Geometric reasoning and simple proofs

Mathematicians do not just find angles: they explain why. Here you will give reasons and follow simple proofs.

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Angle facts as reasons

Each step of a solution needs a reason:
Angles on a straight line add to 180°. Angles around a point add to 360°.
Vertically opposite angles are equal.
Alternate angles are equal. Corresponding angles are equal. Co-interior angles add to 180°.
Angles in a triangle add to 180°; in a quadrilateral, 360°.

Proof: angles in a triangle add to 180°

Draw a line through the top vertex, parallel to the base.
The two angles next to the top angle are alternate to the base angles, so they are equal to them.
The three angles at the top lie on a straight line, so they add to 180°.
So the three angles of the triangle add to 180°. This works for every triangle.

The exterior angle of a triangle

Extend one side. The exterior angle equals the sum of the two interior opposite angles.
Why: exterior + adjacent interior = 180° (straight line), and all three interior angles = 180°. So the exterior angle = the other two interior angles.
Worked example

An isosceles triangle has a top angle of 40°. Find each base angle, and the exterior angle at the base.

  1. Base angles are equal: (180 − 40) ÷ 2 = 70° (angles in a triangle)
  2. Exterior angle: 180 − 70 = 110° (angles on a straight line)
  3. Check: 40 + 70 = 110 (exterior angle = interior opposites)

Answer: 70° each, exterior 110°

Key idea

Give a reason for every step. Measuring one drawing is not a proof; a proof works for every case. The exterior angle of a triangle equals the two interior opposite angles.

Key words
Exterior angle
The angle between one side of a polygon and the next side extended outwards. The exterior angles of any polygon add to 360°.
Vertically opposite angles
Angles opposite each other where two straight lines cross. They are equal.
Alternate angles
Angles on opposite sides of a line that crosses two parallel lines, inside the parallel lines (a Z shape). They are equal.
Corresponding angles
Angles in the same position at each place where a line crosses two parallel lines (an F shape). They are equal.
Co-interior angles
Angles between two parallel lines on the same side of the crossing line (a C or U shape). They add up to 180°.
Vertex
A corner of a shape, where edges or sides meet. The plural is vertices.

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2 more questions

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  1. A triangle has interior angles of 48° and 77°. Find the exterior angle next to the third angle.

    Show the answer

    Answer: 125 °

    The exterior angle equals the sum of the two opposite interior angles: 48 + 77 = 125°.

  2. A line is drawn through the top corner of a triangle, parallel to the base. Why does each base angle equal one of the new angles at the top?

    1. They are alternate angles, which are equal
    2. They are corresponding angles, which are equal
    3. They are co-interior angles, which are equal
    4. They are vertically opposite angles
    Show the answer

    Answer: They are alternate angles, which are equal

    Each side of the triangle crosses the two parallel lines, making a Z: alternate angles.

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Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.

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