Question

In the figure 11.34, ABCD is a quadrilateral in which AB = AD and BC = DC. Diagonals AC and BD intersect each other at O. Show that-
(i) ∆ ABC is congruent to ∆ ADC
(ii) ∆ AOB is congruent to ∆ AOD
(iii) AC is perpendicular to BD
(iv) AC bisects BD


I know the first two just tell the 3 and 4
If you want tell 1 & 2 also
I can get an idea of solving it in another way
And please hurry....I have exam
And please don't send unwanted reply​

Answer 1

Proved below.

Step-by-step explanation:

Given:

ABCD is a quadrilateral and AB = AD and BC = DC.

Also diagonals AC and BD intersect each other at O.

To prove:

(i) ∆ ABC is congruent to ∆ ADC

(ii) ∆ AOB is congruent to ∆ AOD

(iii) AC is perpendicular to BD

(iv) AC bisects BD

Proof:

(i) In ΔABC and ΔADC, we have    

AB = CD                  (given)  

AC = AC                  (common side)

∴ Δ ABC ≅ Δ ADC   (By SSS congruency)

By CPCT, we get    

∠DAC = ∠BAC

or  ∠DAO = ∠BAO    (since AOC is a staright line)     [1]

(ii) In Δ AOB and Δ AOD, we have          

AD = BC           (given)          

AO = AO     (common side)  

∠DAO = ∠BAO     (from (1)

∴ ΔAOB ≅ ΔAOD  (By ASA congruency)                  [2]

(iii) As ΔAOB ≅ ΔAOD              [ from 2 ]

Then by CPCT, we get    

∠ AOD =∠ AOB                         [3]

∠AOD + ∠AOB = 180°      [ linear pair property]

∠AOD + ∠AOD = 180°    (from (3)                  

2 ∠AOD = 180°                      

∠AOD = 90°

This proves that AC⊥BD

(iv) Also, since we have  

ΔAOB ≅ ΔAOD

So by CPCT, we get  DO = OB

This proves that AC bisects BD.

Hence proved.

Answer 2

Answer:

attached above,............................ ..