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Angles in a Triangle & Quadrilateral

The angles inside a triangle always total 180 degrees, and inside any four-sided shape they total 360. With those two totals you can hunt down almost any missing angle.

MYP 4Standard MathsGeometryCriteria A · C · D~10 min read

Written by Fayek Jawad Haider · MYP student

Angles in a triangle

Whatever its shape, the three interior angles of any triangle add up to \(180^\circ\). This is the single most useful fact in the whole topic.

\[ a + b + c = 180^\circ \]
Worked example

A triangle has angles of \(43^\circ\) and \(78^\circ\). Find the third angle.

1
The three angles add to \(180^\circ\): \(43 + 78 + c = 180\).
2
Add the two known angles: \(43 + 78 = 121\).
3
Subtract from \(180\): \(c = 180 - 121\).
\(c = 59^\circ\)

Sanity check the size

Every angle in a triangle must be less than \(180^\circ\), and the three must total exactly \(180^\circ\). If your missing angle comes out at zero or negative, one of the given angles has been misread.

Using the type of triangle

The name of a triangle hands you extra angle facts for free.

TriangleAngle fact
Equilateralall three angles equal \(60^\circ\)
Isoscelesthe two base angles are equal
Right-angledone angle is \(90^\circ\)
Worked example

An isosceles triangle has an apex angle of \(40^\circ\). Find each of the two equal base angles.

1
The two base angles are equal. Call each one \(x\), so \(40 + x + x = 180\).
2
The two base angles together take up \(180 - 40 = 140^\circ\).
3
Share that equally between them: \(x = 140 \div 2\).
Each base angle is \(70^\circ\)

Angles in a quadrilateral

A quadrilateral is any shape with four straight sides. Split it into two triangles with a single diagonal and you get \(2 \times 180 = 360\), so the four interior angles always add up to \(360^\circ\).

\[ a + b + c + d = 360^\circ \]
Worked example

A quadrilateral has angles of \(100^\circ\), \(85^\circ\) and \(95^\circ\). Find the fourth angle.

1
The four angles add to \(360^\circ\): \(100 + 85 + 95 + d = 360\).
2
Add the knowns: \(100 + 85 + 95 = 280\).
3
Subtract: \(d = 360 - 280\).
\(d = 80^\circ\)

Putting it together

Harder questions chain the rules. You might use the straight-line rule to find one angle inside a triangle, then the triangle sum to finish. Work one angle at a time and write the reason beside each, so a marker can follow your route.

Where this is assessed

Applying the correct angle sum is Criterion A (Knowing and understanding). Setting out each step with a reason is Criterion C (Communicating). Multi-step problems that mix triangle, quadrilateral and straight-line rules sit in Criterion D (Applying maths in context).

Check yourself

1. A triangle has angles \(55^\circ\) and \(65^\circ\). Find the third angle. +

The angles add to \(180^\circ\): \(180 - 55 - 65 = \mathbf{60^\circ}\).

2. An isosceles triangle has an apex angle of \(50^\circ\). Find each base angle. +

The two equal base angles share what is left of \(180^\circ\): \((180 - 50) \div 2 = 130 \div 2 = \mathbf{65^\circ}\) each.

3. A quadrilateral has angles \(90^\circ\), \(90^\circ\) and \(130^\circ\). Find the fourth angle. +

The four angles add to \(360^\circ\): \(360 - 90 - 90 - 130 = \mathbf{50^\circ}\).

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