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Properties of a Parallelogram and Related Theorems - Quadrilaterals, Class 9, Mathematics | Extra Documents & Tests for Class 9 PDF Download

ANGLE SUM PROPERTY OF A QUADRILATERAL

THEOREM-I : The sum of the four angles of a quadrilateral is 360°.
Given : A quadrilateral ABCD.
To Prove : ∠A + ∠B + ∠C + ∠D = 360°
NCRT,Question and Answer,Important,Class 9 Mathematics,CBSE Class 9
Construction : Join AC. 
Proof :

STATEMENTREASON

1. In ΔABC
∠1+ ∠4+ ∠6 = 180°

Sum of the all angles of triangle is equal to 180°

2. In ΔADC
∠2 + ∠3 + ∠5 = 180°
Sum of the all angles of triangle is equal to 180°

3. (∠1 +∠2) + (∠3 +∠4)

∠5 +∠6 = 180° + 180°

Adding (1) & (2), we get
4. ∠A + ∠C + ∠D + ∠B = 360° 
5. ∠A + ∠B + ∠C +∠D = 360° 

Hence, proved.

Ex.1 Three angles of a quadrilateral measure 56°, 100° and 88°. Find the measure of the fourth angle.
Sol. Let the measure of the fourth angle be x°.
∴  56° + 100° + 88° + x° = 360°  ∴ [Sum of all the angles of quadrileteral is 360°]
⇒  244+x=360 
⇒   x = 360 – 244 = 116
Hence, the measure of the fourth angle is 116°.

Ex.2 The angles of a quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles of the quadrilateral.
Sol. Let the four angles of the quadrilateral be 3x, 5x, 9x and 13x . [NCERT]
∴  3x + 5x + 9x + 13x = 360°      ∴ [Sum of all the angles of quadrileteral is 360°]
⇒ 30x = 360°
⇒ x =12°
Hence, the angles of the quadrilateral are 3 × 12° = 36°, 5 × 12° = 60°, 9 × 12° = 108

PARALLELOGRAM:- A quadrilateral in which both pairs of opposite Csides are parallel is called a parallelogram.
NCRT,Question and Answer,Important,Class 9 Mathematics,CBSE Class 9
In the figure, ABCD is a quadrilateral in which AB ⊥ DC, BC ⊥ AD.
∴ quadrilateral ABCD is a parallelogram.

PROPERTIES OF A PARALLELOGRAM

THEOREM-1 A diagonal of a parallelogram divides it into two congruent triangles.
Given : ABCD is a parallelogram and AC is a diagonal which forms two triangles CAB and ACD.
To Prove : ΔACD ≌ ΔCAB
NCRT,Question and Answer,Important,Class 9 Mathematics,CBSE Class 9
Proof :

STATEMENT

REASON

1. ∵ AB ⊥ DC and AD ⊥ BC

ABCD is a parallelogram

2. (i) ∵ AB ⊥DC and AC is a transversal
             ∠ACD = ∠CAB
(ii) ∵ AD ⊥ BC and AC is a transversal
    ∴ ∠CAD = ∠ACB

Alternate angles

Alternate angles

3. In ΔACD and ΔCAB,
      ∠ACD = ∠CAB
        AC = AC
         ∠CAD = ∠ACB
Therefore, ΔACD  ≌ ΔCAB

 

From (2)

Common

From (3)

By ASA congruence rule

Hence, proved.

THEOREM-2 In a parallelogram, opposite sides are equal.
NCRT,Question and Answer,Important,Class 9 Mathematics,CBSE Class 9
Given : ABCD is a parallelogram.
To Prove : AB = CD and BC = DA

Construction : Join AC.
Proof :

STATEMENTREASON
1. AB ⊥  DC and AD ⊥ BCSince ABCD is a parallelogram

2. In ΔABC and ΔCDA

     ∠BAC = ∠DCA
      AC = AC
    ∠ACB = ∠CAD
     ΔABC ≌ ΔCDA
∴ AB = CD and BC = DA

Alternate angles

Common

Alternate angles

By ASA congruence rule

C.P.C.T.

