Angle Properties | FractionRush

Welcome to Angle Properties!

Angle properties are the rules that describe how angles relate to each other in geometric figures. Whether working with parallel lines, triangles, polygons or intersecting lines — these rules let you find unknown angles and write clear geometric proofs with reasons. IGCSE examiners always want reasons stated.

Key skill: Always write a geometric reason after every angle calculation | e.g. "angles on a straight line"

Angles at a Point

360° around a point

Straight Line & VOA

180° and vertically opposite

Parallel Lines

Alternate, co-interior, corresponding

Triangles

Angle sum and exterior angle

Quadrilaterals

Angle sum = 360°

Polygons

Interior and exterior angle formulas

1. Angles at a Point

All the angles that meet at a single point — on all sides — add up to exactly 360°. Think of making a full turn.

Angles at a point = 360°

a | b ---+--- c a + b + c + d = 360° | d

Proof: A full rotation is 360°. Any set of angles that fill the space around a point without overlap must sum to 360°. This is an axiom of Euclidean geometry.

Example: Three angles at a point are 110°, 95° and x. Find x.
110 + 95 + x = 360 ⟹ x = 360 − 205 = 155°
Reason: angles at a point sum to 360°

Cambridge examiners expect the reason "angles at a point sum to 360°" stated explicitly for every mark.

2. Angles on a Straight Line

All angles formed on one side of a straight line — sharing the same point — add up to 180°. This is called a supplementary pair when there are just two angles.

Angles on a straight line = 180°

a b ----+---- a + b = 180° (straight line)

Proof: A straight line represents half a full turn (180°). Any angles sitting on top of it must fill that half-turn exactly.

Example: Angles on a straight line are 2x and 3x. Find x.
2x + 3x = 180 ⟹ 5x = 180 ⟹ x = 36°
Reason: angles on a straight line sum to 180°

3. Vertically Opposite Angles

When two straight lines cross, they form two pairs of vertically opposite angles. Each pair is equal.

Vertically opposite angles are equal

a | b ---X--- a = c and b = d c | d

Proof: a + b = 180° (straight line) and c + b = 180° (straight line), so a = c. Similarly b = d.

Example: Two lines cross. One angle is 73°. Find the vertically opposite angle.
Vertically opposite = 73°
Reason: vertically opposite angles are equal

The four angles at a crossing always sum to 360°. The two pairs of vertically opposite angles each sum to 180° with their neighbours.

4. Parallel Lines — Three Key Angle Pairs

When a transversal (a crossing line) cuts two parallel lines, three special pairs of angles are created. You must know these for IGCSE.

4a. Alternate Angles (Z-angles)

Alternate angles are EQUAL | Reason: "alternate angles, AB ∥ CD"

------a------ ← line 1 \ \ (transversal) \ ------b------ ← line 2 a = b (they form a Z shape)

Alternate angles lie on opposite sides of the transversal, between the parallel lines. They form a Z (or N) shape. They are always equal when lines are parallel.

Proof: a = b follows from the parallel postulate — if they were unequal, the lines would eventually meet.

Example: Alternate angle to 65° = 65° (alternate angles, lines parallel)

4b. Co-interior Angles (C-angles / Allied Angles)

Co-interior angles add up to 180° | Reason: "co-interior angles, AB ∥ CD"

------a------ ← line 1 / / (transversal) / ------b------ ← line 2 a + b = 180° (they form a C shape)

Co-interior angles lie on the same side of the transversal, between the parallel lines. They form a C (or U) shape. They are supplementary (sum to 180°) when lines are parallel.

Proof: alternate angles give a = c (the angle on the other side). Then a + b = c + b = 180° (angles on a straight line).

Example: Co-interior to 112° = 180 − 112 = 68° (co-interior angles, lines parallel)

4c. Corresponding Angles (F-angles)

Corresponding angles are EQUAL | Reason: "corresponding angles, AB ∥ CD"

---a--------- ← line 1 | | (transversal) | ---b--------- ← line 2 a = b (they form an F shape)

Corresponding angles are in the same position relative to each parallel line and the transversal. They form an F shape. They are always equal when lines are parallel.

