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Geometry - Angle properties (parallel lines and polygons)

Grade 11IGCSE

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

🔑Concepts

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Parallel Lines: When a transversal intersects two parallel lines, corresponding angles are equal, alternate angles are equal, and co-interior angles sum to 180°.

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Vertically Opposite Angles: These are formed by two intersecting lines and are always equal.

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Sum of Interior Angles: The total of all interior angles in any n-sided polygon.

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Exterior Angles: The sum of exterior angles of any convex polygon is always 360°.

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Regular Polygons: A polygon where all sides and all interior angles are equal.

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Interior and Exterior Angle Relationship: At any vertex of a polygon, the interior angle and the exterior angle sum to 180°.

📐Formulae

Sum of interior angles=(n−2)×180∘\text{Sum of interior angles} = (n - 2) \times 180^\circ

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

Sum of exterior angles=360∘\text{Sum of exterior angles} = 360^\circ

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

Interior angle+Exterior angle=180∘\text{Interior angle} + \text{Exterior angle} = 180^\circ

💡Examples

Problem 1:

Two parallel lines are intersected by a transversal. If one of the co-interior angles is 115∘115^\circ, find the value of its pair.

Solution:

180∘−115∘=65∘180^\circ - 115^\circ = 65^\circ

Explanation:

Co-interior angles (also known as allied angles) between parallel lines are supplementary, meaning they add up to 180°.

Problem 2:

Calculate the number of sides of a regular polygon if each interior angle is 144∘144^\circ.

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, use the formula n=360/exterior anglen = 360 / \text{exterior angle}. Therefore, n=360/36=10n = 360 / 36 = 10.

Problem 3:

A pentagon has four interior angles of 100∘,110∘,120∘,100^\circ, 110^\circ, 120^\circ, and 90∘90^\circ. Find the fifth interior angle.

Solution:

120∘120^\circ

Explanation:

First, find the total sum of interior angles for a pentagon (n=5n=5): (5−2)×180∘=540∘(5-2) \times 180^\circ = 540^\circ. Subtract the known angles from the total: 540−(100+110+120+90)=540−420=120∘540 - (100 + 110 + 120 + 90) = 540 - 420 = 120^\circ.