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State Board · Class 9 · Mathematics Part 2

Lesson 5: Quadrilaterals

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

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The parallelogram and its properties

From daily life

When a sliding iron gate is opened or closed, the lengths of the sides of its quadrilateral stay the same but the angles change. Still, the opposite sides always remain parallel and equal. Such a quadrilateral is a parallelogram.

Simple definition

A quadrilateral whose both pairs of opposite sides are parallel is called a parallelogram. Its four properties: (1) opposite sides are equal, (2) opposite angles are equal, (3) adjacent angles are supplementary (sum 180°), (4) the diagonals bisect each other.

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The parallelogram and its properties

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Rule 1: Opposite sides and angles are equal — AB = DC and AD = BC. Also ∠A = ∠C and ∠B = ∠D. So knowing one side gives the opposite side.

  • Easy: In ▱PQRS, PQ = 6, QR = 4, ∠P = 70°: SR = 6, PS = 4 and ∠R = 70°.
  • Medium: The perimeter of a parallelogram with sides 7 and 5 is 2 × (7 + 5) = 24.
  • Exam type: Perimeter 60 and one side 6 more than the other. Smaller side x: 2(x + x + 6) = 60 ⇒ 4x + 12 = 60 ⇒ x = 12. The sides are 12 and 18.

Rule 2: Adjacent angles are supplementary: sum 180° — Subtract a given angle from 180° to get the adjacent angle. The opposite angles are equal.

Equation from the angles of a parallelogramAdjacent angles of a parallelogram are supplementaryABCD(3x+10)°(2x+20)°(3x+10)°(2x+20)°(3x + 10) + (2x + 20) = 180 ⇒ 5x = 150 ⇒ x = 30∠A = ∠C = 100° and ∠B = ∠D = 80°

In the figure: Write the equation for the sum of two adjacent angles; we get x = 30.

  • Easy: If ∠A = 65°, then ∠B = 180° − 65° = 115°; ∠C = 65°, ∠D = 115°.
  • Medium: Adjacent angles are in the ratio 2 : 3. Total 5 parts; one part is 180 ÷ 5 = 36. The angles are 72° and 108°; all four are 72°, 108°, 72°, 108°.
  • Exam type: ∠A = (3x + 10)° and ∠B = (2x + 20)°: 5x + 30 = 180 ⇒ x = 30. ∠A = 100° and ∠B = 80°.

Rule 3: The diagonals bisect each other — The point where the two diagonals meet is the midpoint of each. So AO = OC and BO = OD.

  • Easy: AO = 5 ⇒ OC = 5 and AC = 10.
  • Medium: BD = 14 ⇒ BO = OD = 7.
  • Exam type: AO = (3x − 1) and OC = (x + 9): 3x − 1 = x + 9 ⇒ x = 5. AO = 14 and AC = 28.
Remember

In a parallelogram opposite sides and opposite angles are equal; adjacent angles are supplementary; the diagonals bisect each other.

Common mistake

Thinking adjacent angles are equal. They are not; their sum is 180° (only in a rectangle both are 90°).

Shortcut

The diagonals need not be equal in length. They only bisect each other. Memory clue: Parallelogram: opposite sides equal, opposite angles equal, diagonals bisect each other.

Spot the difference

PropertyIn a parallelogram
Opposite sidesequal and parallel
Opposite anglesequal
Adjacent anglessum 180°
Diagonalsbisect each other (not necessarily equal)

Practice questions

Easy
  1. 1In ▱PQRS, if PQ = 8, what is SR?
  2. 2In ▱ABCD, if ∠A = 110°, what is ∠C?
  3. 3If ∠A = 60°, what is ∠B?
Medium
  1. 4The perimeter of a parallelogram is 54 and one side is 15. Find the other side.
  2. 5If adjacent angles are in the ratio 1 : 2, find them.
  3. 6Diagonals AC = 18 and BD = 12 meet at O. Find AO and BO. If AB = 10, find the perimeter of ∆AOB.
Challenging
  1. 7∠A = (7x + 5)° and ∠B = (3x + 25)° are adjacent angles. Find x and both angles.
  2. 8AO = (2x + 1), OC = (4x − 9), BO = (3y − 2), OD = (y + 8). Find x, y, AC and BD.
Show answers (try them yourself first)
  1. 1SR = 8. 2) ∠C = 110°. 3) ∠B = 180° − 60° = 120°.
  2. 4The sum of the two sides is 54 ÷ 2 = 27. The other side is 27 − 15 = 12. 5) Total 3 parts; one part is 60°. The angles are 60° and 120°. 6) AO = 9, BO = 6. Perimeter 9 + 6 + 10 = 25.
  3. 710x + 30 = 180 ⇒ x = 15. ∠A = 110° and ∠B = 70°. 8) 2x + 1 = 4x − 9 ⇒ x = 5, AO = 11, AC = 22. 3y − 2 = y + 8 ⇒ y = 5, BO = 13, BD = 26.