Proof: Corresponding angles equal alternate angles (both equal to the angle the transversal makes), by the parallel postulate.

Example: Corresponding angle to 48° = 48° (corresponding angles, lines parallel)

Memory aid:
Z-angles (Alternate) → Equal
C-angles (Co-interior) → add to 180°
F-angles (Corresponding) → Equal
"Z and F are fine (equal); C adds to 180°"

5. Angles in a Triangle

5a. Angle Sum of a Triangle

Angles in a triangle sum to 180°

Proof: Draw line XY through vertex A parallel to BC.
Angle XAB = angle ABC (alternate angles, XY ∥ BC)
Angle YAC = angle BCA (alternate angles, XY ∥ BC)
Angle XAB + angle BAC + angle YAC = 180° (angles on a straight line)
Therefore angle ABC + angle BAC + angle BCA = 180° ✓

Example: Two angles of a triangle are 47° and 68°. Find the third.
Third = 180 − 47 − 68 = 65° (angle sum of triangle = 180°)

5b. Exterior Angle Theorem

Exterior angle of a triangle = sum of the two non-adjacent interior angles

A /\ / \ / \ B /______\ C ---- D Exterior angle ACD = angle BAC + angle ABC

Proof: Interior angles of triangle: A + B + C = 180°.
Angle ACD + angle ACB = 180° (straight line).
So angle ACD = 180° − C = A + B ✓

Example: Interior angles A = 55°, B = 72°. Exterior angle at C = 55 + 72 = 127°
Reason: exterior angle of a triangle equals the sum of the two non-adjacent interior angles

6. Angles in a Quadrilateral

Angles in a quadrilateral sum to 360°

Proof: Any quadrilateral can be split into 2 triangles by drawing a diagonal. Each triangle gives 180°, so total = 2 × 180° = 360°.

Example: A quadrilateral has angles 85°, 110°, 95° and x. Find x.
85 + 110 + 95 + x = 360 ⟹ x = 360 − 290 = 70°
Reason: angles in a quadrilateral sum to 360°

This works for ANY quadrilateral: square, rectangle, trapezium, parallelogram, rhombus, kite, or irregular. The sum is always 360°.

7. Interior and Exterior Angles of Polygons

7a. Sum of Interior Angles of any Polygon

Sum of interior angles = (n − 2) × 180°

Proof: A polygon with n sides can be divided into (n − 2) triangles by drawing diagonals from one vertex. Each triangle contributes 180°, so the total = (n − 2) × 180°.

Example — hexagon (n = 6):
Sum = (6 − 2) × 180 = 4 × 180 = 720°

Example — pentagon (n = 5):
Sum = (5 − 2) × 180 = 3 × 180 = 540°

7b. Interior Angle of a Regular Polygon

Each interior angle of a regular polygon = (n − 2) × 180° ÷ n

In a regular polygon all sides and angles are equal, so divide the total by n.

Regular hexagon (n = 6): (6−2)×180÷6 = 720÷6 = 120°
Regular pentagon (n = 5): (5−2)×180÷5 = 540÷5 = 108°
Regular octagon (n = 8): (8−2)×180÷8 = 1080÷8 = 135°

7c. Exterior Angles of Polygons

Sum of exterior angles of any polygon = 360° | Each exterior angle of regular polygon = 360° ÷ n

Proof: As you walk around any polygon, making one exterior angle turn at each vertex, you complete exactly one full turn = 360°. This is true for any convex polygon, regardless of n.

