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Quadrilaterals - What Exactly is a Quadrilateral?

Grade 9CBSE

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

🔑Concepts

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A quadrilateral is a closed figure with four sides, four vertices, and four angles. It is formed by joining four points in an order such that no three points are collinear.

A general quadrilateral ABCD with four sides and four vertices.
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The sum of the interior angles of a quadrilateral is always 360∘360^\circ. This is known as the Angle Sum Property of a quadrilateral. It can be verified by dividing the quadrilateral into two triangles using a diagonal; since each triangle's angle sum is 180∘180^\circ, the total sum is 2×180∘=360∘2 \times 180^\circ = 360^\circ.

Quadrilateral divided into two triangles by a diagonal.
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Consecutive sides are sides that share a common vertex (e.g., ABAB and BCBC). Opposite sides are sides that do not share a vertex (e.g., ABAB and CDCD). Similarly, opposite angles are those not sharing a common side (e.g., ∠A\angle A and ∠C\angle C).

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A convex quadrilateral has all its interior angles less than 180∘180^\circ and both its diagonals lie entirely inside the figure. In a concave (re-entrant) quadrilateral, at least one interior angle is reflex (>180∘> 180^\circ) and one diagonal lies outside.

📐Formulae

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

∠A+∠B+∠C+∠D=360∘\angle A + \angle B + \angle C + \angle D = 360^\circ

Perimeter of Quadrilateral ABCD=AB+BC+CD+DA\text{Perimeter of Quadrilateral } ABCD = AB + BC + CD + DA

💡Examples

Problem 1:

The three angles of a quadrilateral are 75∘75^\circ, 90∘90^\circ, and 75∘75^\circ. Find the measure of the fourth angle.

Quadrilateral with angles labeled 75, 90, 75 and x.

Solution:

Let the fourth angle be xx. According to the angle sum property of a quadrilateral: 75∘+90∘+75∘+x=360∘75^\circ + 90^\circ + 75^\circ + x = 360^\circ 240∘+x=360∘240^\circ + x = 360^\circ x=360∘−240∘x = 360^\circ - 240^\circ x=120∘x = 120^\circ The fourth angle is 120∘120^\circ.

Explanation:

We use the fundamental property that all four internal angles of any quadrilateral must sum up to exactly 360∘360^\circ. By subtracting the sum of the known angles from 360∘360^\circ, we find the unknown angle.

Problem 2:

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

Quadrilateral with angles represented in terms of x based on ratios.

Solution:

Let the angles be 3x,5x,9x, and 13x3x, 5x, 9x, \text{ and } 13x. By Angle Sum Property: 3x+5x+9x+13x=360∘3x + 5x + 9x + 13x = 360^\circ 30x=360∘30x = 360^\circ x=360∘30=12∘x = \frac{360^\circ}{30} = 12^\circ Now find the individual angles: Angle 1=3×12∘=36∘\text{Angle 1} = 3 \times 12^\circ = 36^\circ Angle 2=5×12∘=60∘\text{Angle 2} = 5 \times 12^\circ = 60^\circ Angle 3=9×12∘=108∘\text{Angle 3} = 9 \times 12^\circ = 108^\circ Angle 4=13×12∘=156∘\text{Angle 4} = 13 \times 12^\circ = 156^\circ

Explanation:

When angles are given in ratio, we represent them as multiples of a common variable xx. Using the angle sum property (360∘360^\circ), we solve for xx and kemudian multiply by each ratio part to get the actual angles.