Tests for a parallelogram

From daily life

If a carpenter takes wooden strips of equal length for both pairs of opposite sides, the frame automatically becomes a parallelogram. We have to decide from measurements rather than from the look, which is why we need tests.

Simple definition

Reading the properties the other way round gives the tests. If any one of the following is true in a quadrilateral, it is a parallelogram.

Four tests that show a quadrilateral is a parallelogramFour tests for a parallelogramABCDBoth pairs ofopposite sides equalααββABCDBoth pairs ofopposite angles equalABCDDiagonals bisecteach otherABCDOne pair of oppositesides equal and parallel

In the figure: Marks for the four tests: equal marks mean equal lengths or equal angles.

Rule 1: Tests 1 and 2: opposite sides or opposite angles equal — If both pairs of opposite sides are equal, or both pairs of opposite angles are equal, the quadrilateral is a parallelogram.

  • Easy: AB = DC = 5 and AD = BC = 3: opposite sides are equal, so ABCD is a parallelogram.
  • Medium: ∠A = ∠C = 70° and ∠B = ∠D = 110°: opposite angles are equal (70 + 110 + 70 + 110 = 360), so it is a parallelogram.
  • Exam type: Sides in order 5, 6, 5, 7: AB = 5 and CD = 5 but BC = 6 and DA = 7 are unequal. So this test does not prove it a parallelogram.

Rule 2: Test 3: the diagonals bisect each other — If the meeting point of the diagonals is the midpoint of both diagonals, it is a parallelogram.

  • Easy: AO = OC and BO = OD: a parallelogram.
  • Medium: AO = OC = 6 and BO = OD = 8 (AC = 12, BD = 16): a parallelogram.
  • Exam type: AO = 5, OC = 5, BO = 4, OD = 6: AC is bisected but BD is not, so it is not a parallelogram.

Rule 3: Test 4: one pair of opposite sides equal and parallel — It is enough that one pair of opposite sides is both parallel and equal.

  • Easy: AB ∥ DC and AB = DC = 9: a parallelogram.
  • Medium: In ▱ABCD, E is the midpoint of AB and F of DC. AE = ½AB = ½DC = FC and AE ∥ FC. So AECF is a parallelogram.
  • Exam type: AB ∥ DC, AB = DC = 9 and ∠A = 90°. It is a parallelogram, and since one angle is 90°, a rectangle. If AD = 12, the diagonal AC = 15 (9² + 12² = 15²).
Remember

Any one of the four tests is enough. But the words "both pairs" must be satisfied (except in the 4th test).

Common mistake

Declaring a parallelogram because only one pair of sides is equal. Equal is not enough; they must also be parallel.

Shortcut

Pick the test from what is given: sides, angles, diagonals or one pair of sides. Memory clue: Test: "one pair of sides equal and parallel" is the easiest.

Spot the difference

What is givenTest to use
all four side lengthsboth pairs of opposite sides equal?
all four anglesboth pairs of opposite angles equal?
parts of diagonalsdo the diagonals bisect each other?
one pair of sidesequal and parallel?

Practice questions

Easy
  1. 1What is the quadrilateral ABCD with AB = DC = 7 and BC = AD = 4?
  2. 2What is a quadrilateral whose diagonals bisect each other?
  3. 3Is a quadrilateral with sides 5, 6, 5, 7 in order a parallelogram?
Medium
  1. 4By which test is a quadrilateral with angles 80°, 100°, 80°, 100° in order a parallelogram?
  2. 5Is a quadrilateral with angles 80°, 80°, 100°, 100° in order a parallelogram by this test?
  3. 6AO = 5, OC = 5, BO = 4, OD = 6. Is it a parallelogram?
Challenging
  1. 7AB ∥ DC, AB = DC = 9, AD = 12 and ∠A = 90°. Which figure is it? Find AC.
  2. 8In ▱ABCD, E and F are the midpoints of AB and DC. Why is AECF a parallelogram?
Show answers (try them yourself first)
  1. 1A parallelogram. 2) A parallelogram. 3) No: BC = 6 and AD = 7 are unequal.
  2. 4Opposite angles equal (test 2). 5) No: the opposite angles 80° and 100° are unequal. 6) No: BD is not bisected (4 ≠ 6).
  3. 7One pair equal and parallel ⇒ parallelogram; ∠A = 90° ⇒ rectangle. AC = √(9² + 12²) = 15. 8) AE = ½AB = ½DC = FC and AE ∥ FC. One pair is equal and parallel, so it is a parallelogram (test 4).

