Any number of angles sitting on one side of a straight line always add up to exactly 180°. This is sometimes called a "straight angle".
Where does 180° come from? A full turn (rotating all the way round back to where you started) is 360°. A straight line is exactly half of a full turn — if you stand facing one direction and turn to face the exact opposite direction, you have turned through half of 360°, which is 180°. Any angles that sit on one side of a straight line, added together, must fill that same half-turn — so however many angles there are, they always add up to 180°.
Example 1 — Two angles on a straight line are 65° and 47°. Find the third angle.
Step 1
Write the rule: angles on a straight line sum to 180°
Step 2
Add the known angles: \(65° + 47° = 112°\)
Step 3
Subtract from 180°: \(180° - 112° = \mathbf{68°}\)
Example 2 — Angles on a straight line are \(3x\), \(2x\), and \(x\). Find \(x\).
Step 1
Write the equation: \(3x + 2x + x = 180°\)
Step 2
Simplify: \(6x = 180°\)
Step 3
Divide: \(x = 180° \div 6 = \mathbf{30°}\)
Tip: Straight line = 180°. Always check your angles add to exactly 180°.
✓ Quick Check 1 — Angles on a Straight Line
Question 1 of 10
Angles Around a Point = 360°; Vertically Opposite Angles
Angles around a point = 360° | Vertically opposite angles are equal
When two straight lines cross, they make four angles. Opposite angles (called vertically opposite) are always equal. All four angles together sum to 360°.
Example 1 — Two lines cross. One angle is 72°. Find the other three angles.
Step 1
Opposite angle = 72° (vertically opposite angles are equal)
Step 2
Adjacent angles: \(180° - 72° = 108°\) (angles on a straight line)
Step 3
All four angles: 72°, 108°, 72°, 108°
Step 4
Check: \(72° + 108° + 72° + 108° = 360°\) ✓
Example 2 — Four angles around a point: 90°, 95°, 80°, and \(x\). Find \(x\).
Step 1
Write the equation: \(90° + 95° + 80° + x = 360°\)
Step 2
Add known angles: \(265° + x = 360°\)
Step 3
Subtract: \(x = 360° - 265° = \mathbf{95°}\)
Vertically opposite angles are formed when two straight lines cross. They are always equal. This is sometimes called "X angles" because of the X shape formed at the crossing point.
✓ Quick Check 2 — Angles Around a Point and Vertically Opposite
Question 1 of 10
Combining Rules: Multi-Step Problems
Many geometry problems require you to use both the straight-line rule and the vertically opposite rule together. Work step by step, labelling each angle as you find it.
Example 1 — Line AB is crossed by line CD at P. Angle APC = 55°. Find angles APD, BPC, and BPD.
Step 1
APD = 180° − 55° = 125° (APC and APD are on a straight line)
Step 2
BPC = 55° (vertically opposite to APC)
Step 3
BPD = 125° (vertically opposite to APD)
Step 4
Check: \(55° + 125° + 55° + 125° = 360°\) ✓
Example 2 (Real-life) — A ship sails on a bearing of 060°. It turns to face the opposite direction. What bearing is it now facing?
Step 1
Opposite direction = current bearing + 180°
Step 2
\(060° + 180° = 240°\)
Step 3
New bearing = 240°
Warning: When lines cross, you get 4 angles. Use vertically opposite AND straight-line rules together — don't just guess.
✓ Quick Check 3 — Multi-Step and Real-Life Problems
Question 1 of 10
Practice: Angles Practice
Work out the missing angle in each problem. Type your answer (numbers only, in degrees), then click Check All.
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Question
Answer
Result
1
Angles on a line: 75°, 60°, and ?
2
Angles on a line: \(2x\) and \(x\). Find \(x\).
3
Around a point: 90°, 90°, 90°, and ?
4
Around a point: 100°, 80°, 120°, and ?
5
Two crossing lines, one angle = 40°. Vertically opposite angle = ?
6
Two crossing lines, one angle = 40°. Adjacent angle = ?
7
Angles on a line: 110° and ?
8
Around a point: 150°, 100°, 70°, and ?
9
Angles on a line: \(3x\), \(2x\), \(x\). Find \(x\).
10
Around a point: 95°, 85°, 100°, and ?
⚠ Things to Watch Out For
Angles on a straight line sum to 180°, not 360°: A straight line is half a full turn (180°). Use 360° only for angles around a point.
Vertically opposite angles — confusing adjacent and opposite: Vertically opposite angles are the PAIR across from each other at a crossing point. Adjacent angles at a crossing point are supplementary (add to 180°), not equal.
Setting up the equation incorrectly: "Find x if 3x + 45 + x = 180". Collect like terms first: 4x + 45 = 180 → 4x = 135 → x = 33.75°. Always form an equation and solve algebraically.
Assuming all angles around a point are equal: They sum to 360° but are not necessarily equal. Only state they are equal if there is a specific reason (e.g. a regular arrangement).
Not checking the answer: Substitute your answer back into all the angles and verify they sum to 180° or 360°.