krit.club logo

Understanding Quadrilaterals - Angle Sum Property

Grade 8CBSE

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

🔑Concepts

•

Angle Sum Property: The sum of the interior angles of any quadrilateral is exactly 360∘360^{\circ}. This can be visualized by drawing a diagonal from one vertex to the opposite vertex, which divides the quadrilateral into two distinct triangles. Since the sum of angles in each triangle is 180∘180^{\circ}, the total sum for the quadrilateral is 180∘+180∘=360∘180^{\circ} + 180^{\circ} = 360^{\circ}.

•

Interior Angles: A quadrilateral has four interior angles. If we denote the vertices as A,B,C,A, B, C, and DD, then the relationship is expressed as ∠A+∠B+∠C+∠D=360∘\angle A + \angle B + \angle C + \angle D = 360^{\circ}.

•

Convex vs. Concave Quadrilaterals: In a convex quadrilateral, all interior angles are less than 180∘180^{\circ}, and the figure bulges outwards. In a concave quadrilateral, at least one interior angle is greater than 180∘180^{\circ} (a reflex angle), making the figure look 'caved in' at one vertex. Remarkably, the sum of interior angles remains 360∘360^{\circ} for both types.

•

Exterior Angle Sum: The sum of the measures of the exterior angles (taken in order) of any convex polygon, including quadrilaterals, is always 360∘360^{\circ}. Imagine extending each side of the quadrilateral; the angles formed outside the shape at each vertex will add up to a full circle.

•

Relationship with Side Count: The sum of interior angles is related to the number of sides nn. For any polygon, the formula is (n−2)×180∘(n - 2) \times 180^{\circ}. For a quadrilateral where n=4n = 4, this calculation gives (4−2)×180∘=360∘(4 - 2) \times 180^{\circ} = 360^{\circ}.

•

Regular Quadrilateral: A square is considered a regular quadrilateral because all its sides and all its interior angles are equal. Visually, this means it has perfect four-fold symmetry, and each interior angle is calculated as 360∘÷4=90∘360^{\circ} \div 4 = 90^{\circ}.

📐Formulae

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

General sum of interior angles for nn sides: S=(n−2)×180∘S = (n - 2) \times 180^{\circ}

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

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

Measure of each exterior angle of a regular nn-sided polygon: 360∘n\frac{360^{\circ}}{n}

💡Examples

Problem 1:

Three angles of a quadrilateral are 110∘110^{\circ}, 70∘70^{\circ}, and 80∘80^{\circ}. Find the measure of the fourth angle.

Solution:

  1. Let the measure of the fourth angle be xx.
  2. According to the angle sum property of a quadrilateral: 110∘+70∘+80∘+x=360∘110^{\circ} + 70^{\circ} + 80^{\circ} + x = 360^{\circ}.
  3. Add the known angles: 260∘+x=360∘260^{\circ} + x = 360^{\circ}.
  4. Subtract 260∘260^{\circ} from both sides: x=360∘−260∘x = 360^{\circ} - 260^{\circ}.
  5. x=100∘x = 100^{\circ}.

Explanation:

We use the property that the sum of all four interior angles must equal 360∘360^{\circ}. By setting up a simple linear equation with the unknown angle xx, we can solve for its value.

Problem 2:

The angles of a quadrilateral are in the ratio 3:5:9:133 : 5 : 9 : 13. Find the measure of each angle.

Solution:

  1. Let the common ratio factor be xx. The four angles are 3x3x, 5x5x, 9x9x, and 13x13x.
  2. Using the angle sum property: 3x+5x+9x+13x=360∘3x + 5x + 9x + 13x = 360^{\circ}.
  3. Combine like terms: 30x=360∘30x = 360^{\circ}.
  4. Solve for xx: x=360∘30=12∘x = \frac{360^{\circ}}{30} = 12^{\circ}.
  5. Calculate each angle:
    • First angle: 3×12∘=36∘3 \times 12^{\circ} = 36^{\circ}
    • Second angle: 5×12∘=60∘5 \times 12^{\circ} = 60^{\circ}
    • Third angle: 9×12∘=108∘9 \times 12^{\circ} = 108^{\circ}
    • Fourth angle: 13×12∘=156∘13 \times 12^{\circ} = 156^{\circ}.

Explanation:

When angles are given in a ratio, we represent them as multiples of a variable xx. We then sum these expressions and set them equal to 360∘360^{\circ} to find the value of xx, which allows us to determine the actual measure of each angle.