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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 simple closed figure or polygon formed by four line segments. It has exactly four sides, four vertices, and four interior angles. Visually, it is any four-sided flat shape like a square, rectangle, or a random four-cornered kite.

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The four line segments that form the boundary are called the 'Sides'. In a quadrilateral ABCDABCD, the sides are AB,BC,CD,AB, BC, CD, and DADA. Sides that meet at a common vertex are called 'Adjacent Sides' (e.g., ABAB and BCBC), while sides that do not meet are called 'Opposite Sides' (e.g., ABAB and CDCD).

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The points where two sides meet are called 'Vertices'. For a quadrilateral ABCDABCD, the vertices are the points A,B,C,A, B, C, and DD. The angles formed at these corners are called interior angles, denoted as ∠A,∠B,∠C,\angle A, \angle B, \angle C, and ∠D\angle D.

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Angles in a quadrilateral can be classified as 'Adjacent' or 'Opposite'. 'Adjacent Angles' share a common side (e.g., ∠A\angle A and ∠B\angle B share side ABAB). 'Opposite Angles' are those that do not share a side and face each other across the interior (e.g., ∠A\angle A and ∠C\angle C).

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A 'Diagonal' is a line segment that connects two non-adjacent (opposite) vertices. Every quadrilateral has exactly two diagonals. Visually, if you draw a line from one corner to the corner directly across from it, you have created a diagonal (ACAC and BDBD in quadrilateral ABCDABCD).

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When naming a quadrilateral, the vertices must be listed in a cyclic order, either clockwise or counter-clockwise. For example, if the vertices are P,Q,R,P, Q, R, and SS, you can name it PQRSPQRS or PSRQPSRQ, but you cannot name it PRQSPRQS because PP and RR are opposite vertices, not adjacent ones.

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The region containing all points inside the boundary of the quadrilateral is called the 'Interior'. The 'Boundary' consists of the four sides, and any point not on the boundary or in the interior is in the 'Exterior'. Visually, imagine a fence; the grass inside is the interior, the fence itself is the boundary, and the road outside is 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.