Key relationship: Interior angle + Exterior angle = 180° (angles on a straight line)

Regular decagon (n = 10): Exterior = 360÷10 = 36° | Interior = 180−36 = 144°
Regular hexagon (n = 6): Exterior = 360÷6 = 60°

Finding n from an exterior angle: If exterior angle = 40°, then n = 360÷40 = 9 (nonagon)

Quick summary table:
Triangle (n=3): Interior sum 180° | Each interior 60°
Quadrilateral (n=4): Interior sum 360° | Each interior 90°
Pentagon (n=5): Interior sum 540° | Each interior 108°
Hexagon (n=6): Interior sum 720° | Each interior 120°
Octagon (n=8): Interior sum 1080° | Each interior 135°
Decagon (n=10): Interior sum 1440° | Each interior 144°

8. Writing Geometric Reasons (IGCSE Exam Skill)

Cambridge IGCSE requires you to state a reason for every angle you calculate. Reasons must be specific geometric facts — not just descriptions.

Accepted reasons (use these exact phrases):
• Angles at a point sum to 360°
• Angles on a straight line sum to 180°
• Vertically opposite angles are equal
• Alternate angles are equal (AB ∥ CD)
• Corresponding angles are equal (AB ∥ CD)
• Co-interior angles sum to 180° (AB ∥ CD)
• Angle sum of a triangle is 180°
• Exterior angle of a triangle = sum of non-adjacent interior angles
• Angle sum of a quadrilateral is 360°
• Sum of interior angles = (n−2) × 180°
• Sum of exterior angles of a polygon = 360°

For parallel line reasons, always name the parallel lines: e.g. "alternate angles, PQ ∥ RS". Without naming the lines you may lose the mark.

Example 1 — Angles at a Point and on a Straight Line

Three angles at a point are 2x, 3x and x + 60°. Find x and all three angles.

Set up equation: 2x + 3x + (x + 60) = 360

6x + 60 = 360 ⟹ 6x = 300 ⟹ x = 50°

Angles: 2(50) = 100°, 3(50) = 150°, 50 + 60 = 110°. Check: 100 + 150 + 110 = 360 ✓

Reason: angles at a point sum to 360°

Example 2 — Vertically Opposite and Supplementary Angles

Two lines cross. One angle is (3x + 15)° and the vertically opposite angle is (5x − 9)°. Find x.

Vertically opposite angles are equal: 3x + 15 = 5x − 9

15 + 9 = 5x − 3x ⟹ 24 = 2x ⟹ x = 12°

Angle = 3(12) + 15 = 51°. The adjacent angle = 180 − 51 = 129°.

Reason: vertically opposite angles are equal; angles on a straight line sum to 180°

Example 3 — Parallel Lines: Alternate, Corresponding, Co-interior

AB and CD are parallel. A transversal crosses AB at P and CD at Q. Angle APQ = 68°. Find angles PQD, PQC and the co-interior angle on the same side.

Alternate angle: Angle PQD = 68° (alternate angles, AB ∥ CD)

Reason: alternate angles are equal, AB ∥ CD

Corresponding angle: Angle corresponding to APQ at Q on the same side = 68°

Reason: corresponding angles are equal, AB ∥ CD

Co-interior angle: Angle PQC = 180 − 68 = 112°

Reason: co-interior angles sum to 180°, AB ∥ CD

Example 4 — Triangle Angles with Exterior Angle

In triangle ABC: angle A = (x + 20)°, angle B = (2x − 10)°, angle C = x°. Find all angles and the exterior angle at C.

Angle sum: (x + 20) + (2x − 10) + x = 180

4x + 10 = 180 ⟹ 4x = 170 ⟹ x = 42.5°

Angles: A = 62.5°, B = 75°, C = 42.5°. Check: 62.5 + 75 + 42.5 = 180 ✓

Reason: angle sum of a triangle is 180°

Exterior angle at C = A + B = 62.5 + 75 = 137.5°

Reason: exterior angle of a triangle = sum of the two non-adjacent interior angles

Example 5 — Regular Polygon Interior and Exterior Angles

A regular polygon has an exterior angle of 24°. (a) How many sides does it have? (b) What is each interior angle? (c) What is the sum of interior angles?

