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CBSE · Class 9 · Mathematics

Chapter 12: Quadrilaterals

5 concepts · 40 practice questions with answers · Free to read

This lesson is also in: मराठी · हिंदी

Quadrilaterals and their angles

From daily life

A kite, a window frame and a field with four sides are quadrilaterals. A diagonal splits each into two triangles.

Simple definition

A quadrilateral has 4 sides, 4 angles and 2 diagonals. The angle sum is 360°, because a diagonal splits it into 2 triangles of 180° each.

Formula: ∠A + ∠B + ∠C + ∠D = 360°

ABCDA+B+C+D = 360

In the figure: The red diagonal makes two triangles, so the four angles add to 2 × 180° = 360°.

Rule 1: Find the missing angle — Subtract the three given angles from 360°.

  • Easy: 70°, 90°, 100°: fourth = 100°.
  • Medium: Angles in ratio 1 : 2 : 3 : 4: 36°, 72°, 108°, 144°.
  • Exam-type: Three angles 80° each: fourth 120°.

Rule 2: Exterior angles — The four exterior angles (one at each vertex) add to 360°.

  • Easy: Interior 110° ⇒ exterior 70°.
  • Medium: Exterior angles 80°, 90°, 100°: fourth 90°.
  • Exam-type: A convex quadrilateral has each interior angle less than 180°.
Remember

The angle sum of every simple quadrilateral is 360°, whatever its shape.

Common mistake

Using 180° (the triangle sum) for a quadrilateral.

Memory clue

"Four corners, three-sixty."

Practice questions

Easy
  1. 1How many diagonals does a quadrilateral have?
  2. 2Three angles of a quadrilateral are 50°, 75° and 125°. What is the fourth angle?
  3. 3The angles of a quadrilateral are in the ratio 1:1:2:2. What is the measure of the largest angle?
Medium
  1. 4The angles of a quadrilateral are x°, (x + 30)°, (x + 20)° and (x + 10)°. Find the smallest angle.
  2. 5Three exterior angles of a quadrilateral (one at each vertex) are 100°, 110° and 120°. What is the fourth exterior angle?
  3. 6Three exterior angles of a quadrilateral (one at each vertex) are 50°, 70° and 120°. What is the fourth exterior angle?
Challenging
  1. 7In a quadrilateral ABCD, ∠A = ∠C = 92° and ∠B = 101°. Find ∠D.
  2. 8In quadrilateral ABCD, ∠A = ∠B = x°, ∠C = (x + 24)° and ∠D = (2x − 39)°. Find ∠D.
Show answers (try them yourself first)
  1. 12
  2. 2110°
  3. 3120°
  4. 475°
  5. 530°
  6. 6120°
  7. 775°
  8. 8111°

Properties of a parallelogram

From daily life

A folding gate and a railway track bend into parallelograms. Opposite sides stay parallel and equal as it folds.

Simple definition

A parallelogram has both pairs of opposite sides parallel. Then: opposite sides are equal, opposite angles are equal, adjacent angles add to 180° and diagonals bisect each other.

Formula: AB = DC, AD = BC; ∠A = ∠C; ∠A + ∠B = 180°; AO = OC

OABCD

In the figure: The diagonals meet at O, the midpoint of both, so AO = OC and BO = OD.

Rule 1: Use the properties — Opposite equal, adjacent supplementary, diagonals bisect.

  • Easy: ∠A = 65° ⇒ ∠C = 65°, ∠B = ∠D = 115°.
  • Medium: AB = 7 cm, AD = 4 cm: perimeter 22 cm.
  • Exam-type: Diagonal AC = 10: AO = 5.

Rule 2: Algebra in a parallelogram — Set equal the parts that the property says are equal.

  • Easy: AB = 3x + 1 and DC = 2x + 5: x = 4.
  • Medium: ∠A = 2x, ∠B = x + 30: 3x + 30 = 180, x = 50.
  • Exam-type: Diagonals of a parallelogram are not always equal.
Remember

A diagonal splits a parallelogram into two congruent triangles.

Common mistake

Assuming the diagonals are equal or perpendicular in every parallelogram.

Memory clue

"Opposites equal, diagonals share the middle."

