Geometry: Shapes and Angles

Geometry is the study of shapes, sizes, and positions. In the FSCE 11+ exam, you need to know the properties of 2D shapes, understand angle rules, and calculate missing angles. This lesson covers all the key facts and techniques, with clear worked examples that show you exactly how to set out your reasoning — an essential skill for the FSCE exam.

Properties of 2D Shapes

Triangles

Type	Properties	Angles
Equilateral	All 3 sides equal, all 3 angles equal	Each angle is 60°
Isosceles	2 sides equal, 2 angles equal	The two base angles are equal
Scalene	All sides different, all angles different	No equal angles
Right-angled	Has one angle of exactly 90°	One angle is 90°

Key fact: The angles in any triangle add up to 180°.

Quadrilaterals

Shape	Sides	Angles	Other Properties
Square	4 equal sides	4 right angles (90°)	2 pairs of parallel sides, 4 lines of symmetry
Rectangle	2 pairs of equal sides	4 right angles (90°)	2 pairs of parallel sides, 2 lines of symmetry
Parallelogram	2 pairs of equal sides	Opposite angles are equal	2 pairs of parallel sides, no lines of symmetry
Rhombus	4 equal sides	Opposite angles are equal	2 pairs of parallel sides, 2 lines of symmetry
Trapezium	1 pair of parallel sides	Angles vary	Only 1 pair of parallel sides
Kite	2 pairs of adjacent equal sides	1 pair of opposite angles are equal	1 line of symmetry

Key fact: The angles in any quadrilateral add up to 360°.

Regular Polygons

A regular polygon has all sides equal and all angles equal.

Shape	Sides	Sum of Interior Angles	Each Interior Angle
Equilateral triangle	3	180°	60°
Square	4	360°	90°
Regular pentagon	5	540°	108°
Regular hexagon	6	720°	120°
Regular octagon	8	1080°	135°

Formula: Sum of interior angles = (n - 2) x 180°, where n is the number of sides.

Each interior angle of a regular polygon = sum of angles ÷ number of sides.

Angle Rules

Rule 1: Angles on a Straight Line

Angles on a straight line add up to 180°.

If one angle is 130°, the other is 180° - 130° = 50°.

Rule 2: Angles at a Point

Angles around a full point add up to 360°.

Rule 3: Vertically Opposite Angles

When two straight lines cross, the angles opposite each other are equal. These are called vertically opposite angles.

Rule 4: Angles in a Triangle

The three interior angles of a triangle add up to 180°.

Rule 5: Angles in a Quadrilateral

The four interior angles of a quadrilateral add up to 360°.

Rule 6: Angles on Parallel Lines

When a line crosses two parallel lines (a transversal):

• Corresponding angles are equal (they are in the same position at each intersection — like an F shape).
• Alternate angles are equal (they are on opposite sides of the transversal between the parallel lines — like a Z shape).
• Co-interior angles (also called allied angles) add up to 180° (they are on the same side between the parallel lines — like a C or U shape).

Worked Examples

Worked Example 1: Missing Angle in a Triangle

Question: In triangle ABC, angle A = 47° and angle B = 68°. Find angle C.

Step-by-step solution:

1. State the rule: Angles in a triangle add up to 180°.
2. A + B + C = 180°
3. 47° + 68° + C = 180°
4. 115° + C = 180°
5. C = 180° - 115° = 65°

Answer: Angle C = 65°

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Worked Example 2: Missing Angle in a Quadrilateral

Question: A quadrilateral has angles of 85°, 110°, and 72°. Find the fourth angle.

Step-by-step solution:

1. State the rule: Angles in a quadrilateral add up to 360°.
2. 85° + 110° + 72° + x = 360°
3. 267° + x = 360°
4. x = 360° - 267° = 93°

Answer: The fourth angle is 93°.

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Worked Example 3: Vertically Opposite Angles

Question: Two straight lines cross. One of the angles formed is 124°. Find the other three angles.

Step-by-step solution:

1. The angle vertically opposite to 124° is also 124° (vertically opposite angles are equal).
2. The angle on a straight line with 124° is 180° - 124° = 56°.
3. The angle vertically opposite to 56° is also 56°.
4. Check: 124° + 56° + 124° + 56° = 360°. Correct (angles at a point add to 360°).

Answer: The four angles are 124°, 56°, 124°, and 56°.

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Worked Example 4: Angles on Parallel Lines

Question: Two parallel lines are crossed by a transversal. One of the angles is 65°. It is a corresponding angle to angle x, and an alternate angle to angle y. Find x and y.

Step-by-step solution:

1. Corresponding angles are equal, so x = 65°.
2. Alternate angles are equal, so y = 65°.
3. If asked for the co-interior angle: 180° - 65° = 115° (co-interior angles add up to 180°).

Answer: x = 65°, y = 65°.

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Worked Example 5: Interior Angle of a Regular Polygon

Question: Find the interior angle of a regular decagon (10 sides).

Step-by-step solution:

1. Sum of interior angles = (n - 2) x 180° = (10 - 2) x 180° = 8 x 180° = 1,440°.
2. Each interior angle = 1,440° ÷ 10 = 144°.

Answer: Each interior angle is 144°.

Angle Rules Decision Flowchart

flowchart TD
    A[You need to find a missing angle] --> B{What information do you have?}
    B -->|Angles on a straight line| C[They add up to 180°]
    B -->|Angles at a point| D[They add up to 360°]
    B -->|Angles in a triangle| E[They add up to 180°]
    B -->|Angles in a quadrilateral| F[They add up to 360°]
    B -->|Two crossing lines| G[Vertically opposite angles are equal]
    B -->|Parallel lines with transversal| H{What type?}
    H -->|Corresponding F-shape| I[They are equal]
    H -->|Alternate Z-shape| J[They are equal]
    H -->|Co-interior C-shape| K[They add up to 180°]
    B -->|Regular polygon| L[Use n-2 times 180° divided by n]

Common Mistakes