Rectangle, rhombus and square

From daily life

A book is a rectangle, a carrom board surface is a square, and a kite or diamond shape looks like a rhombus. All of these are special kinds of parallelograms.

Simple definition

Rectangle: a parallelogram with all angles 90°. Rhombus: a parallelogram with all sides equal. Square: all angles 90° and all sides equal, that is both a rectangle and a rhombus.

Rectangle and rhombus are parallelograms; a square is bothThe family of quadrilateralsQuadrilateralParallelogramopposite sides parallelRectangleall angles 90°Rhombusall sides equalSquare90° angles and equal sides

In the figure: Family tree: the rectangle and rhombus come from the parallelogram, and the square comes from both.

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Rectangle, rhombus and square

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Rule 1: Rectangle: diagonals are equal and bisect each other — Since all angles of a rectangle are 90°, Pythagoras works on the diagonal: diagonal² = length² + breadth².

  • Easy: Rectangle 8 × 6: diagonal = √(64 + 36) = 10.
  • Medium: Diagonal 13 and one side 5: the other side is √(169 − 25) = √144 = 12.
  • Exam type: Perimeter 34 and diagonal 13. l + b = 17 and l² + b² = 169. (l + b)² = 289 ⇒ 2lb = 289 − 169 = 120 ⇒ lb = 60. The sides are 12 and 5 (12 + 5 = 17, 12 × 5 = 60).

Rule 2: Rhombus: the diagonals are perpendicular and bisect each other — All four sides are equal. The diagonals meet at a right angle, so half-diagonals and a side form a right triangle. Area = ½ × diagonal₁ × diagonal₂.

  • Easy: The perimeter of a rhombus of side 5 is 4 × 5 = 20.
  • Medium: Diagonals 16 and 12: halves 8 and 6, side √(64 + 36) = 10. Area ½ × 16 × 12 = 96.
  • Exam type: Side 13 and one diagonal 24: half is 12, the other half is √(169 − 144) = 5, so the other diagonal is 10. Area ½ × 24 × 10 = 120.

Rule 3: Square: properties of both the rectangle and the rhombus — The diagonals are equal, perpendicular and bisect each other. Area of a square = ½ × diagonal².

  • Easy: A square of side 7 has area 7 × 7 = 49 and perimeter 28.
  • Medium: A square with diagonal 12 has area ½ × 12 × 12 = 72.
  • Exam type: A rhombus has diagonals 6 and 8. Side √(9 + 16) = 5. The diagonals are unequal, so it is not a square. It would be a square if the diagonals were equal.
Remember

A square is both a rectangle and a rhombus. But not every rectangle is a square.

Common mistake

Thinking the diagonals of a rhombus are equal. They are perpendicular, not equal (except in a square).

Shortcut

Rectangle: diagonals equal; rhombus: diagonals perpendicular; square: both. Memory clue: Rectangle = parallelogram + one right angle; rhombus = parallelogram + adjacent sides equal.

Spot the difference

RectangleRhombusSquare
All angles 90°yesnoyes
All sides equalnoyesyes
Diagonals equalyesnoyes
Diagonals perpendicularnoyesyes

Practice questions

Easy
  1. 1If the sides of a rectangle are 6 and 8, what is the diagonal?
  2. 2What is the perimeter of a rhombus of side 9?
  3. 3What is the perimeter of a square of side 7?
Medium
  1. 4A rectangle has diagonal 20 and one side 12. Find the other side and the area.
  2. 5The diagonals of a rhombus are 10 and 24. Find the side and the area.
  3. 6If the diagonal of a square is 16, what is its area?
Challenging
  1. 7A rhombus has side 15 and one diagonal 18. Find the other diagonal and the area.
  2. 8What is the perimeter of a rhombus of side 30 cm? If one of its angles is 90°, which quadrilateral is it?
Show answers (try them yourself first)
  1. 1√(36 + 64) = 10. 2) 4 × 9 = 36. 3) 4 × 7 = 28.
  2. 4√(400 − 144) = 16. Area 12 × 16 = 192. 5) Halves 5 and 12 ⇒ side √(25 + 144) = 13. Area ½ × 10 × 24 = 120. 6) ½ × 16 × 16 = 128.
  3. 7Half is 9; the other half is √(225 − 81) = √144 = 12. The other diagonal is 24. Area ½ × 18 × 24 = 216. 8) Perimeter 4 × 30 = 120 cm. If one angle is 90°, all are 90°, so it is a square.

