Angle Sum of a Triangle
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Theory
The three angles inside any triangle add to \(180^\circ\). This is the angle sum of a triangle. To find a missing angle, add the angles you know and subtract from \(180^\circ\). This Year 7 maths page shows why the rule works and how to use it with right angles, isosceles triangles, straight lines and pronumerals.
For a triangle with angles \(a^\circ\), \(b^\circ\) and \(c^\circ\): \(a + b + c = 180\). Reason: angle sum of a triangle is \(180^\circ\).
Finding a missing angle. Write the equation with its reason, add the known angles, then subtract from \(180\): \(x + 70 + 45 = 180\), so \(x = 180 - 115 = 65\).
Special triangles. Right-angled: the right angle is \(90^\circ\), so the other two angles add to \(90^\circ\). Isosceles: the two base angles are equal, so call both \(x^\circ\). Equilateral: each angle is \(60^\circ\).
Why it is true. Draw a line \(DE\) through \(A\) parallel to \(BC\). Alternate angles on parallel lines are equal, so the angles of the triangle can be copied to \(A\). There they make a straight line, which is \(180^\circ\).
Other angle facts still work. Angles on a straight line add to \(180^\circ\), vertically opposite angles are equal, and angles on parallel lines (alternate, corresponding, co-interior) can give an angle inside the triangle first.
The rule, and what it gives for special triangles:
| Triangle | Fact |
|---|---|
| right-angled | the other two angles add to \(90^\circ\) |
| isosceles | base angles equal: \(x + x + \text{top} = 180\) |
| equilateral | each angle is \(60^\circ\) |
How to find a missing angle in a triangle
- Find any angle you need first (a straight line, a right angle, equal base angles).
- Write the equation: the three angles \(= 180\), with the reason.
- Add the known angles and subtract from \(180\).
- Check that the three angles add to \(180\).
How to solve when the angles have pronumerals
- Add the three angles and make them equal to \(180\).
- Collect like terms, then solve for the pronumeral.
- Substitute back to find each angle.
| \(x + 38 + 103\) | \(=\) | \(180\) |
| \(x + 141\) | \(=\) | \(180\) |
| \(x\) | \(=\) | \(180 - 141\) |
| \(\) | \(=\) | \(39\) |
\(x = 39\) (angle sum of a triangle is \(180^\circ\)). Check: \(39 + 38 + 103 = 180\).
| \(x + 61 + 58\) | \(=\) | \(180\) |
| \(x + 119\) | \(=\) | \(180\) |
| \(x\) | \(=\) | \(61\) |
\(x = 61\), so \(\angle ABC = 61^\circ\). Two angles are equal, so the triangle is isosceles.
The triangle is isosceles, so both base angles are \(x^\circ\).
| \(x + x + 52\) | \(=\) | \(180\) |
| \(2x\) | \(=\) | \(128\) |
| \(x\) | \(=\) | \(64\) |
\(x = 64\). Check: \(64 + 64 + 52 = 180\).
(i) If the \(30^\circ\) angle is between the two equal sides, find each base angle.
(ii) If the \(30^\circ\) angle is a base angle, find the third angle.
| \(180 - 30\) | \(=\) | \(150\) |
| \(150 \div 2\) | \(=\) | \(75\) |
| \(180 - 30 - 30\) | \(=\) | \(120\) |
(i) \(75^\circ\) each
(ii) \(120^\circ\)
| \(\angle ACB + 110^\circ\) | \(=\) | \(180^\circ\) |
| \(\angle ACB\) | \(=\) | \(70^\circ\) |
| \(\angle BAC + 70^\circ + 70^\circ\) | \(=\) | \(180^\circ\) |
| \(\angle BAC\) | \(=\) | \(40^\circ\) |
\(\angle BAC = 40^\circ\). The first step uses angles on a straight line; the second uses the angle sum of a triangle.
The wall, the ground and the ladder make a right-angled triangle, so the two other angles add to \(90^\circ\).
| \(90 - 63\) | \(=\) | \(27\) |
\(27^\circ\)
| \(2x + 3x + 55\) | \(=\) | \(180\) |
| \(5x\) | \(=\) | \(125\) |
| \(x\) | \(=\) | \(25\) |
\(x = 25\). The angles are \(50^\circ\), \(75^\circ\) and \(55^\circ\) (check: \(50 + 75 + 55 = 180\)). The largest is \(75^\circ\), so the triangle is acute-angled.
Common pitfalls
Frequently asked questions
What do the angles of a triangle add up to?
\(180^\circ\), for every triangle.
How do you find the third angle of a triangle?
Add the two angles you know and subtract from \(180\). For \(38^\circ\) and \(103^\circ\): \(180 - 141 = 39\), so the third angle is \(39^\circ\).
Why do the angles of a triangle add to 180 degrees?
Draw a line through one vertex parallel to the opposite side. Alternate angles copy the other two angles to that vertex, where the three angles make a straight line.
What are the angles of a right-angled isosceles triangle?
\(90^\circ\), \(45^\circ\) and \(45^\circ\).
Can a triangle have two right angles?
No. Two right angles already make \(180^\circ\), leaving nothing for the third angle.