Geometry & Angles

Geometry questions are a major part of the GL 11+ exam. You need to know the properties of 2D shapes, understand angle rules, and be able to identify different types of triangles and quadrilaterals. This lesson covers the key facts and techniques.

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Types of Angles

Type of angle	Size	Diagram hint
Acute	Less than 90°	Small and sharp
Right angle	Exactly 90°	Square corner
Obtuse	Between 90° and 180°	Wide opening
Straight line	Exactly 180°	Flat line
Reflex	Between 180° and 360°	More than a half turn
Full turn	Exactly 360°	Complete circle

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Angle Rules

Angles on a Straight Line

Angles on a straight line add up to 180°.

Worked Example: Two angles on a straight line are 115° and x°. Find x.

• 115 + x = 180
• x = 180 - 115 = 65°

Angles at a Point

Angles around a point add up to 360°.

Worked Example: Three angles meet at a point: 120°, 90°, and x°. Find x.

• 120 + 90 + x = 360
• x = 360 - 210 = 150°

Vertically Opposite Angles

When two straight lines cross, the vertically opposite angles are equal.

    a / \ b
     /   \
    /     \
   b \   / a
      \ /

If a = 55°, then the angle opposite it is also 55°.

Angles in a Triangle

The three angles in any triangle add up to 180°.

Worked Example: A triangle has angles of 47° and 68°. Find the third angle.

• 47 + 68 + x = 180
• x = 180 - 115 = 65°

Angles in a Quadrilateral

The four angles in any quadrilateral add up to 360°.

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Types of Triangles

Triangle type	Sides	Angles
Equilateral	All 3 sides equal	All 3 angles = 60°
Isosceles	2 sides equal	2 angles equal (base angles)
Scalene	All 3 sides different	All 3 angles different
Right-angled	Has one 90° angle	The other two add up to 90°

Isosceles Triangle — Worked Example

An isosceles triangle has an angle of 40° at the top. Find the two base angles.

• The two base angles are equal. Call each one x.
• 40 + x + x = 180
• 40 + 2x = 180
• 2x = 140
• x = 70°

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Types of Quadrilaterals

Quadrilateral	Properties
Square	4 equal sides, 4 right angles, 2 pairs of parallel sides
Rectangle	2 pairs of equal sides, 4 right angles, 2 pairs of parallel sides
Parallelogram	2 pairs of equal and parallel sides, opposite angles equal
Rhombus	4 equal sides, opposite angles equal, 2 pairs of parallel sides
Trapezium	Exactly 1 pair of parallel sides
Kite	2 pairs of adjacent sides equal, 1 pair of opposite angles equal

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Properties of Polygons

A polygon is a 2D shape with straight sides.

Shape	Sides	Sum of interior angles
Triangle	3	180°
Quadrilateral	4	360°
Pentagon	5	540°
Hexagon	6	720°
Heptagon	7	900°
Octagon	8	1,080°