Diagonal of a Parallelogram Which Bisects One Angle It Bisects The Second Angles


 
 
Concept Explanation
 

Diagonal of a Parallelogram Which Bisects One Angle It Bisects The Second Angles

Diagonal of a Parallelogram Which Bisects One Angle Bisects The other Angle: We know that a parallelogram is a quadrilateral in which pair of opposite side is equal and parallel, but its diagonal bisects the angles of the parallelogram then that parallelogram is a rhombus.

Theorem: If diagonal of a parallelogram bisects one of the angles of the parallelogram, it also bisects the second angles. Also, prove that it is a rhombus.

GIVEN  A parallelogram ABCD in which diagonal AC bisects <A.

To prove AC bisects <C

Proof  Since ABCD is a parallelogram. Therefore,  AB large parallel DC

Now, AB large parallel DC and BD intersects them,

large therefore    <2 = <4        .....(i)  [Alternate interior angles]

Again, AD large parallel BC and BD intersects them

large therefore   <3 = <6         ....(ii)  [Alternate interior angles]

But, it is given that BD is the bisector of <B. Therefore,

      <2 = <6         ....(iii)

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

     <3 = <4          .....(iv)Hence, BD bisects <D.

In a parallelogram  <B = <D    [Opposite angles are equal]

therefore frac{1}{2}angle B = frac{1}{2}angle D

     < 2 = <3        

 large Rightarrow   AD = AB          [ large because  Angles opposite to equal sides are equal]

But, AB = DC anf BC = AD             [large because  ABCD is a parallelogram]

large therefore   AB = BC = CD = DA

Hence, ABCD is a rhombus.

 

ILLUSTRATION: ABCD is a rhombus with angle ABC= 56^0, then  find the value of angle ACD

Solution:  ABCD is a rhombus and each rhombus is a parallelogram

AB || CD and BC is the transversal 

therefore ;; angle ABC +angle BCD= 180^0      ....... [ Cointerior angles are supplementary]

Rightarrow ;; 56^0 +angle BCD= 180^0             ......[ Substituting the value of angle ABC

Rightarrow ;; angle BCD= 180^0-56^0=124^0

Now we know that the diagonal bisects both the angles of a rhombus

angle BCA = angle ACD                           ................[2]

From the figure 

angle BCA +angle ACD = angle BCD

angle ACD +angle ACD = 126^0

2times angle ACD = 126^0

angle ACD = frac{126^0}{2}= 63^0

 

Sample Questions
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Question : 1

In the following figure, the diagonal AC bisects angle A. So it must bisects _____ .

Right Option : B
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Question : 2

In the following figure, ABCD is a parallelogram. The diagonal AC bisects angle A and BD bisects angle D then AC and BD must bisects which angles?

Right Option : C
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Question : 3

ABCD is a rhombus with angle BAD=30^{0}, then  find the value of angle DAC .

Right Option : A
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Explanation
 
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