(a) n = 360 ÷ 24 = 15 sides (regular pentadecagon)

Reason: sum of exterior angles = 360°

(b) Interior angle = 180 − 24 = 156°

Reason: interior + exterior = 180° (angles on a straight line)

(c) Sum of interior angles = (15 − 2) × 180 = 13 × 180 = 2340°

Reason: sum of interior angles = (n − 2) × 180°

Example 6 — Multi-step: Parallel Lines and Triangle

AB ∥ CD. Triangle PQR has P on AB and R on CD. Angle QPR = 55° and angle PRQ = 70°. Find angle PQR and the angle that QR makes with CD at R.

Angle PQR: 55 + 70 + angle PQR = 180

Angle PQR = 180 − 125 = 55°

Reason: angle sum of a triangle is 180°

Angle QRD (between QR extended and CD): This is the exterior angle at R = 55 + 55 = 110°. Or: angle QRC = 180 − 70 = 110° (straight line), and angle QRD = 180 − 110 = 70°...

More carefully: angle PRQ = 70° is the angle inside the triangle at R. The angle between QR and CD on the other side = 180 − 70 = 110° (angles on straight line), so angle QRD = 70° (vertically opposite to angle PRsomething — use alternate: QR meets CD, and AB ∥ CD, angle at P with AB = angle at R with CD = alternate).

Angle that QR makes below CD at R = alternate angle to angle RQP = 55°

Reason: alternate angles are equal, AB ∥ CD

Regular Polygon Angle Visualiser

Drag the slider to change the number of sides. The polygon is drawn on the canvas and all angle values are calculated and displayed.

Sides (n): 6

Adjust the slider to see the polygon.

Parallel Lines Angle Explorer

Set the transversal angle to see how all eight angles at the two intersections are related.

Transversal angle: 55°

Adjust the slider to explore angle relationships.

Exercise 1 — Angles at a Point and on a Straight Line

Find the unknown angle. Give your answer as an integer number of degrees.

1. Three angles at a point are 120°, 95° and x°. Find x.

2. Two angles on a straight line are 67° and x°. Find x.

3. Four angles at a point are 80°, 100°, 95° and x°. Find x.

4. Angles on a straight line are 3x° and 2x°. Find x.

5. Angles on a straight line are x°, 2x° and 3x°. Find x.

6. Three angles at a point are 140°, 130° and x°. Find x.

Exercise 2 — Vertically Opposite Angles

Two straight lines cross. Find the unknown angle.

1. One angle at the crossing is 48°. What is the vertically opposite angle?

2. One angle at the crossing is 112°. What is the adjacent angle on the straight line?

3. Two vertically opposite angles are (4x + 10)° and (6x − 8)°. Find x.

4. Two vertically opposite angles are (2x + 30)° and (3x + 10)°. Find the size of each angle (enter the angle in degrees, not x).

5. One angle at a crossing is 37°. Find all four angles at the crossing. Enter the largest angle.

6. Four angles at a crossing are: 2x°, (x+15)°, 2x°, (x+15)°. Find x.

Exercise 3 — Parallel Lines

AB and CD are parallel. Find each unknown angle.

1. A transversal makes an angle of 70° with AB. Find the alternate angle with CD.

2. A transversal makes an angle of 55° with AB. Find the corresponding angle at CD.

3. A transversal makes an angle of 80° with AB. Find the co-interior angle at CD.

4. Co-interior angles are (2x + 10)° and (3x − 30)°. Find x.

5. Alternate angles are (4x − 5)° and (3x + 12)°. Find x.

6. Corresponding angles are (5x + 8)° and (2x + 44)°. Find x.

Exercise 4 — Triangles and Exterior Angles

Find the unknown angle in each triangle.

1. A triangle has angles 60°, 70° and x°. Find x.

2. An isosceles triangle has base angles each 52°. Find the apex angle.

3. A triangle's angles are (x + 10)°, (2x − 5)° and (x + 15)°. Find x.

4. An exterior angle of a triangle is 115°. One non-adjacent interior angle is 48°. Find the other non-adjacent interior angle.