The trapezium

From daily life

A seat built beside a road, or the slanted side of a bucket, is wide at the bottom and narrow at the top. Only the top and bottom sides are parallel; the two side edges are slanted.

Simple definition

A quadrilateral with exactly one pair of parallel opposite sides is a trapezium. The parallel sides are the bases and the slanted sides are the legs. If the legs are equal it is an isosceles trapezium. The segment joining the midpoints of the legs is the median.

A trapezium, its median and an isosceles trapeziumA trapezium and its medianABCDPQAB = 14 (bottom), DC = 8 (top), AB ∥ DCMedian PQ = (14 + 8) ÷ 2 = 11 (PQ ∥ AB)AD = BC, so it is isosceles: ∠A = ∠B

In the figure: AB ∥ DC. The median PQ is parallel to the bases and equals half the sum of the two bases.

Rule 1: Angles on the same leg are supplementary — If AB ∥ DC then ∠A + ∠D = 180° and ∠B + ∠C = 180° (angles on the same leg).

  • Easy: AB ∥ DC and ∠A = 70° ⇒ ∠D = 180° − 70° = 110°.
  • Medium: If ∠A = 95° then ∠D = 85°; if ∠B = 100° then ∠C = 80°.
  • Exam type: ∠A = (3x + 5)° and ∠D = (2x + 25)°: 5x + 30 = 180 ⇒ x = 30. ∠A = 95° and ∠D = 85°.

Rule 2: Isosceles trapezium: base angles are equal — If legs AD = BC then ∠A = ∠B and ∠C = ∠D. Its diagonals are also equal.

  • Easy: Bases 12 and 8, legs 5 and 5: perimeter 12 + 8 + 5 + 5 = 30.
  • Medium: ∠A = ∠B = 70° ⇒ ∠C = ∠D = 180° − 70° = 110°. Sum 70 + 70 + 110 + 110 = 360.
  • Exam type: In an isosceles trapezium ∠A = (5x + 10)° and ∠B = (7x − 30)°: equal ⇒ 5x + 10 = 7x − 30 ⇒ x = 20. ∠A = ∠B = 110° and ∠C = ∠D = 70°.

Rule 3: Median = half the sum of the two bases — The median is parallel to the bases. Length = (base₁ + base₂) ÷ 2.

  • Easy: Bases 14 and 8: median (14 + 8) ÷ 2 = 11.
  • Medium: Median 15 and one base 18: the other base is 2 × 15 − 18 = 12.
  • Exam type: Bases (3x + 5) and (x + 7), median 12: (4x + 12) ÷ 2 = 12 ⇒ 4x = 12 ⇒ x = 3. The bases are 14 and 10.
Remember

In a trapezium the two angles on the same leg are supplementary. In an isosceles one the base angles are equal.

Common mistake

Taking all opposite sides of a trapezium to be parallel. Only the bases are parallel, not the legs.

Shortcut

For the median, add the two bases and divide by 2. The height does not matter. Memory clue: Median = (base + base) ÷ 2: "two bases, half".

Spot the difference

ParallelogramTrapezium
Pairs of parallel sidestwoonly one
Opposite anglesequalusually not equal
Diagonalsbisect each otherusually do not bisect
Median—half the sum of the bases

Practice questions

Easy
  1. 1In a trapezium AB ∥ DC and ∠A = 80°. Find ∠D.
  2. 2What is the median of a trapezium with bases 20 and 12?
  3. 3How many pairs of opposite sides are parallel in a trapezium?
Medium
  1. 4In an isosceles trapezium ∠A = 65°. Find the other angles.
  2. 5The median is 18 and one base is 26. Find the other base.
  3. 6A trapezium has bases 15 and 9 and legs 5 and 5. Find its perimeter.
Challenging
  1. 7AB ∥ DC, ∠A = (6x + 15)° and ∠D = (3x + 30)°. Find x and both angles.
  2. 8Bases (2x + 3) and (x + 9), median 12. Find x and both bases.
Show answers (try them yourself first)
  1. 1∠D = 100°. 2) (20 + 12) ÷ 2 = 16. 3) Only one pair.
  2. 4∠B = 65°, ∠C = ∠D = 180° − 65° = 115°. 5) The other base is 2 × 18 − 26 = 10. 6) 15 + 9 + 5 + 5 = 34.
  3. 79x + 45 = 180 ⇒ x = 15. ∠A = 105° and ∠D = 75°. 8) (3x + 12) ÷ 2 = 12 ⇒ 3x = 12 ⇒ x = 4. The bases are 11 and 13.

