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Geometry - Angle Properties (Lines, Triangles, Polygons)

Grade 10IGCSE

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

🔑Concepts

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Angles on a straight line sum to 180°.

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Angles around a point sum to 360°.

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Vertically opposite angles are equal.

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

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Triangles: Interior angles sum to 180°. The exterior angle of a triangle is equal to the sum of the two opposite interior angles.

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Isosceles Triangles: Two sides are equal, and the angles opposite those sides are equal.

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Polygons: The sum of exterior angles of any convex polygon is 360°.

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Regular Polygons: All sides and all interior angles are equal.

📐Formulae

Sum of interior angles of a polygon: (n−2)×180∘(n - 2) \times 180^\circ

Sum of exterior angles of any polygon: 360∘360^\circ

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

Individual exterior angle of a regular polygon: 360∘n\frac{360^\circ}{n}

Interior angle + Exterior angle = 180∘180^\circ

💡Examples

Problem 1:

A regular polygon has an interior angle of 144°. Calculate the number of sides (n) the polygon has.

Solution:

n = 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 is 360/36=10360 / 36 = 10.

Problem 2:

In triangle ABC, angle A = 40° and angle B is twice the size of angle C. Find the value of angle B.

Solution:

Angle B = 93.33°

Explanation:

Let angle C = xx. Then angle B = 2x2x. The sum of angles is 40+x+2x=18040 + x + 2x = 180. Simplifying gives 3x=1403x = 140, so x=46.67∘x = 46.67^\circ. Therefore, angle B = 2×46.67=93.33∘2 \times 46.67 = 93.33^\circ.

Problem 3:

Two parallel lines are intersected by a transversal. If a pair of co-interior angles are (2x+10)(2x + 10) and (3x−20)(3x - 20), find the value of x.

Solution:

x = 38

Explanation:

Co-interior angles between parallel lines sum to 180°. Therefore, (2x+10)+(3x−20)=180(2x + 10) + (3x - 20) = 180. Combining like terms: 5x−10=180⇒5x=190⇒x=385x - 10 = 180 \Rightarrow 5x = 190 \Rightarrow x = 38.