5. The exterior angle of a triangle is (4x + 6)° and the two non-adjacent interior angles are x° and (2x + 12)°. Find x.

6. An equilateral triangle has all angles equal. What is each interior angle?

Exercise 5 — Polygons

Use the polygon angle formulas to find each answer.

1. Find the sum of interior angles of a pentagon (n = 5).

2. Find each interior angle of a regular hexagon (n = 6).

3. Find each exterior angle of a regular octagon (n = 8).

4. A regular polygon has an exterior angle of 30°. How many sides does it have?

5. Find each interior angle of a regular decagon (n = 10).

6. A regular polygon has each interior angle = 150°. How many sides does it have?

Practice — 20 Questions

Mixed practice covering all angle property skills. Give all answers as integers (degrees or number of sides as appropriate).

1. Three angles at a point are 110°, 140° and x°. Find x.

2. Angles on a straight line are 72° and x°. Find x.

3. Two lines cross. One angle is 65°. Find the vertically opposite angle.

4. Two lines cross. One angle is 65°. Find the adjacent supplementary angle.

5. AB ∥ CD. The alternate angle to 82° is ? (degrees)

6. AB ∥ CD. The co-interior angle to 82° is ? (degrees)

7. AB ∥ CD. The corresponding angle to 82° is ? (degrees)

8. A triangle has angles 45°, 90° and x°. Find x.

9. An exterior angle of a triangle is 130°. One non-adjacent interior angle is 74°. Find the other.

10. A quadrilateral has angles 80°, 95°, 105° and x°. Find x.

11. Sum of interior angles of a heptagon (n = 7) in degrees.

12. Each interior angle of a regular nonagon (n = 9) in degrees.

13. Each exterior angle of a regular hexagon (n = 6) in degrees.

14. A regular polygon has exterior angle 20°. How many sides?

15. A regular polygon has interior angle 162°. How many sides?

16. Co-interior angles are x° and (x + 40)°. Find x.

17. A triangle has angles 2x°, 3x° and 4x°. Find x.

18. An isosceles triangle has apex angle 40°. Find each base angle.

19. The sum of interior angles of a polygon is 1440°. How many sides?

20. Each interior angle of a regular polygon is 135°. How many sides?

Challenge — 8 Questions

Multi-step problems. Read carefully and apply multiple angle rules in sequence.

1. AB ∥ CD ∥ EF. A transversal crosses all three lines, making angles 58° with AB and some angle with CD. If the angle between CD and EF with the same transversal is (3x + 7)° and the alternate angle with CD is 58°, and the co-interior angle between CD and EF is (2x + 54)°, find x. (Hint: co-interior with EF and CD must sum to 180°)

2. In triangle PQR, angle P = (3x − 10)°, angle Q = (x + 20)° and angle R = (2x + 30)°. Find the exterior angle at R.

3. A polygon has n sides with angle sum 2520°. Find n.

4. Two regular polygons share an edge. One is a regular hexagon (interior angle 120°) and one is a regular square (interior angle 90°). The angle between their outer edges at the shared vertex = 360 − 120 − 90 = ? degrees.

5. AB ∥ CD. Point P lies between the lines. Angle PAB = 35° and angle PCD = 55°. Find angle APC. (Draw an auxiliary line through P parallel to AB and CD.)

6. In quadrilateral ABCD, angle A = (x + 20)°, angle B = (2x − 5)°, angle C = (x + 35)° and angle D = (3x − 10)°. Find angle B in degrees.

7. A regular polygon has each interior angle measuring 156°. How many sides does it have?

8. In the diagram, AB ∥ CD. The transversal meets AB at X with angle AXY = 48° and meets CD at Y. A triangle XYZ has Z between the parallel lines with angle XZY = 90°. Find angle XYZ.