Mathematics
Grade-VIII(CBSE)
Quadrilaterals
What is Quadrilateral?
Properties:
The sum of interior angles of a quadrilateral is always 360°.
It has two diagonals.
Types of Quadrilaterals
Type |
Description |
Properties |
Parallelogram |
Opposite sides are parallel and equal. |
Opposite angles are equal, Diagonals bisect each
other. |
Rectangle |
A
parallelogram with all angles 90°. |
Opposite
sides equal, Diagonals equal and bisect each other. |
Square |
A rectangle with all sides equal. |
All angles 90°, All sides equal, Diagonals equal and
perpendicular bisectors. |
Rhombus |
A
parallelogram with all sides equal. |
Opposite
angles equal, Diagonals bisect each other at right angles. |
Trapezium |
One pair of opposite sides is parallel. |
Non-parallel sides called legs. |
Kite |
Two
pairs of adjacent sides are equal. |
Diagonals
intersect at right angles, one diagonal bisects the other. |
Angle Sum Property
The sum of the interior angles of any
quadrilateral is always 3600
∠A
+ ∠B
+ ∠C
+ ∠D
= 360°
Properties of Special Quadrilaterals
Quadrilateral |
Opp. Sides |
Opp. Angles |
Diagonals |
Angle |
Parallelogram |
Equal & Parallel |
Equal |
Bisect each other |
Not necessarily 90° |
Rectangle |
Equal
& Parallel |
Equal |
Equal
and bisect |
All
90° |
Square |
Equal & Parallel |
Equal |
Equal, perpendicular, bisect |
All 90° |
Rhombus |
Equal
& Parallel |
Equal |
Perpendicular
and bisect |
Not
necessarily 90° |
Trapezium |
One pair parallel |
No specific property |
No specific property |
Varies |
Kite |
Adjacent
sides equal |
One
pair equal |
One
bisects the other at 90° |
Varies |
Important Theorems & Rules
1. Angle Sum Property: Sum of interior
angles = 360°.
2. Diagonals of a rectangle are equal and bisect each other.
3. Diagonals of a square are equal and perpendicular bisectors.
4. Diagonals of a rhombus are perpendicular but not necessarily equal.
Practice Questions
Section
A: Objective Type Questions
Q1. The sum of all interior angles of a
quadrilateral is:
(a) 90°
(b)
180° (c) 270° (d) 360°
Q2. A square is also a:
(a) Parallelogram (b) Rhombus
(c) Rectangle (d)
All of these
Q3. The diagonals of a rhombus:
(a) Are equal (b)
Bisect at 90° (c) Do not bisect
each other (d) Are not perpendicular
Section
B: Very Short Answer Type
Q4. Name the quadrilateral which has only
one pair of parallel sides.
Q5. Find the fourth angle of a
quadrilateral if three angles are 110°, 85°, and 95°.
Section
C: Short Answer Type
Q6. In a parallelogram, if one angle is
70°, find the other three angles.
Q7. A quadrilateral has angles 90°, 80°,
and 110°. Find the fourth angle.
Q8. List the properties of a rectangle.
Section
D: Long Answer Type
Q9. Prove that the diagonals of a rhombus
bisect each other at right angles.
Q10. In a quadrilateral ABCD, ∠A = 75°, ∠B = 85°, ∠C = 110°. Find ∠D. Is the
quadrilateral cyclic?
Solutions
Section A Answers:
Q1.
(d) 360°
Q2. (d) All of these
Q3. (b) Bisect each other at 90°
Section B Solutions:
Q4.
Trapezium
Q5.
360° − (110° + 85° + 95°) = 70°
Section C Solutions:
Q6.
One angle = 70°, then others are 110°, 70°, and 110°.
Q7.
360 − (90 + 80 + 110) = 80°
Q8.
- Opposite sides equal & parallel
- All angles 90°
- Diagonals equal and bisect each otherSection
D Solutions:
Q9.
Let ABCD be a rhombus.
Diagonals AC and BD intersect at O.
All sides are equal. (definition of
rhombus)
Triangles AOB and COB are congruent (by
SSS).
∠AOB
= ∠COB,
(cpct)
hence diagonals bisect at 90°.
Q10.
∠D
= 360 − (75 + 85 + 110) = 90°
Since ∠A
+ ∠C
= 185° ≠ 180°, the quadrilateral is not cyclic.
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