Question 5
ABCD is a quadrilateral in which AB = BC and AD = CD. Show that BD bisects both the angles ABC and ADC.

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Solution

AB = BC (given)
$∴$ $∠ 2 = ∠ 1$ ...(i)
[angles opposite to equal sides are equal]
and AD = CD [given]
$\Rightarrow ∠ 4 = ∠ 3$ ...(ii)
[angles opposite to equal sides are equal]
On adding Eqs. (i) and (ii), we get
$∠ 2 + ∠ 4 = ∠ 1 + ∠ 3 \Rightarrow ∠ B C D = ∠ B A D . . . (i i i) I n Δ B A D a n d Δ B C D,$
AB = BC [given]
$∠ B A D = ∠ B C D [f r o m E q . (i i i)]$
and AD = CD [given]
$∴ Δ B A D ≅ Δ B C D [b y S A S c o n g r u e n c e r u l e]$
Hence, $∠ A B D = ∠ C B D a n d ∠ A D B = ∠ C D B$
$i . e .,$ BD bisects the angles ABC and ADC.

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Question 5 ABCD is a quadrilateral in which AB = BC and AD = CD. Show that BD bisects both the angles ABC and ADC.

Question 5
ABCD is a quadrilateral in which AB = BC and AD = CD. Show that BD bisects both the angles ABC and ADC.