Geometry & Measures
Grade 1-3Angle Facts & Rules
Angle facts are the building blocks for almost every other topic in geometry, from parallel lines to circle theorems, which is why examiners expect you to recall and apply them quickly and without a calculator. This lesson covers angles on a straight line, angles around a point, vertically opposite angles, and the angle sums of triangles and quadrilaterals.
Written by Asad, Co-Founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Angles on a straight line and around a point
Angles on a straight line always add up to 180°. If you know all but one of the angles on a straight line, subtract the ones you know from 180° to find the missing one.
Angles around a point always add up to 360°, whether there are two, three, four or more angles meeting there. The same approach applies: add up the angles you know and subtract from 360°.
A straight line has two angles of 110° and x. Since they sum to 180°, x = 180 - 110 = 70°. Four angles of 100°, 90°, 120° and x meet at a point, so x = 360 - 100 - 90 - 120 = 50°.
Vertically opposite angles
When two straight lines cross, they form two pairs of vertically opposite angles, and each pair is equal. You can spot vertically opposite angles because they are directly across the crossing point from each other, forming an X shape.
Two lines cross so that one angle is 65° and the angle directly opposite it is x. Since vertically opposite angles are equal, x = 65°, with no calculation needed.
Angles in triangles
The three angles in any triangle always add up to 180°. In an isosceles triangle, the two base angles opposite the equal sides are also equal to each other.
A triangle has angles of 50°, 60° and x, so x = 180 - 50 - 60 = 70°. An isosceles triangle has a top angle of 40°, so each base angle is (180 - 40) ÷ 2 = 70°.
Angles in quadrilaterals
The four angles in any quadrilateral always add up to 360°. This follows from splitting the quadrilateral into two triangles, each contributing 180°.
A quadrilateral has angles of 95°, 80°, 120° and x, so x = 360 - 95 - 80 - 120 = 65°.
Worked Examples
Three exam-style questions, fully solved.
The diagram shows a straight line with two angles, 55° and x, and a third angle of 70° also on the line. Find the size of angle x.
Easy- 1.Angles on a straight line sum to 180°: x + 55 + 70 = 180
Answer: x = 55°
Triangle ABC is isosceles, with AB = AC. Angle A = 40°. Find the size of angle B.
Medium- 1.The base angles of an isosceles triangle are equal, so angles B and C are equal
- 2.Angles in a triangle sum to 180°, so B and C together make 180 - 40 = 140°
- 3.Divide this equally between the two base angles: 140 ÷ 2
Answer: Angle B = 70°
In a triangle, two of the angles are 80° and 60°. The third angle of the triangle is also one of three angles meeting at a point, together with an angle of 150° and angle y. Find the size of angle y.
Hard- 1.Find the third angle of the triangle: 180 - 80 - 60 = 40°
- 2.This 40° angle is one of three angles meeting at a point with 150° and y, so they sum to 360°: 40 + 150 + y = 360
Answer: y = 170°
Avoid These
The most common mistakes students make.
Using 360° instead of 180° for angles on a straight line, or the other way round for angles around a point.
Assuming two angles that look equal in a diagram are vertically opposite, when they are actually adjacent on a straight line instead.
In an isosceles triangle, forgetting to divide by 2 after subtracting the known angle from 180°, giving the sum of both base angles instead of one of them.
Mixing up the angle sum of a quadrilateral (360°) with the angle sum of a triangle (180°).
Not showing the angle fact used as a reason, such as "angles on a straight line sum to 180°", which can cost a mark even when the final numerical answer is correct.
FAQ
Questions parents and students ask.
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