. One angle is 75°. Find ∠b, ∠c and ∠d.

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$\angle c = 75°$ (vertically opposite to 75°)

$\angle b = 180° - 75° = 105°$ (angles on straight line)

$\angle d = 105°$ (vertically opposite to ∠b)

Answer: ∠b = 105°, ∠c = 75°, ∠d = 105°

Practice 3: Find angles p, q and r

Two straight lines cross. One angle is 42°. Find ∠p, ∠q and ∠r.

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$\angle q = 42°$ (vertically opposite to 42°)

$\angle p = 180° - 42° = 138°$ (angles on straight line)

$\angle r = 138°$ (vertically opposite to ∠p)

Answer: ∠p = 138°, ∠q = 42°, ∠r = 138°

Key Takeaways

When two lines intersect, they form 4 angles
Vertically opposite angles are always equal
There are 2 pairs of vertically opposite angles
You can combine this with "angles on a straight line = 180°" to find all angles