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Geometry - Angle Properties (Lines, Triangles, Quadrilaterals)

Grade 9IGCSE

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

🔑Concepts

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

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

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Vertically opposite angles are equal when two straight lines intersect.

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Parallel Line Properties: Alternate angles are equal (Z-shape), Corresponding angles are equal (F-shape), and Allied/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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Isosceles triangles have two equal sides and two equal base angles; Equilateral triangles have all angles equal to 60∘60^{\circ}.

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The sum of interior angles in any quadrilateral is 360∘360^{\circ}.

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Properties of special quadrilaterals: In a parallelogram, opposite angles are equal; in a rhombus, diagonals bisect at 90∘90^{\circ}.

📐Formulae

Angles on a line: a+b+c=180∘a + b + c = 180^{\circ}

Sum of angles in a triangle: ∠A+∠B+∠C=180∘\angle A + \angle B + \angle C = 180^{\circ}

Exterior angle of triangle: ext∠=int∠opp1+int∠opp2ext\angle = int\angle_{opp1} + int\angle_{opp2}

Sum of angles in a quadrilateral: ∠A+∠B+∠C+∠D=360∘\angle A + \angle B + \angle C + \angle D = 360^{\circ}

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

Co-interior angles: x+y=180∘x + y = 180^{\circ} (where lines are parallel)

💡Examples

Problem 1:

In triangle ABC, angle A is 40∘40^{\circ} and angle B is 85∘85^{\circ}. Calculate the exterior angle at vertex C.

Solution:

125∘125^{\circ}

Explanation:

Using the exterior angle theorem, the exterior angle is equal to the sum of the two opposite interior angles. Therefore, Exterior ∠C=∠A+∠B=40∘+85∘=125∘\angle C = \angle A + \angle B = 40^{\circ} + 85^{\circ} = 125^{\circ}.

Problem 2:

Two parallel lines are intersected by a transversal. If one of the co-interior angles is 72∘72^{\circ}, find the value of the other co-interior angle.

Solution:

108∘108^{\circ}

Explanation:

Co-interior (allied) angles between parallel lines are supplementary, meaning they add up to 180∘180^{\circ}. Calculation: 180∘−72∘=108∘180^{\circ} - 72^{\circ} = 108^{\circ}.

Problem 3:

A quadrilateral has three angles measuring 110∘110^{\circ}, 80∘80^{\circ}, and 75∘75^{\circ}. Find the size of the fourth angle.

Solution:

95∘95^{\circ}

Explanation:

The sum of angles in a quadrilateral is always 360∘360^{\circ}. Sum of known angles: 110+80+75=265∘110 + 80 + 75 = 265^{\circ}. Fourth angle: 360∘−265∘=95∘360^{\circ} - 265^{\circ} = 95^{\circ}.