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Basic Geometrical Ideas - Quadrilaterals

Grade 6CBSE

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

🔑Concepts

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A quadrilateral is a polygon with four sides, four vertices, and four interior angles. It is named by listing its vertices in a cyclic order (either clockwise or counter-clockwise), such as ABCDABCD.

A general quadrilateral ABCD showing four sides and four vertices.
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Two sides of a quadrilateral are called adjacent sides if they have a common endpoint. Two sides are called opposite sides if they do not have a common endpoint. In quadrilateral ABCDABCD, ABAB and BCBC are adjacent, while ABAB and CDCD are opposite.

Diagram highlighting adjacent sides sharing a vertex.
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Diagonals are line segments connecting opposite vertices of a quadrilateral. Every quadrilateral has exactly two diagonals. For example, in ABCDABCD, the diagonals are ACAC and BDBD.

Quadrilateral ABCD with interior diagonals AC and BD.
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The interior of a quadrilateral is the region enclosed by its boundary. Any point lying inside this boundary is in the interior, while points on the edges are on the boundary, and points outside are in the exterior.

📐Formulae

Number of sides in a quadrilateral = 44

Number of vertices in a quadrilateral = 44

Number of diagonals in a quadrilateral = 22

Sum of interior angles of a quadrilateral = 360∘360^\circ

Angle Sum Property: ∠A+∠B+∠C+∠D=360∘\angle A + \angle B + \angle C + \angle D = 360^\circ

💡Examples

Problem 1:

In a quadrilateral PQRSPQRS, the measures of three angles are ∠P=75∘\angle P = 75^\circ, ∠Q=85∘\angle Q = 85^\circ, and ∠R=110∘\angle R = 110^\circ. Find the measure of the fourth angle ∠S\angle S.

Solution:

  1. We know the Angle Sum Property of a quadrilateral states: ∠P+∠Q+∠R+∠S=360∘\angle P + \angle Q + \angle R + \angle S = 360^\circ
  2. Substitute the given values: 75∘+85∘+110∘+∠S=360∘75^\circ + 85^\circ + 110^\circ + \angle S = 360^\circ
  3. Add the known angles: 270∘+∠S=360∘270^\circ + \angle S = 360^\circ
  4. Subtract 270∘270^\circ from both sides: ∠S=360∘−270∘\angle S = 360^\circ - 270^\circ
  5. ∠S=90∘\angle S = 90^\circ

Explanation:

To find the missing angle, we use the fact that the sum of all four interior angles in any quadrilateral must always equal 360∘360^\circ.

Problem 2:

Given a quadrilateral with vertices K,L,M,NK, L, M, N in order, list all the pairs of opposite sides and all the pairs of adjacent angles.

Solution:

  1. Opposite Sides: These are sides that do not share a vertex. In KLMNKLMN, the pairs are (KLKL and MNMN) and (LMLM and NKNK).
  2. Adjacent Angles: These are angles that share a common side. The pairs are: (∠K,∠L\angle K, \angle L), (∠L,∠M\angle L, \angle M), (∠M,∠N\angle M, \angle N), and (∠N,∠K\angle N, \angle K).

Explanation:

Opposite sides are like the parallel or non-touching sides of a box. Adjacent angles are simply the corners that are next to each other along the same line segment.

Problem 3:

In the quadrilateral WXYZWXYZ shown below, identify: (a) A pair of adjacent sides meeting at vertex XX, and (b) The two diagonals.

Quadrilateral WXYZ for identification of sides and diagonals.

Solution:

(a) The sides meeting at vertex XX are WXWX and XYXY. Therefore, WXWX and XYXY are adjacent sides. (b) The diagonals are the segments joining opposite vertices, which are WYWY and XZXZ.

Explanation:

Adjacent sides always share a common vertex. Diagonals connect vertices that are not next to each other.

Problem 4:

Calculate the missing angle xx in the quadrilateral ABCDABCD where ∠A=120∘\angle A = 120^\circ, ∠B=60∘\angle B = 60^\circ, and ∠C=100∘\angle C = 100^\circ.

Quadrilateral with three given angles and one unknown angle x.

Solution:

The sum of the interior angles of a quadrilateral is 360∘360^\circ. Sum of given angles = 120∘+60∘+100∘=280∘120^\circ + 60^\circ + 100^\circ = 280^\circ. x=360∘−280∘=80∘x = 360^\circ - 280^\circ = 80^\circ. Therefore, ∠D=80∘\angle D = 80^\circ.

Explanation:

We use the Angle Sum Property of a quadrilateral: ∠A+∠B+∠C+∠D=360∘\angle A + \angle B + \angle C + \angle D = 360^\circ.