Practice questions

Easy
  1. 1In parallelogram ABCD, AB = 14 cm and BC = 16 cm. What is the perimeter of ABCD?
  2. 2In parallelogram ABCD, ∠A = 110°. What is ∠C?
  3. 3In parallelogram ABCD, AB = (3x + 2) cm and CD = (1x + 8) cm. Find the length of AB.
Medium
  1. 4Two adjacent angles of a parallelogram are in the ratio 1 : 3. What is the measure of the larger angle?
  2. 5The diagonals of parallelogram ABCD meet at O. AC = 30 cm and BD = 8 cm. What is OA + OB?
  3. 6The perimeter of a parallelogram is 54 cm and one side is 14 cm. What is the length of the adjacent side?
Challenging
  1. 7The diagonals of parallelogram ABCD meet at O, with OA = (3x − 6) cm and OC = (x + 6) cm. Find the length of diagonal AC.
  2. 8The diagonals of parallelogram ABCD meet at O. AB = 11 cm, AC = 14 cm and BD = 20 cm. Find the perimeter of triangle AOB.
Show answers (try them yourself first)
  1. 160 cm
  2. 2110°
  3. 311 cm
  4. 4135°
  5. 519 cm
  6. 613 cm
  7. 724 cm
  8. 828 cm
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Properties of a parallelogram

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Tests for parallelograms and special types

From daily life

A carpenter checks if a frame is a parallelogram by measuring the diagonals' halves. No need to test everything.

Simple definition

A quadrilateral is a parallelogram if: both pairs of opposite sides are equal, or both pairs of opposite angles are equal, or the diagonals bisect each other, or one pair of opposite sides is equal and parallel. A rectangle has all angles 90°; a rhombus has all sides equal; a square is both.

Formula: Rectangle: equal diagonals. Rhombus: perpendicular diagonals.

ParallelogramRectangleRhombusSquareSquare =rectangle+ rhombus

In the figure: Every square is a rectangle and a rhombus, and every rectangle or rhombus is a parallelogram.

Rule 1: Which type? — Use the diagonal facts: equal ⇒ rectangle, perpendicular ⇒ rhombus, both ⇒ square.

  • Easy: Diagonals bisect each other and are equal: rectangle.
  • Medium: Diagonals bisect at right angles: rhombus.
  • Exam-type: Equal, perpendicular, bisecting diagonals: square.

Rule 2: Is it a parallelogram? — One test is enough; you need not check all properties.

  • Easy: Opposite sides 5, 7, 5, 7: yes.
  • Medium: Angles 60°, 120°, 60°, 120° in order: yes.
  • Exam-type: Sides 5, 5, 7, 7 in order (kite): not a parallelogram.
Remember

Every rhombus is a parallelogram, but a parallelogram is a rhombus only when all sides are equal.

Common mistake

Thinking equal opposite sides in a different order (5, 5, 7, 7) make a parallelogram.

Memory clue

"Square = rectangle + rhombus."

Practice questions

Easy
  1. 1In quadrilateral PQRS, PQ = 8 cm, QR = 8 cm, RS = 7 cm and SP = 7 cm. Is PQRS necessarily a parallelogram?
  2. 2The diagonals of quadrilateral ABCD meet at O with OA = 10 cm, OC = 10 cm, OB = 14 cm and OD = 7 cm. Is ABCD a parallelogram?
  3. 3The diagonals of quadrilateral ABCD meet at O with OA = 9 cm, OC = 9 cm, OB = 9 cm and OD = 12 cm. Is ABCD a parallelogram?
Medium
  1. 4In quadrilateral ABCD the diagonals meet at O, OB = OD and OA = (3x − 6) cm, OC = (x + 4) cm. For which value of x is ABCD a parallelogram?
  2. 5The diagonals of a rhombus are 20 cm and 48 cm. Find its side.
  3. 6The diagonals of a rhombus are 24 cm and 18 cm. Find its side.
Challenging
  1. 7A rectangle has sides 3 cm and 4 cm. Find the length of its diagonal.
  2. 8In rhombus ABCD, ∠A = 80°. What is ∠ABD (the angle that diagonal BD makes with side AB)?
Show answers (try them yourself first)
  1. 1No, it may be a kite
  2. 2No
  3. 3No
  4. 45
  5. 526 cm
  6. 615 cm
  7. 75 cm
  8. 850°
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Tests for parallelograms and special types

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Midpoint theorem and medians

From daily life

A rope joining the middles of two sides of a triangular tent is parallel to the base and exactly half as long.

Simple definition

Midpoint theorem: the segment joining the midpoints of two sides of a triangle is parallel to the third side and half of it. Converse: a line through the midpoint of one side parallel to another side bisects the third. A median joins a vertex to the midpoint of the opposite side.

Formula: DE ∥ BC and DE = ½ BC

ABCDEDE = BC/2D, E midpoints

In the figure: D and E are midpoints of AB and AC, so DE is parallel to BC and half as long.

Rule 1: Apply the theorem — Midpoint segment = half the base, and parallel to it.

