Angles

Spec reference: Geometry and measures - Properties of angles and shapes
Key idea: Use angle facts for triangles, polygons, parallel lines and circles.

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Basic angle facts

Fact	Rule
Angles on a straight line	Sum = $180°$
Angles around a point	Sum = $360°$
Vertically opposite angles	Equal
Angles in a triangle	Sum = $180°$
Angles in a quadrilateral	Sum = $360°$

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Angles in polygons

$$\text{Sum of interior angles}=(n-2)\times 180°$$

where $n$ is the number of sides.

$$\text{Each interior angle (regular polygon)}=\frac{(n-2)\times 180°}{n}$$$$\text{Each exterior angle (regular polygon)}=\frac{360°}{n}$$

Example

Find the sum of interior angles of a hexagon.

$(6-2)\times 180°=4\times 180°=\boxed{{\displaystyle 720°}}$

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Angles in parallel lines

When a transversal crosses parallel lines:

• Alternate angles are equal (Z-angles)
• Corresponding angles are equal (F-angles)
• Co-interior angles add up to $180°$ (C-angles)

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Angles in a circle (circle theorems)

Theorem	Rule
Angle at centre	Twice the angle at circumference on same arc
Angle in semicircle	$90°$
Angles in same segment	Equal
Opposite angles in cyclic quadrilateral	Sum = $180°$
Tangent to radius	$90°$
Tangents from external point	Equal length
Alternate segment theorem	Angle between tangent and chord = angle in alternate segment

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Bearing angles

Bearings are measured clockwise from North, written as 3 digits.

Example

N is $000°$, E is $090°$, S is $180°$, W is $270°$

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Exam tips

Watch out for

• Always give a reason for each angle calculation in geometry proofs
• Exterior angle of a triangle = sum of the two non-adjacent interior angles
• Co-interior angles are supplementary (add to 180°), not equal

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Test Yourself

Question 1 of 5

Sum of interior angles of a pentagon?