Question

ABCDis a rhombus. show that diagonal ACbisect angle Aas well as angle C and diagonal BD bisects angle Bas well as angle D
​

Answer 1

Answer:

Hey there!

The answer is given below:

Step-by-step explanation:

Given:

 1.  ABCD  is a rhombus.

 2.   Diagonals are AC and BD.

To prove:

•   Diagonal AC bisect angle A as well as angle C

•    Diagonal BD bisects angle B as well as angle D.

Proof:

We know,

In a rhombus, all sides are equal and diagonals are different but intersect at 90 degrees.  

⇒   AB = BC = CD = DA          ------------------(since all sides are equal)

⇒    AC ⊥ BD                         -----------------(diagonals intersect at 90 degrees)

In Δ ABC ,  (from the figure given below)

⇒  AB = BC

⇒ ∴∠CAB = ∠ACB

⇒   AO = OC

∴  ∠ABO = ∠CBO

and

Similarly,  in ΔADC

⇒AD=DC

⇒AO=OC

∴Corresponding angles are equal

⇒∠DAC  = ∠DCA

and ∠CDO =  ∠ADO

Therefore,

AC bisects ∠A and ∠C and BD bisects ∠B and ∠D

​  

Hence proved.

Thanks!