Understanding Quadrilaterals Class 8 Maths NCERT

What are Quadrilaterals?

In geometry, a quadrilateral is a polygon that has exactly four sides, four vertices (corners), and four interior angles. The word 'quadrilateral' comes from the Latin words 'quadri' (meaning four) and 'latus' (meaning side). Think of shapes like a blackboard, a window, or even the screen you're reading this on – most of them are quadrilaterals! These shapes are fundamental in our everyday lives and in many areas of mathematics and engineering.

Quadrilaterals can be broadly classified into two types: Convex and Concave.

• Convex Quadrilateral: All interior angles are less than 180 degrees, and all diagonals lie entirely inside the figure. Most of the quadrilaterals you'll study, like squares, rectangles, and parallelograms, are convex.
• Concave Quadrilateral: At least one interior angle is greater than 180 degrees (a reflex angle), and at least one part of a diagonal lies outside the figure. Imagine a dart shape or an arrowhead – that's often a concave quadrilateral.

Understanding these basic definitions is the first step to mastering the fascinating world of quadrilaterals. Remember, a quadrilateral is always a closed figure formed by four line segments.

Types of Quadrilaterals and Their Properties

Trapezium (or Trapezoid)

A quadrilateral with at least one pair of parallel sides. The parallel sides are called bases, and the non-parallel sides are called legs.

Kite

A quadrilateral where two pairs of equal-length sides are adjacent to each other. Its diagonals are perpendicular to each other, and one diagonal bisects the other.

Parallelogram

A quadrilateral with two pairs of parallel sides. Key properties include: opposite sides are equal, opposite angles are equal, consecutive angles are supplementary (add up to 180°), and diagonals bisect each other.

Rhombus

A parallelogram with all four sides equal in length. In addition to parallelogram properties, its diagonals bisect each other at right angles (90°) and also bisect the angles of the rhombus.

Rectangle

A parallelogram with all four angles equal to 90 degrees. Its diagonals are equal in length and bisect each other. Opposite sides are equal.

Square

A rectangle with all four sides equal in length (or a rhombus with all four angles equal to 90 degrees). It possesses all properties of a parallelogram, rhombus, and rectangle. Its diagonals are equal, bisect each other at 90°, and bisect the angles.

Understanding the Angle Sum Property of a Quadrilateral

1. Step 1: Draw a Quadrilateral — Start by drawing any simple convex quadrilateral, let's call its vertices A, B, C, and D. Label its interior angles as ∠A, ∠B, ∠C, and ∠D.
2. Step 2: Draw a Diagonal — Now, draw one of its diagonals. For instance, draw the diagonal connecting vertex A to vertex C. This diagonal divides the quadrilateral ABCD into two distinct triangles: ΔABC and ΔADC.
3. Step 3: Recall the Angle Sum Property of a Triangle — Remember from previous classes that the sum of the interior angles of any triangle is always 180 degrees. This fundamental property will help us find the sum of angles in our quadrilateral.
4. Step 4: Sum the Angles of the Triangles — For ΔABC, the sum of its angles is ∠BAC + ∠B + ∠BCA = 180°. For ΔADC, the sum of its angles is ∠DAC + ∠D + ∠DCA = 180°. The sum of all interior angles of the quadrilateral ABCD is the sum of the angles of these two triangles: (∠BAC + ∠B + ∠BCA) + (∠DAC + ∠D + ∠DCA). Notice that (∠BAC + ∠DAC) forms ∠A of the quadrilateral, and (∠BCA + ∠DCA) forms ∠C of the quadrilateral. Thus, ∠A + ∠B + ∠C + ∠D = 180° + 180° = 360°. So, the sum of the interior angles of any quadrilateral is always 360 degrees.