The midpoint theorem of a triangle

From daily life

In a triangular field two pegs are fixed at the midpoints of two sides. The rope between the two pegs is parallel to the third side and half as long, whatever the shape of the field.

Simple definition

Theorem: The segment joining the midpoints of any two sides of a triangle is parallel to the third side and half its length. Converse: a line drawn through the midpoint of one side parallel to another side bisects the third side.

Live figure

The midpoint theorem of a triangle

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Rule 1: PQ = ½ BC and PQ ∥ BC — If P and Q are the midpoints of AB and AC, then PQ is half of BC and parallel to it. Given BC we get PQ, and given PQ we get BC.

  • Easy: BC = 16 ⇒ PQ = 8.
  • Medium: PQ = 9 ⇒ BC = 2 × 9 = 18.
  • Exam type: ∠ABC = 70° and PQ ∥ BC ⇒ corresponding angle ∠APQ = 70°. Also ∠AQP = ∠ACB.

Rule 2: Joining all three midpoints halves every side — Joining the midpoints of all three sides gives a small triangle whose sides are half those of the big one. So the perimeter is also half.

Triangle obtained by joining the midpoints of the sidesThe small triangle formed by joining the midpointsABCPQRBig triangle sides 10, 12, 14 ⇒ small triangle sides 5, 6, 7Perimeters 36 and 18: the small one has half the perimeter

In the figure: The midpoint triangle of a triangle with sides 10, 12, 14 has sides 5, 6, 7.

  • Easy: Sides 14, 16, 18: the small triangle has sides 7, 8, 9; perimeter 24.
  • Medium: P, Q, R are midpoints of AB, BC, CA and PQ = 5, QR = 6, RP = 7. AC = 2 × 5 = 10, AB = 2 × 6 = 12, BC = 2 × 7 = 14.
  • Exam type: If the big triangle has perimeter 36, the small one has 36 ÷ 2 = 18. Its area, however, is one quarter.

Rule 3: Joining midpoints of the sides of a quadrilateral gives a parallelogram — If the midpoints of the four sides of any quadrilateral are joined in order, we always get a parallelogram. Its sides are half the diagonals of the big quadrilateral.

  • Easy: A quadrilateral has diagonals 10 and 6: the midpoint quadrilateral has sides 5 and 3; perimeter 2 × (5 + 3) = 16.
  • Medium: Diagonals 16 and 12: sides 8 and 6; perimeter 2 × (8 + 6) = 28.
  • Exam type: If the diagonals of the big quadrilateral are perpendicular, adjacent sides of the midpoint quadrilateral are perpendicular, so it becomes a rectangle.
Remember

PQ is half of BC only when both P and Q are midpoints.

Common mistake

Drawing any line parallel to BC and taking it as half. The half length comes only when the midpoints are joined.

Shortcut

Side of the small triangle = the opposite side of the big triangle ÷ 2. Memory clue: Midpoint theorem: the joining line is parallel to the third side and half of it.

Spot the difference

Midpoint theorem
Joinsmidpoints of two sides
Directionparallel to the third side
Lengthhalf of the third side
Joining all three midpointshalf the perimeter, one quarter of the area

Practice questions

Easy
  1. 1If BC = 20, what is PQ? (P, Q are midpoints of AB, AC)
  2. 2If PQ = 6, what is BC?
  3. 3If AB = 18 and P is the midpoint of AB, what is AP?
Medium
  1. 4A triangle has sides 18, 20, 22. Find the perimeter of the triangle formed by joining the midpoints.
  2. 5If PQ ∥ BC and ∠ABC = 55°, find ∠APQ.
  3. 6P is the midpoint of AB and PQ ∥ BC, with AC = 20. Find AQ.
Challenging
  1. 7P, Q, R are midpoints of AB, BC, CA. PQ = 5, QR = 6, RP = 7. Find the sides and the perimeter of the big triangle.
  2. 8A quadrilateral has diagonals 16 and 12. What quadrilateral is formed by joining the midpoints of its sides? What is its perimeter? What is it if the diagonals are perpendicular?
Show answers (try them yourself first)
  1. 1PQ = 10. 2) BC = 12. 3) AP = 9.
  2. 4Sides 9, 10, 11; perimeter 30. 5) Corresponding angles: ∠APQ = 55°. 6) By the converse, Q is the midpoint of AC ⇒ AQ = 10.
  3. 7AC = 10, AB = 12, BC = 14. Perimeter 36. 8) A parallelogram; sides 8 and 6; perimeter 28. If the diagonals are perpendicular it is a rectangle.