# Quadrilaterals - Class 9 : Notes

**(1) Prove that sum of the angles of a quadrilateral is .
**

**Given:**Quadrilateral ABCD

**To Prove:**

**Construction:**Join AC

**Proof:**In , We have

..........(i)

In , we have

...............(ii)

Adding (i) and (ii), we get

**(2) Prove that a diagonal of a parallelogram divides it into two congruent triangles.
**

**Given:**A parallelogram ABCD

**To Prove:**A diagonal, say, AC, of parallelogram ABCD divides it into congruent triangles ABC and CDA i.e.

**Construction:**Join AC

**Proof:**Since ABCD is a parallelogram. Therefore,

and

Now, and transversal AC intersects them at A and C respectively.

â€¦â€¦.(i)Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â [Alternate interior angles]

Again, and transversal AC intersects them at A and C respectively. Therefore,

Â â€¦â€¦(ii) Â Â Â Â Â Â Â Â [Alternate interior angles]

Now, in ABC and CDA, we have

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â [From (i)]

So, by ASA congruence criterion, we have

**(3) Prove that two opposite angles of a parallelogram are equal.
**

**Given:**A parallelogram ABCD

**To prove:**and

**Proof:**Since ABCD is a parallelogram. Therefore,

and

Now, and transversal AD intersects them at A and D respectively.

Â â€¦â€¦.(i)Â Â [Sum of Consecutive interior anglesis ]

Again, and DC intersects them at D and C respectively.

â€¦.. (ii)Â Â Â Â [Sum of Consecutive interior angles is ]

From (i) and (ii), we get

.

Similarly, .

Hence, and

**(4) Prove that the diagonals of a parallelogram bisect each other.
**

**Given:**A parallelogram ABCD such that its diagonals AC and BD intersect at O.

**To prove:**and

**Proof:**Since ABCD is a parallelogram. Therefore,

and

Now, and transversal AC intersects them at A and C respectively.

â€¦â€¦..(i)

Again, and BD intersects them at B and D respectively.

â€¦â€¦..(ii)

Now, in AOB and COD, we have

and,

So, by ASA congruence criterion

and

Hence, and

**(5) Prove that in a parallelogram, the bisectors of any two consecutive angles intersect at right angle.
**

**Given:**A parallelogram ABCD such that the bisectors of consecutive angles A and B intersect at P.

**To prove:**

**Proof:**Since ABCD is a parallelogram. Therefore,

Now, and transversal AB intersects them.

â€¦.(i)

AP is the bisector of and BP is the bisector of then Â and

In , we have

Â Â Â Â Â [From (i)]

**(6) Prove that if a diagonal of a parallelogram bisects one of the angles of the parallelogram it also bisects the second angle.
**

**Given:**A parallelogram ABCD in which diagonal AC bisects .

**To prove:**AC bisects

**Proof:**Since ABCD is a parallelogram. Therefore,

Now, and AC intersects them.

Â Â â€¦â€¦(i) [Alternate interior angles]

Again, and AC intersects them.

Â â€¦â€¦(ii) [Alternate interior angles]

But, it is given that AC is the bisector of . Therefore,

Â Â Â Â Â Â Â Â â€¦â€¦..(iii)

From (i), (ii) and (iii), we get

Â Â Â Â Â Â Â â€¦â€¦â€¦(iv)

Hence, AC bisects .

From (ii) and (iii), we have

Â Â Â Â Â [Angles opposite to equal sides are equal]

But, and Â Â Â [ ABCD is a parallelogram]

Hence, ABCD is a rhombus.

**(7) Prove that the angles bisectors of a parallelogram form a rectangle.
**

**Proof:**Since ABCD is a parallelogram. Therefore,

Now, and transversal AB intersects them at A and B respectively. Therefore,

Â [Sum of consecutive interior angles is ]

Â â€¦.(i)Â Â [AS and BS are bisectors of and respectively]

But, in , we have

[Sum of the angle of a triangle is ]

[ and are vertically opposite angles ]

Similarly, we can prove that

, and

Hence, PQRS is a rectangle.

