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Geometry - Angle Properties of Lines, Triangles, and Polygons

Grade 8IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Angles on a straight line always sum to 180∘180^\circ.

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Angles around a point always sum to 360∘360^\circ.

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Vertically opposite angles are equal when two straight lines intersect.

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Parallel Lines: Alternate angles are equal (Z-shape), Corresponding angles are equal (F-shape), and Co-interior angles sum to 180∘180^\circ (C-shape).

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The sum of interior angles in any triangle is 180∘180^\circ.

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An exterior angle of a triangle is equal to the sum of the two opposite interior angles.

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Types of Triangles: Isosceles (two equal angles/sides) and Equilateral (all angles 60∘60^\circ).

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The sum of exterior angles of any convex polygon is 360∘360^\circ.

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For regular polygons, all interior angles are equal and all exterior angles are equal.

📐Formulae

Sum of interior angles of an nn-sided polygon = (n−2)×180∘(n - 2) \times 180^\circ

Individual interior angle of a regular nn-sided polygon = (n−2)×180∘n\frac{(n - 2) \times 180^\circ}{n}

Individual exterior angle of a regular nn-sided polygon = 360∘n\frac{360^\circ}{n}

Interior angle + Exterior angle = 180∘180^\circ (at any vertex)

💡Examples

Problem 1:

A regular polygon has an interior angle of 144∘144^\circ. Calculate the number of sides nn of the polygon.

Solution:

n=10n = 10

Explanation:

First, find the exterior angle: 180∘−144∘=36∘180^\circ - 144^\circ = 36^\circ. Since the sum of exterior angles is 360∘360^\circ, the number of sides n=360∘36∘=10n = \frac{360^\circ}{36^\circ} = 10. The polygon is a decagon.

Problem 2:

In a triangle ABCABC, angle A=40∘A = 40^\circ and angle B=75∘B = 75^\circ. Find the exterior angle at vertex CC.

Solution:

115∘115^\circ

Explanation:

Using the exterior angle theorem, the exterior angle at CC is equal to the sum of the opposite interior angles AA and BB. Therefore, 40∘+75∘=115∘40^\circ + 75^\circ = 115^\circ.

Problem 3:

Two parallel lines are intersected by a transversal. If one of the co-interior angles is 70∘70^\circ, find the value of the other co-interior angle.

Solution:

110∘110^\circ

Explanation:

Co-interior angles (also known as allied angles) are supplementary, meaning they add up to 180∘180^\circ. Calculation: 180∘−70∘=110∘180^\circ - 70^\circ = 110^\circ.