The origin of the word "quadrilateral" is the two Latin words quadri, a variant of four, and latus, meaning "side".

Quadrilaterals are simple (not self-intersecting) or complex (self-intersecting), also called crossed. Simple quadrilaterals are either convex or concave.

The interior angles of a simple (and planar) quadrilateral ABCD add up to 360 degrees of arc, that is

This is a special case of the n-gon interior angle sum formula (n - 2) × 180°. In a crossed quadrilateral, the four interior angles on either side of the crossing add up to 720°.

All convex quadrilaterals tile the plane by repeated rotation around the midpoints of their edges.

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**Exercise 8.1 : ** Solutions of Questions on Page Number : **146**

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**Exercise 8.2 : ** Solutions of Questions on Page Number : **150**

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**Popular Articles**

Q1 :
**
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The angles of quadrilateral are in the ratio 3: 5: 9: 13. Find all the angles of the quadrilateral.

**Answer :**

Let the common ratio between the angles be *x*. Therefore,
the angles will be 3*x*, 5*x*, 9*x*, and
13*x* respectively.

As the sum of all interior angles of a quadrilateral is 360º,

∴ 3*x* + 5*x* +
9*x* + 13*x* = 360º

30*x* = 360º

*x* = 12º

Hence, the angles are

3*x* = 3 x 12 =
36º

5*x* = 5 x 12 =
60º

9*x* = 9 x 12 =
108º

13*x* = 13 x 12 =
156º

Answer needs Correction? Click Here

Q2 :
**
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If the diagonals of a parallelogram are equal, then show that it is a rectangle.

**Answer :**

Let ABCD be a parallelogram. To show that ABCD is a rectangle, we have to prove that one of its interior angles is 90º.

In ΔABC and ΔDCB,

AB = DC (Opposite sides of a parallelogram are equal)

BC = BC (Common)

AC = DB (Given)

∴ ΔABC ≅ ΔDCB (By SSS Congruence rule)

⇒ ∠ ABC = ∠ DCB

It is known that the sum of the measures of angles on the same side of transversal is 180º.

∠ ABC + ∠ DCB = 180º (AB || CD)

⇒ ∠ ABC + ∠ ABC = 180º

⇒ 2∠ ABC = 180º

⇒ ∠ ABC = 90º

Since ABCD is a parallelogram and one of its interior angles is 90º, ABCD is a rectangle.

Answer needs Correction? Click Here

Q3 :
**
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Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.

**Answer :**

Q4 :
**
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Show that the diagonals of a square are equal and bisect each other at right angles.

**Answer :**

Q5 :
**
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Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.

**Answer :**

Q6 :
**
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Diagonal AC of a parallelogram ABCD bisects ∠ A (see the given figure). Show that

(i) It bisects ∠ C also,

(ii) ABCD is a rhombus.

**Answer :**

Q7 :
**
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ABCD is a rhombus. Show that diagonal AC bisects ∠ A as well as ∠ C and diagonal BD bisects ∠ B as well as ∠ D.

**Answer :**

Q8 :
**
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ABCD is a rectangle in which diagonal AC bisects ∠ A as well as ∠ C. Show that:

(i) ABCD is a square (ii) diagonal BD bisects ∠ B as well as ∠ D.

**Answer :**

Q9 :
**
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In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see the given figure). Show that:

(i) ΔAPD ≅ ΔCQB

(ii) AP = CQ

(iii) ΔAQB ≅ ΔCPD

(iv) AQ = CP

(v) APCQ is a parallelogram

**Answer :**

Q10 :
**
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ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD (See the given figure). Show that

(i) ΔAPB ≅ ΔCQD

(ii) AP = CQ

**Answer :**

Q11 :
**
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In ΔABC and ΔDEF, AB = DE, AB || DE, BC = EF and BC || EF. Vertices A, B and C are joined to vertices D, E and F respectively (see the given figure). Show that

(i) Quadrilateral ABED is a parallelogram

(ii) Quadrilateral BEFC is a parallelogram

(iii) AD || CF and AD = CF

(iv) Quadrilateral ACFD is a parallelogram

(v) AC = DF

(vi) ΔABC ≅ ΔDEF.

**Answer :**

Q12 :
**
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ABCD is a trapezium in which AB || CD and AD = BC (see the given figure). Show that

(i) ∠ A = ∠ B

(ii) ∠ C = ∠ D

(iii) ΔABC ≅ ΔBAD

(iv) diagonal AC = diagonal BD

[*Hint*: Extend AB and draw a line through C parallel to DA
intersecting AB produced at E.]

**Answer :**

Q1 :
**
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ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see the given figure). AC is a diagonal. Show that:

(i) SR || AC and SR = AC

(ii) PQ = SR

(iii) PQRS is a parallelogram.

**Answer :**

Q2 :
**
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ABCD is a rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rectangle.

**Answer :**

Q3 :
**
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ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rhombus.

**Answer :**

Q4 :
**
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ABCD is a trapezium in which AB || DC, BD is a diagonal and E is the mid - point of AD. A line is drawn through E parallel to AB intersecting BC at F (see the given figure). Show that F is the mid-point of BC.

**Answer :**

Q5 :
**
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In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively (see the given figure). Show that the line segments AF and EC trisect the diagonal BD.

**Answer :**

Q6 :
**
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Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.

**Answer :**

Q7 :
**
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ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB and parallel to BC intersects AC at D. Show that

(i) D is the mid-point of AC

(ii) MD ⊥ AC

(iii)

**Answer :**

Maths : CBSE ** NCERT ** Exercise Solutions for Class 9th for ** Quadrilaterals ** ( Exercise 8.1, 8.2 ) will be available online in PDF book form soon. The solutions are absolutely Free. Soon you will be able to download the solutions.

- Chapter 1 - Number Systems Class 9
- Chapter 2 - Polynomials Class 9
- Chapter 6 - Lines and Angles Class 9
- Chapter 7 - Triangles Class 9
- Chapter 12 - Heron's Formula Class 9
- Chapter 3 - Coordinate Geometry Class 9
- Chapter 5 - Introduction to Euclid's Geometry Class 9
- Chapter 13 - Surface Areas and Volumes Class 9
- Chapter 9 - Areas of Parallelograms and Triangles Class 9
- Chapter 4 - Linear Equations in Two Variables Class 9