Angle reasoning with reasons

When a question says "give reasons", each step needs the angle fact written out. Learn these as phrases:

• Angles on a straight line add up to 180°.
• Angles around a point add up to 360°.
• Vertically opposite angles are equal.
• Angles in a triangle add up to 180°. Angles in a quadrilateral add up to 360°.
• Base angles of an isosceles triangle are equal.
• Alternate angles are equal. Corresponding angles are equal. Co-interior angles add up to 180°.
• Exterior angles of a polygon add up to 360°.

"Z angles" or "F angles" do not score: write alternate or corresponding.

1. triangle ABC, BC extended to D; angle BAC = 50°, angle ACD = 125°
2. angle ACB = 180 − 125 = 55° (angles on a straight line add up to 180°)
3. angle ABC = 180 − 50 − 55 = 75° (angles in a triangle add up to 180°)
4. check: 125 = 50 + 75

Write the angle with its three letters (angle ACB, not just "C") so the examiner knows which one you mean.

Find the parallel lines first (arrows on the lines), then look for the shape the angles make:

• a Z shape: alternate angles, equal
• an F shape: corresponding angles, equal
• a C or U shape between the lines: co-interior angles, which add up to 180°

The usual slip: calling co-interior angles alternate. Alternate angles are on opposite sides of the crossing line; co-interior angles are on the same side.

When angles are given as expressions, use a fact to write an equation, solve it, then substitute back.

1. (3x + 10)° and (2x − 5)° on a straight line
2. 3x + 10 + 2x − 5 = 180
3. 5x + 5 = 180, so x = 35
4. the angles are 115° and 65°

The reason for the equation is the angle fact you used.