  • Easy: BC = 12 ⇒ DE = 6.
  • Medium: DE = 7 ⇒ BC = 14.
  • Exam-type: Triangle formed by three midpoints has half the sides of the original, perimeter half.

Rule 2: Medians — A triangle has three medians; they meet at the centroid, which divides each in the ratio 2 : 1.

  • Easy: Median 9 cm: centroid is 6 cm from the vertex and 3 cm from the side.
  • Medium: A median halves the area of the triangle.
  • Exam-type: In a right triangle, the median to the hypotenuse equals half the hypotenuse.
Remember

The midpoint theorem fails if the points are not midpoints.

Common mistake

Saying DE = BC instead of DE = ½ BC.

Memory clue

"Mid to mid: parallel and half."

Practice questions

Easy
  1. 1In triangle ABC, D and E are the midpoints of AB and AC. If BC = 26 cm, what is DE?
  2. 2In triangle ABC, D and E are the midpoints of AB and AC and DE = 6 cm. What is BC?
  3. 3The sides of triangle ABC are 26 cm, 8 cm and 22 cm. The midpoints of its three sides are joined to form a smaller triangle. What is the perimeter of the smaller triangle?
Medium
  1. 4In triangle ABC, D and E are the midpoints of AB and AC. DE = (2x + 1) cm and BC = (5x − 6) cm. Find DE.
  2. 5In triangle ABC, D and E are the midpoints of AB and AC and ∠B = 75°. What is ∠ADE?
  3. 6The midpoints of the sides of a triangle of area 28 cm² are joined to form a smaller triangle. What is the area of the smaller triangle?
Challenging
  1. 7The medians of a triangle meet at the centroid G. The median AD is 36 cm long. What is the length AG?
  2. 8The diagonals of a quadrilateral are 24 cm and 14 cm. The midpoints of its four sides are joined in order to form a new quadrilateral. What is the perimeter of the new quadrilateral?
Show answers (try them yourself first)
  1. 113 cm
  2. 212 cm
  3. 328 cm
  4. 417 cm
  5. 575°
  6. 67 cm²
  7. 724 cm
  8. 838 cm
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Midpoint theorem and medians

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Tiling and angles at a point

From daily life

Bathroom tiles fit without gaps because the corners meeting at a point add to exactly 360°. Squares and hexagons tile; regular pentagons do not.

Simple definition

Angles around a point add to 360°. A shape tiles the plane if copies fit with no gaps or overlaps, so the angles meeting at each vertex sum to 360°. Every quadrilateral tiles the plane because its four angles make 360°.

Formula: Sum of angles at a point = 360°

90901008090+90+100+80 = 360

In the figure: The four angles around the point add up to 360°, so shapes fit there with no gap.

Rule 1: Find an angle at a point — Subtract the known angles from 360°.

  • Easy: 100°, 120°, 90°: remaining 50°.
  • Medium: Four equal angles: 90° each.
  • Exam-type: Six equilateral triangles meet at a point: 6 × 60° = 360°.

Rule 2: Will it tile? — Check whether whole copies of the corner angle make 360°.

  • Easy: Square: 4 × 90° = 360°, tiles.
  • Medium: Regular hexagon (120°): 3 × 120° = 360°, tiles.
  • Exam-type: Regular pentagon (108°): 360 ÷ 108 is not a whole number, does not tile.
Remember

Only triangles, quadrilaterals and hexagons can tile as regular shapes: 6, 4 and 3 of them meet at a point.

Common mistake

Forgetting that the full turn around a point is 360°, not 180°.

Memory clue

"Full turn, three-sixty."

Practice questions

Easy
  1. 1Which regular polygon has an interior angle of 120°?
  2. 2What is each interior angle of a regular decagon?
  3. 3Can copies of a regular dodecagon alone tile a flat surface with no gaps?
Medium
  1. 4At a vertex of a tiling, two tiles with corner angles 108° and 90° meet and the remaining space is filled by one more tile corner. What is the angle of that corner?
  2. 5A rectangular floor is 20 m by 12 m. It is covered with square tiles of side 4 m. How many tiles are needed?
  3. 6Each exterior angle of a regular polygon is 18°. How many sides does the polygon have?
Challenging
  1. 7Regular polygons meet at a vertex without gaps. The corners of all but one are 90°, 120°, 90°. The remaining polygon is regular. How many sides does it have?
  2. 8Each exterior angle of a regular polygon is 15°. What is the sum of all its interior angles?
Show answers (try them yourself first)
  1. 1hexagon
  2. 2144°
  3. 3No
  4. 4162°
  5. 515
  6. 620
  7. 73
  8. 83960°
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Tiling and angles at a point

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