krit.club logo

Quadrilaterals - Properties of a Parallelogram

Grade 9CBSE

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

🔑Concepts

•

Definition of a Parallelogram: A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. Visually, this means the top side is parallel to the bottom side, and the left side is parallel to the right side, forming a shape where opposite sides never meet.

•

Congruent Triangles Property: A diagonal of a parallelogram divides it into two congruent triangles. For a parallelogram ABCDABCD, drawing diagonal ACAC creates two triangles, △ABC\triangle ABC and △CDA\triangle CDA, which are identical in shape and size (△ABC≅△CDA\triangle ABC \cong \triangle CDA).

•

Opposite Sides Property: In a parallelogram, opposite sides are equal in length. This means if you measure the parallel sides, you will find AB=CDAB = CD and AD=BCAD = BC. Visually, the lengths of the parallel rails of the shape are always the same.

•

Opposite Angles Property: In a parallelogram, opposite angles are equal. Specifically, ∠A=∠C\angle A = \angle C and ∠B=∠D\angle B = \angle D. If you look at the corners diagonally across from each other, they will have the exact same degree measure.

•

Consecutive Interior Angles: The adjacent (consecutive) angles of a parallelogram are supplementary, meaning they add up to 180∘180^{\circ}. For example, ∠A+∠B=180∘\angle A + \angle B = 180^{\circ} and ∠B+∠C=180∘\angle B + \angle C = 180^{\circ}. Visually, any two angles located along the same side of the parallelogram total a straight line's angle.

•

Diagonals Bisect Each Other: The diagonals of a parallelogram intersect at a point that is the midpoint of both diagonals. If diagonal ACAC and BDBD intersect at point OO, then AO=OCAO = OC and BO=ODBO = OD. Visually, the two crossing lines 'cut each other in half'.

•

Conditions for a Parallelogram: A quadrilateral is a parallelogram if one pair of opposite sides is both equal and parallel. This is a shortcut to prove a shape is a parallelogram without checking all four sides and angles.

📐Formulae

Perimeter of a Parallelogram=2(a+b)\text{Perimeter of a Parallelogram} = 2(a + b), where aa and bb are lengths of adjacent sides.

Area of a Parallelogram=base×height=b×h\text{Area of a Parallelogram} = \text{base} \times \text{height} = b \times h

Sum of adjacent angles=180∘\text{Sum of adjacent angles} = 180^{\circ}

Sum of all interior angles=360∘\text{Sum of all interior angles} = 360^{\circ}

💡Examples

Problem 1:

In a parallelogram ABCDABCD, two adjacent angles are in the ratio 3:23:2. Find the measure of all the angles of the parallelogram.

Solution:

  1. Let the two adjacent angles be ∠A=3x\angle A = 3x and ∠B=2x\angle B = 2x.
  2. Since adjacent angles in a parallelogram are supplementary: 3x+2x=180∘3x + 2x = 180^{\circ}.
  3. Simplify the equation: 5x=180∘5x = 180^{\circ}.
  4. Divide by 5: x=180∘5=36∘x = \frac{180^{\circ}}{5} = 36^{\circ}.
  5. Calculate the angles: ∠A=3×36∘=108∘\angle A = 3 \times 36^{\circ} = 108^{\circ} and ∠B=2×36∘=72∘\angle B = 2 \times 36^{\circ} = 72^{\circ}.
  6. Use the property that opposite angles are equal: ∠C=∠A=108∘\angle C = \angle A = 108^{\circ} and ∠D=∠B=72∘\angle D = \angle B = 72^{\circ}.
  7. Therefore, the angles are 108∘,72∘,108∘,72∘108^{\circ}, 72^{\circ}, 108^{\circ}, 72^{\circ}.

Explanation:

This problem uses the supplementary property of adjacent angles to set up an algebraic equation, then applies the opposite angles property to find the remaining measures.

Problem 2:

In a parallelogram PQRSPQRS, the diagonals PRPR and QSQS intersect at OO. If PO=3x+2PO = 3x + 2 and OR=17OR = 17, and QO=4y−1QO = 4y - 1 and OS=15OS = 15, find the values of xx and yy.

Solution:

  1. In a parallelogram, diagonals bisect each other, so the intersection point OO is the midpoint.
  2. For diagonal PRPR: PO=ORPO = OR. Thus, 3x+2=173x + 2 = 17.
  3. Solve for xx: 3x=17−2⇒3x=15⇒x=53x = 17 - 2 \Rightarrow 3x = 15 \Rightarrow x = 5.
  4. For diagonal QSQS: QO=OSQO = OS. Thus, 4y−1=154y - 1 = 15.
  5. Solve for yy: 4y=15+1⇒4y=16⇒y=44y = 15 + 1 \Rightarrow 4y = 16 \Rightarrow y = 4.
  6. The values are x=5x = 5 and y=4y = 4.

Explanation:

This solution relies on the 'Diagonals Bisect Each Other' property, which allows us to set the segments of each diagonal equal to one another to solve for the unknown variables.