Worked Examples: Applying Quadrilateral Properties

• Example 1: Finding an Unknown Angle The angles of a quadrilateral are 70°, 90°, and 110°. Find the measure of the fourth angle. Solution: We know that the sum of the interior angles of a quadrilateral is 360°. Let the fourth angle be x. So, 70° + 90° + 110° + x = 360° 270° + x = 360° x = 360° - 270° x = 90° Therefore, the fourth angle is 90°.
• Example 2: Using Parallelogram Properties In a parallelogram ABCD, if ∠A = 75°, find the measures of ∠B, ∠C, and ∠D. Solution: In a parallelogram, opposite angles are equal, and consecutive angles are supplementary. 1. Find ∠C: Since opposite angles are equal, ∠C = ∠A = 75°. 2. Find ∠B: Since consecutive angles are supplementary, ∠A + ∠B = 180°. 75° + ∠B = 180° ∠B = 180° - 75° = 105°. 3. Find ∠D: Since opposite angles are equal, ∠D = ∠B = 105°. Thus, ∠B = 105°, ∠C = 75°, and ∠D = 105°.
• Example 3: Diagonals of a Rhombus The diagonals of a rhombus are 16 cm and 12 cm. Find the length of each side of the rhombus. Solution: Let the rhombus be ABCD, and its diagonals be AC and BD. Let them intersect at O. In a rhombus, diagonals bisect each other at right angles. So, AO = AC/2 = 16/2 = 8 cm. BO = BD/2 = 12/2 = 6 cm. Also, ∠AOB = 90°. Consider the right-angled triangle ΔAOB. By Pythagoras theorem: AB² = AO² + BO² AB² = 8² + 6² AB² = 64 + 36 AB² = 100 AB = √100 AB = 10 cm. Since all sides of a rhombus are equal, each side of the rhombus is 10 cm.

Exam Tip: Avoiding Common Mistakes

While understanding quadrilaterals, students often make a few common mistakes. One frequent error is confusing the properties of different types of quadrilaterals, especially between a rectangle and a rhombus, or thinking all parallelograms are rectangles. Remember that a square is both a rectangle and a rhombus, but a rectangle is not always a square, and a rhombus is not always a square. Always draw a clear diagram for the given problem, even if it's not provided, as it helps visualise the properties. Carefully read the question to identify which specific type of quadrilateral is mentioned, as its unique properties will be key to solving the problem. Double-check your calculations, especially when dealing with angle sums or using the Pythagorean theorem for diagonal problems.

Practice Questions with Solutions

• Q: The angles of a quadrilateral are in the ratio 2:3:4:1. Find the measure of each angle. A: Step 1: Let the angles be 2x, 3x, 4x, and x. Step 2: The sum of the angles in a quadrilateral is 360°. 2x + 3x + 4x + x = 360° 10x = 360° x = 360° / 10 x = 36° Step 3: Calculate each angle. First angle = 2 36° = 72° Second angle = 3 36° = 108° Third angle = 4 36° = 144° Fourth angle = 1 36° = 36° Final answer: The angles are 72°, 108°, 144°, and 36°.
• Q: One angle of a parallelogram is 60°. Find the measures of the remaining three angles. A: Step 1: In a parallelogram, opposite angles are equal. So, if one angle is 60°, its opposite angle is also 60°. Step 2: Consecutive angles in a parallelogram are supplementary (sum to 180°). Let the given angle be A = 60°. Then B = 180° - 60° = 120°. Step 3: The angle opposite to B is D, so D = 120°. Final answer: The remaining angles are 120°, 60°, and 120°.
• Q: The adjacent sides of a rectangle are 8 cm and 6 cm. Find the length of its diagonal. A: Step 1: In a rectangle, the adjacent sides form a right angle (90°). The diagonal forms the hypotenuse of a right-angled triangle with these sides. Step 2: Apply the Pythagorean theorem: (diagonal)² = (side1)² + (side2)². (diagonal)² = 8² + 6² (diagonal)² = 64 + 36 (diagonal)² = 100 Step 3: Take the square root to find the diagonal length. diagonal = √100 = 10 cm. Final answer: The length of the diagonal is 10 cm.
• Q: A quadrilateral has two opposite angles measuring 100° and 80°. If the other two angles are equal, find the measure of each of the equal angles. A: Step 1: Let the two equal angles be y. Step 2: The sum of the angles in a quadrilateral is 360°. 100° + 80° + y + y = 360° 180° + 2y = 360° Step 3: Solve for y. 2y = 360° - 180° 2y = 180° y = 180° / 2 y = 90° Final answer: Each of the equal angles measures 90°.

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