An angle is an amount of turning, measured in degrees, with a full turn equal to 360°. This topic covers the five types of angles, angles on a straight line and at a point, vertically opposite angles, parallel lines cut by a transversal, and the angle sums of triangles and quadrilaterals.
It is the foundation for all of Sec 1/2 geometry and appears in the G1 exam every year. In G1 Maths, every angle statement in your working must be followed by its reason in brackets, and the reason often carries its own mark.
What these notes cover
- •Types of angles
- •Naming angles with three letters
- •Estimating and measuring with a protractor
- •Angles on a straight line and at a point
- •Vertically opposite angles
- •Parallel lines and a transversal (F, Z and C shapes)
- •Classifying triangles by sides and by angles
- •Angle sum of a triangle and isosceles base angles
- •Angle sum of a quadrilateral
- •The five special quadrilaterals
- •Common mistakes and exam tips
- •Summary
Key definitions
- Angle
- An amount of turning. Two straight arms meet at a point called the vertex, and the angle tells us how far one arm has turned away from the other. Angles are measured in degrees (°); a full turn is 360°.
- Parallel lines
- Lines that never meet, no matter how far they are extended, and stay the same distance apart everywhere. They are marked with matching arrowheads, and AB ∥ CD reads “AB is parallel to CD”.
- Transversal
- A straight line that cuts across two (or more) other lines. When it cuts two parallel lines, eight angles are formed, four at each crossing point.
- Vertically opposite angles
- The two angles directly facing each other across the vertex where two straight lines cross; they are always equal. “Vertically” comes from vertex, not “up and down”.
- Base angles
- In an isosceles triangle, the two equal angles that face the two equal (tick-marked) sides.
Key formulas
Angles on a straight line
a + b = 180°
Use only when all the angles share the same vertex and lie on the same side of the straight line. Reason: (adj. ∠s on a str. line).
Angles at a point
p + q + r + s = 360°
Works for any number of angles that meet at the same point and fill the space around it completely, with no gaps and no overlaps. Reason: (∠s at a pt.).
Vertically opposite angles
a = c and b = d
When two straight lines cross, the angles directly facing each other across the crossing point are equal. Reason: (vert. opp. ∠s).
Parallel lines and a transversal
∠a = ∠e (F), ∠c = ∠e (Z), ∠d + ∠e = 180° (C)
For a transversal cutting AB ∥ CD: corresponding angles (F shape) are equal; alternate angles (Z shape, between the lines on opposite sides of the transversal) are equal; interior angles (C or U shape, between the lines on the same side) add up to 180°. Use only when the lines are marked parallel. Reasons: (corr. ∠s, AB ∥ CD), (alt. ∠s, AB ∥ CD), (int. ∠s, AB ∥ CD).
Angle sum of a triangle
∠A + ∠B + ∠C = 180°
Holds for every triangle. In an equilateral triangle each angle is 180° ÷ 3 = 60°; in an isosceles triangle the two base angles are equal. Reasons: (∠ sum of △), (base ∠s of isos. △).
Angle sum of a quadrilateral
a + b + c + d = 360°
One diagonal splits any quadrilateral into two triangles, so the four interior angles total 2 × 180° = 360°. To find a missing angle, subtract the three known angles from 360°. Reason: (∠ sum of quad.).
Worked example
A roof truss is an isosceles triangle. The apex angle (the top angle, between the two equal sides) is 40°. Find x, the size of each base angle.
- 1The three angles of a triangle add up to 180°, so the sum of the two base angles = 180° − 40° = 140° (∠ sum of △).
- 2The two base angles are equal, so each one is x = 140° ÷ 2 = 70° (base ∠s of isos. △).
- 3Check: 70° + 70° + 40° = 180°, so each base angle is 70°.
Watch out: Base angles are not always at the bottom of the page. Find the two tick-marked sides first; the equal angles are the ones facing those sides.
Common mistakes
Using 180° at a point instead of 360°
Wrong: Three angles meet at a point: x + 150° + 130° = 180°, so x = −100°.
Right: x + 150° + 130° = 360°, so x = 360° − 280° = 80° (∠s at a point). A negative answer, or one larger than 360°, means the wrong total was used.
Assuming lines are parallel without arrow markings
Wrong: Writing y = 70° (corr. ∠s) for two lines that only look parallel on the page.
Right: Use corresponding, alternate or interior angles only when the lines carry matching arrowheads or the question states AB ∥ CD. Otherwise find another route, such as angles in a triangle.
Forgetting the reason in brackets
Wrong: x = 180° − 65° = 115°
Right: x = 180° − 65° = 115° (adj. ∠s on a str. line). Every line of angle working needs its own reason.
Questions students ask
What is the difference between an obtuse angle and a reflex angle?
An obtuse angle is between 90° and 180°, while a reflex angle is between 180° and 360°. Obtuse stays below a straight line; for a reflex angle the arc must sweep past the straight line, going the long way round.
How do I tell alternate, corresponding and interior angles apart?
Trace the letter: an F shape gives corresponding angles, which are equal; a Z shape gives alternate angles, which are equal; a C or U shape gives interior angles, which add up to 180°. The letters may be rotated, stretched or flipped, and the rules only work when the lines are marked parallel.
Why do the angles in a quadrilateral add up to 360°?
Draw one diagonal from a corner to the opposite corner, which splits the quadrilateral into two triangles. The angles in each triangle add up to 180°, so the four angles total 2 × 180° = 360°. This works for every quadrilateral, including ones with no special name.
Do I need to write reasons for angle questions in G1 Maths?
Yes. Write the reason in brackets after every angle statement, using standard short forms such as (adj. ∠s on a str. line), (∠s at a pt.) or (alt. ∠s, AB ∥ CD). A correct answer with no reason can lose marks, and at Sec 1 and 2 the reason often carries its own mark.
In the full PDF
- •Labelled diagrams for every angle type, the eight angles formed by a transversal, and the F, Z and C shapes
- •Further worked examples: classifying and naming angles, the angle between clock hands at 4 o'clock, alternate angles with parallel lines, and parallelogram angles
- •The other three common mistakes: mixing up alternate and corresponding angles, misreading the protractor's inner and outer scale, and assuming a triangle is isosceles
- •Properties table for the five special quadrilaterals: square, rectangle, parallelogram, rhombus and trapezium
- •A complete reasons table pairing each situation with the short-form reason to write in brackets
- •An exam checklist for earning every mark and a one-page summary