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Understanding Quadrilaterals - Angle Sum Property of a Polygon

Grade 8ICSE

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

🔑Concepts

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The Angle Sum Property of a polygon states that the sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n - 2) \times 180^{\circ}. This is derived by dividing the polygon into (n−2)(n - 2) triangles from a single vertex.

A pentagon divided into three triangles from one vertex showing the sum is 3 * 180 degrees.
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Regardless of the number of sides, the sum of the measures of the exterior angles of any convex polygon (taken one at each vertex) is always 360∘360^{\circ}.

Triangle with extended sides showing exterior angles 1, 2, and 3 which sum to 360 degrees.
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In a regular polygon, all interior angles are equal and all exterior angles are equal. This symmetry allows us to find the measure of a single angle by dividing the total sum by the number of sides nn.

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At each vertex of a polygon, the interior angle and its corresponding exterior angle are supplementary, meaning their sum is always 180∘180^{\circ} because they form a linear pair.

📐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}

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

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

Number of sides (nn) of a regular polygon = 360∘Each exterior angle\frac{360^{\circ}}{\text{Each exterior angle}}

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

💡Examples

Problem 1:

Find the sum of the interior angles of a polygon with 1212 sides (a dodecagon).

Solution:

Step 1: Identify the number of sides, n=12n = 12. \nStep 2: Use the formula for the sum of interior angles: S=(n−2)×180∘S = (n - 2) \times 180^{\circ} \nStep 3: Substitute n=12n = 12 into the formula: S=(12−2)×180∘S = (12 - 2) \times 180^{\circ} \nStep 4: Calculate the values: S=10×180∘=1800∘S = 10 \times 180^{\circ} = 1800^{\circ}

Explanation:

The problem asks for the total sum of all angles inside the polygon. By applying the (n−2)×180∘(n-2) \times 180^{\circ} rule, we determine how many triangles the 12-sided figure can be divided into, which is 10.

Problem 2:

Each interior angle of a regular polygon is 144∘144^{\circ}. Find the number of sides of the polygon.

Solution:

Step 1: Find the measure of each exterior angle using the supplementary relationship: Exterior angle=180∘−Interior angle\text{Exterior angle} = 180^{\circ} - \text{Interior angle} Exterior angle=180∘−144∘=36∘\text{Exterior angle} = 180^{\circ} - 144^{\circ} = 36^{\circ} \nStep 2: Use the formula for the number of sides based on the exterior angle: n=360∘Each exterior anglen = \frac{360^{\circ}}{\text{Each exterior angle}} \nStep 3: Substitute the exterior angle value: n=360∘36∘n = \frac{360^{\circ}}{36^{\circ}} \nStep 4: Solve for nn: n=10n = 10

Explanation:

To find the number of sides, it is often easier to find the exterior angle first. Since the interior and exterior angles sum to 180∘180^{\circ}, we find the exterior angle is 36∘36^{\circ}. Dividing the total exterior sum (360∘360^{\circ}) by this value gives the number of sides.

Problem 3:

Find the value of xx in the given quadrilateral where three interior angles are 80∘80^{\circ}, 110∘110^{\circ}, and 120∘120^{\circ}.

A quadrilateral with three labeled angles 80, 110, 120 and one unknown x.

Solution:

  1. For a quadrilateral, n=4n = 4.
  2. Sum of interior angles = (4−2)×180∘=360∘(4 - 2) \times 180^{\circ} = 360^{\circ}.
  3. Sum of given angles = 80∘+110∘+120∘=310∘80^{\circ} + 110^{\circ} + 120^{\circ} = 310^{\circ}.
  4. x=360∘−310∘=50∘x = 360^{\circ} - 310^{\circ} = 50^{\circ}.

Final Answer: x=50∘x = 50^{\circ}

Explanation:

Since the sum of all interior angles of any quadrilateral must be 360∘360^{\circ}, we subtract the sum of the known angles from 360∘360^{\circ} to find the unknown angle.

Problem 4:

Calculate the number of sides of a regular polygon if each of its exterior angles measures 45∘45^{\circ}.

A portion of a regular polygon showing an exterior angle of 45 degrees.

Solution:

  1. We know that for a regular polygon, the number of sides n=360∘Each exterior anglen = \frac{360^{\circ}}{\text{Each exterior angle}}.
  2. Given each exterior angle = 45∘45^{\circ}.
  3. n=360∘45∘n = \frac{360^{\circ}}{45^{\circ}}.
  4. n=8n = 8.

Final Answer: The polygon has 88 sides (it is a regular octagon).

Explanation:

Because the sum of exterior angles of any polygon is always 360∘360^{\circ}, and in a regular polygon all exterior angles are equal, dividing the total sum by the measure of one angle gives the total number of vertices (sides).