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Understanding Quadrilaterals - Angle Sum Property

Grade 8CBSE

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

🔑Concepts

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The Angle Sum Property states that the sum of all interior angles of a quadrilateral is exactly 360∘360^{\circ}. This can be visualized by dividing the quadrilateral into two triangles using a diagonal; since each triangle's angles sum to 180∘180^{\circ}, the total sum is 2×180∘=360∘2 \times 180^{\circ} = 360^{\circ}.

A quadrilateral ABCD divided by diagonal AC into two triangles T1 and T2.
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In a convex polygon, the sum of the measures of the exterior angles (one at each vertex) is always 360∘360^{\circ}, regardless of the number of sides. For a quadrilateral, ∠1+∠2+∠3+∠4=360∘\angle 1 + \angle 2 + \angle 3 + \angle 4 = 360^{\circ}.

A quadrilateral with extended sides showing four exterior angles.
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The interior angle sum of a polygon with nn sides is calculated using the formula S=(n−2)×180∘S = (n - 2) \times 180^{\circ}. For a quadrilateral, n=4n=4, so S=(4−2)×180∘=360∘S = (4-2) \times 180^{\circ} = 360^{\circ}.

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For a regular polygon (where all sides and angles are equal), each interior angle can be found by dividing the total sum by the number of sides: Angle=(n−2)×180∘n\text{Angle} = \frac{(n - 2) \times 180^{\circ}}{n}.

📐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.

Problem 3:

In the given quadrilateral ABCDABCD, ∠A=x∘\angle A = x^{\circ}, ∠B=(2x+10)∘\angle B = (2x + 10)^{\circ}, ∠C=80∘\angle C = 80^{\circ}, and ∠D=90∘\angle D = 90^{\circ}. Find the value of xx and the measures of angles AA and BB.

Quadrilateral ABCD with interior angles labeled as x, 2x+10, 80, and 90 degrees.

Solution:

  1. According to the Angle Sum Property: ∠A+∠B+∠C+∠D=360∘\angle A + \angle B + \angle C + \angle D = 360^{\circ}
  2. Substitute the given values: x+(2x+10)+80+90=360x + (2x + 10) + 80 + 90 = 360
  3. Combine like terms: 3x+180=3603x + 180 = 360
  4. Subtract 180180 from both sides: 3x=1803x = 180
  5. Divide by 33: x=60x = 60
  6. Calculate the angles: ∠A=60∘\angle A = 60^{\circ} ∠B=2(60)+10=130∘\angle B = 2(60) + 10 = 130^{\circ}

Explanation:

We use the property that all interior angles of a quadrilateral add up to 360 degrees. By setting up a linear equation with the unknown variable xx, we solve for xx and kemudian substitute it back into the expressions for the individual angles.

Problem 4:

Find the value of yy in the quadrilateral where three exterior angles are 100∘100^{\circ}, 80∘80^{\circ}, and 110∘110^{\circ} as shown.

Quadrilateral with four exterior angles labeled 100, 80, 110, and y.

Solution:

  1. The sum of exterior angles of any convex polygon is 360∘360^{\circ}.
  2. Let the fourth exterior angle be y∘y^{\circ}.
  3. 100+80+110+y=360100 + 80 + 110 + y = 360
  4. 290+y=360290 + y = 360
  5. y=360−290y = 360 - 290
  6. y=70y = 70 Therefore, the value of yy is 70∘70^{\circ}.

Explanation:

Since the sum of exterior angles is always constant at 360 degrees for any polygon, we simply subtract the sum of the known exterior angles from 360 to find the missing one.