**(8) Prove that a quadrilateral is a parallelogram if its opposite sides are equal.
**

**Given:**A quadrilateral ABCD in which and

**To prove:**ABCD is a parallelogram.

**Construction:**Join AC.

**Proof:**In ACB and CAD, we have

Â Â [Common Side]

So, by SAS criterion of congruence, we have

ACB and CAD

Â Â Â Â â€¦.(i)

And,

Now, line AC intersects AB and DC at A and C, such that

Â Â Â .....(ii)

i.e., alternate interior angles are equal.

Â Â Â â€¦..(iii)

Similarly, line AC intersects BC and AD at C and A such that

i.e., alternate interior angles are equal.

Â Â Â â€¦..(iv)

From (iii) and (iv), we have

and

Hence, ABCD is a parallelogram.

**(9) Prove that a quadrilateral is a parallelogram if its opposite angles are equal.
**

**Given:**A quadrilateral ABCD in which and .

**To prove:**ABCD is a parallelogram.

**Proof:**In quadrilateral ABCD, we have

â€¦â€¦(i)

Â Â â€¦â€¦(ii)

Â â€¦..(iii)

Since sum of the angles of a quadrilateral is

Â â€¦â€¦(iv)

Â â€¦..(v) Â Â []

Now, line AB intersects AD and BC at A and B respectively such that

i.e. the sum of consecutive interior angles is

Â Â Â â€¦â€¦(vi)

Again,

Now, line BC intersects AB and DC at A and C respectively such that

i.e., the sum of consecutive interior angles is .

Â Â Â Â â€¦â€¦(vii)

From (vi) and (vii), we get

and .

Hence, ABCD is a parallelogram.

**(10) Prove that the diagonals of a quadrilateral bisect each other, if it is a parallelogram.
**

**Proof:**In AOD and COB, we have

So, by SAS criterion of congruence, we have

Now, line AC intersects AD and BC at A and C respectively such that

i.e., alternate interior angles are equal.

Similarly,

Hence, ABCD is a parallelogram.

**(11) Prove that a quadrilateral is a parallelogram if its one Pair of opposite sides are equal and parallel.
**

**Given:**A quadrilateral ABCD in which and .

**To prove:**is a parallelogram.

**Construction:**Join AC.

**Proof:**In ABC and CDA, we have

And,

So, by SAS criterion of congruence, we have

Thus, line AC intersects AB and DC at A and C respectively such that

i.e., alternate interior angles are equal.

.

Thus, and .

Hence, quadrilateral ABCD is a parallelogram.

**(12) Prove that each of the four angles of a rectangle is a right angle.
**

**Given:**A rectangle ABCD such that

**To prove:**

**Proof:**Since ABCD is a rectangle.

ABCD is a parallelogram

Now, and line AB intersects them at A and B.

Similarly, we can show that and

Hence,

**(13) Prove that each of the four sides of a rhombus of the same length.
**

**Given:**A rhombus ABCD such that .

**To prove:**.

**Proof:**Since ABCD is rhombus

ABCD is a parallelogram

and

But,

Hence, all the four sides of a rhombus are equal.

**(14) Prove that the diagonals of a rectangle are of equal length.
**

**Given:**A rectangle ABCD with AC and BD as its diagonals.

**To prove:**.

**Proof:**Since ABCD is a rectangle

ABCD is a parallelogram such that one of its angles, say , is a right angle.

and

Now, and AB intersects them at A and B respectively.

]

In ABD and BAC, we have

And,

So, by SAS criterion of congruence, we have

}

Hence,

**(15) Prove that diagonals of a parallelogram are equal if and only if it is a rectangle.
**

**Proof:**In ABC and DCB, we have

And, So, by SAS criterion of congruence, we have

But, and BC cuts them.

Thus, .

Hence, ABCD is a rectangle.