Hence, proved.

THEOREM-3 In a parallelogram, opposite angles are equal.
NCRT,Question and Answer,Important,Class 9 Mathematics,CBSE Class 9
Given : ABCD is a parallelogram.
To Prove : ∠A = ∠C and ∠B = ∠D
Proof :

STATEMENT

REASON

1. AB ⊥ DC and AD ⊥ BC

Since ABCD is a parallelogram

2. AB ⊥  DC and AD is a transversal
∴ ∠A + ∠D = 180°

Sum of consecutive interior angles is 180°

3. AD⊥BC and DC is a transversal
 ∴ ∠D + ∠C = 180°

Sum of consecutive interior angles is 180°

4. ∠A + ∠D = ∠D + ∠C
           ∠A = ∠C

 

From (2) & (3)

 

 

5. Similarly, ∠B = ∠D
              ∠A = ∠C and ∠B = ∠D

Hence proved.

THEOREM-4 : The diagonals of a parallelogram bisect each other.
NCRT,Question and Answer,Important,Class 9 Mathematics,CBSE Class 9
Given : ABCD is a parallelogram, diagonals AC and BD intersect at O. O
To Prove : OA = OC and OB = OD
Proof :

STATEMENT

REASON

1. AB ⊥ DC and AD ⊥ BC

ABCD is a parallelogram

2. In ΔAOB and ΔCOD ,
∴ ∠BAO = ∠DCO
AB = CD
∴ ∠ABO = ∠CDO

Alternate angles
Opposite sides of a parallelogram°
Alternate angles

3. ΔAOB NCRT,Question and Answer,Important,Class 9 Mathematics,CBSE Class 9 ΔCOD

By ASA congruence rule

4. ∴ OA = OC and OB = OD

[C.P.C.T.]

Hence, proved.

 

The document Properties of a Parallelogram and Related Theorems - Quadrilaterals, Class 9, Mathematics | Extra Documents & Tests for Class 9 is a part of the Class 9 Course Extra Documents & Tests for Class 9.
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FAQs on Properties of a Parallelogram and Related Theorems - Quadrilaterals, Class 9, Mathematics - Extra Documents & Tests for Class 9

1. What are the properties of a parallelogram?
Ans. A parallelogram has the following properties: 1. Opposite sides are parallel. 2. Opposite sides are equal in length. 3. Opposite angles are equal in measure. 4. Consecutive angles are supplementary.
2. What is the difference between a parallelogram and a rectangle?
Ans. A rectangle is a type of parallelogram in which all angles are right angles (90 degrees). In addition to the properties of a parallelogram, a rectangle also has the following properties: 1. All angles are right angles (90 degrees). 2. Opposite sides are equal in length.
3. What is the formula for the area of a parallelogram?
Ans. The formula for the area of a parallelogram is: Area = base x height where "base" is the length of one of the parallel sides, and "height" is the perpendicular distance between the two parallel sides.
4. What is the Midpoint Theorem for a parallelogram?
Ans. The Midpoint Theorem for a parallelogram states that the line segment connecting the midpoints of two sides of a parallelogram is parallel to the other two sides and is equal in length to half of the length of the third side.
5. How do you prove that a quadrilateral is a parallelogram?
Ans. A quadrilateral can be proven to be a parallelogram if it satisfies any of the following conditions: 1. Both pairs of opposite sides are parallel. 2. Both pairs of opposite sides are equal in length. 3. One pair of opposite sides is both parallel and equal in length, and the other pair of opposite sides is parallel. 4. Both pairs of opposite angles are equal in measure. 5. One pair of opposite angles is both equal in measure and supplementary, and the other pair of opposite angles is equal in measure.
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