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Geometry - Angle Properties of Lines, Triangles, and Polygons

Grade 8Cambridge (IGCSE)

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

🔑Concepts

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When two parallel lines are intersected by a transversal, corresponding angles are equal, alternate angles are equal, and co-interior angles sum to 180∘180^\circ.

Parallel lines intersected by a transversal showing corresponding and alternate angles.
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The sum of interior angles in any triangle is exactly 180∘180^\circ. For any nn-sided polygon, the sum is (n−2)×180∘(n-2) \times 180^\circ.

Triangle with labeled interior angles x, y, and z.
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The exterior angle of a triangle is equal to the sum of the two opposite interior angles.

Triangle showing exterior angle e and opposite interior angles A and B.
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In a regular polygon, all interior angles are equal and all exterior angles are equal. The sum of all exterior angles for any convex polygon is always 360∘360^\circ.

📐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.

Problem 4:

In the given diagram, lines L1L1 and L2L2 are parallel. Find the value of xx.

Diagram showing parallel lines L1 and L2 with co-interior angles 120 degrees and x.

Solution:

x+120∘=180∘x + 120^\circ = 180^\circ x=180∘−120∘x = 180^\circ - 120^\circ x=60∘x = 60^\circ

Explanation:

Since the two lines are parallel, the angles indicated are co-interior (also known as allied) angles. Co-interior angles between parallel lines always sum to 180∘180^\circ.

Problem 5:

Find the sum of the interior angles of the pentagon shown.

A standard five-sided polygon (pentagon).

Solution:

Sum=(n−2)×180∘Sum = (n - 2) \times 180^\circ Sum=(5−2)×180∘Sum = (5 - 2) \times 180^\circ Sum=3×180∘=540∘Sum = 3 \times 180^\circ = 540^\circ

Explanation:

A pentagon has n=5n = 5 sides. Using the formula for the sum of interior angles (n−2)×180∘(n-2) \times 180^\circ, we substitute 5 for nn.