**(16) Prove that the diagonals of a rhombus are perpendicular to each other.
**

**Given:**A rhombus ABCD whose diagonals AC and BD intersect at O.

**To prove:**4

**Proof:**We know that a parallelogram is a rhombus, if ll of its sides are equal. So, ABCD is a rhombus. This implies that ABCD is a parallelogram such that

Â Â â€¦..(i)

Since the diagonals of a parallelogram bisect each other.

and Â Â Â â€¦..(ii)

Now, in BOC and DOC, we have

So, by SSS criterion of congruence, we have

But,

Similarly,

Hence,

**(17) Prove that diagonals of a parallelogram are perpendicular if and only if it is a rhombus.
**

**Given:**A parallelogram ABCD in which .

**To prove:**Parallelogram ABCD is a rhombus.

**Proof:**Suppose AC and BD intersect at O. Since the diagonals of a parallelogram bisect each other. So, we have

Â Â â€¦â€¦(i)

Now, in AOD and COD, we have

So, by SAS criterion of congruence, we have

Â Â Â â€¦â€¦(ii)

Since ABCD is a parallelogram.

and

Hence, parallelogram ABCD is a rhombus.

**(18) Prove that the diagonals of a square are equal and perpendicular to each other.
**

**Given:**A square ABCD.

**To prove:**and .

**Proof:**In ADB and BCA, we have

And,

So, by SAS criterion of congruence, we have

Now, in AOB and AOD, we have

And,

So, by SSS criterion of congruence, we have

But,

Hence, and

**(19) Prove that if the diagonals of a parallelogram are equal and intersect at right angles, Â then it is square
**

**Given:**A parallelogram ABCD in which and

**To prove:**ABCD is a square.

**Proof:**In AOB and AOD, we have

And

So, by SAS criterion of congruence, we have

But, and

Â Â â€¦â€¦.(i)

Now, in ABD and BAC, we have

And,

So, by SSS criterion of congruence, we have

But,

Â Â â€¦â€¦(ii)

From (i) and (ii), we obtain that ABCD is a parallelogram whose all side are equal and all angles are right angles.

Hence, ABCD is a square.

**(20) Prove that the line segment joining the mid-points of any two sides of a triangle is parallel to the third side and equal to half of it.
**

**Given:**A in which D and E are the mid points of sides AB and AC respectively. DE is joined

**To prove:**and

**Construction:**Produce the line segment DE to F, such that . Join FC

**Proof:**In AED and CEF, we have

And,

So, by SAS criterion of congruence, we have

Â Â Â â€¦(i)

And, Â â€¦â€¦(ii)

Now, D is the mid-point of AB

Â Â â€¦..(iii)

Now, DF intersects AD and FC at D and F respectively such that

i.e. alternate interior angles are equal.

Â Â Â â€¦..(iv)

From (iii) and (iv), we find that DBCF is a quadrilateral such that one pair of sides are equal and parallel.

DBCF is a parallelogram.

and

But, D,E,F are collinear and.

and

**(21) Prove that a line through the mid-point of a side of a triangle parallel to another side bisects theÂ third side.
**

**Proof:**We have to prove that E is the mid-point of AC. If possible, let E be not the mid-point of AC. Let E prime be the mid-point AC. Join DE prime.Now, in , D is the mid-point of AB and E prime is the mid-point of AC. We have,

Â Â â€¦..(i)

Also, Â â€¦.(ii)

From (i) and (ii), we find that two intersecting lines DE and DEâ€™ are both parallel to Line BC.

This is contradiction to the parallel line axiom.

So, our supposition is wrong. Hence, E is the mid-point of AC.

**(22) Prove that the quadrilateral formed by joining the mid-points of the sides of a quadrilateral, in order, is a parallelogram.
**

**Given:**ABCD is a quadrilateral in which P,Q,R,S are the mid-points of sides AB, BC, CD and DA respectively.

**To prove:**PQRS is a parallelogram.

**Proof:**In , P and Q are the mid-points of sides AB and BC respectively.

and Â Â â€¦.(i)

In , R and S are the mid-points of CD and AD respectively.

and Â Â Â â€¦â€¦.(ii)

From (i) and (ii), we have

and

Thus, in quadrilateral PQRS one pair of opposite sides are equal and parallel.

Hence, PQRS is a parallelogram.

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