Lines & Angles • Topic 3 of 3

Angles Around a Point

The full turn around any single point is 360°, so all the angles meeting at a point add to 360°. A closely related fact you will use just as often: the angles on one side of a straight line add to 180° (a straight angle). These two rules let you find a missing angle whenever several angles share a vertex — a common GRE setup with rays fanning out from one point, or several lines crossing at the same spot. When the angles are given as a ratio, split 360° (or 180°) into the total number of parts and multiply. As always, solve from the numbers, because the diagram is not necessarily drawn to scale and a "wide-looking" angle may be the small one.

Four rays from a single point dividing the full turn into angles that sum to 360 degreesAngles around a pointa°b°c°d°a + b + c + d = 360°

✅ Solved examples

1. Three angles meet at a point and measure 120°, 150° and x°. Find x.
Angles around a point sum to 360°: x = 360 − (120 + 150) = 360 − 270 = 90°.
2. Four angles around a point are in the ratio 1 : 2 : 3 : 4. Find the largest angle.
Total parts = 1 + 2 + 3 + 4 = 10, and 360° / 10 = 36° per part. The largest = 4 × 36° = 144°.
3. Three angles lie on one side of a straight line and measure (2x)°, (3x)° and x°. Find x.
Angles on a straight line sum to 180°: 2x + 3x + x = 180 → 6x = 180 → x = 30.
4. Around a point, the angles are x°, 2x°, 3x° and 90°. Find x.
Sum around a point is 360°: x + 2x + 3x + 90 = 360 → 6x = 270 → x = 45.

✏️ Practice — try these, take hints as needed

1. Four angles around a point are 90°, 90°, 100° and x°. Find x.
They sum to 360°.
Add the three known angles.
360 − 280.
80°
2. Two angles lie on a straight line; one is 65°. Find the other.
Angles on a line sum to 180°.
Subtract from 180.
180 − 65.
115°
3. Three angles around a point are in the ratio 2 : 3 : 4. Find the largest angle.
Total parts = 9.
360 / 9 = 40° per part.
Largest = 4 parts.
160°
4. Three angles on a straight line measure (x + 10)°, (2x)° and (x − 10)°. Find x.
They sum to 180°.
The +10 and −10 cancel.
4x = 180.
x = 45
5. Four angles around a point are x°, (x + 20)°, (x + 40)° and (x + 60)°. Find the largest angle.
Sum = 360°.
4x + 120 = 360.
x = 60, largest = x + 60.
120°

📝 Topic test — 8 questions

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