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

In $△ A B D$ and $△ C B D$ ,
$A D = C D$ [ given ]
$B D = B D$ [ Common ]
$A B = B C$ [ given ]
Hence, $△ A B D ≅ △ C B D$ [ SSS ]
$∠ A B D = ∠ 1$
$∠ C B D = ∠ 2$
$∠ 1 = ∠ 2$ [ CPCT ]
Also ,
$∠ 3 = ∠ 4$ [ CPCT ]
Hence ,
$B D$ bisects both the angles $A B C$ and $A D E$

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

In △ABD and △CBD,AD=CD [ given ]BD=BD [ Common ]AB=BC [ given ]Hence, △ABD≅△CBD [ SSS ]∠ABD=∠1∠CBD=∠2∠1=∠2 [ CPCT ]Also ,∠3=∠4 [ CPCT ]Hence ,BD bisects both the angles ABC and ADE

In $△ A B D$ and $△ C B D$ ,
$A D = C D$ [ given ]
$B D = B D$ [ Common ]
$A B = B C$ [ given ]
Hence, $△ A B D ≅ △ C B D$ [ SSS ]
$∠ A B D = ∠ 1$
$∠ C B D = ∠ 2$
$∠ 1 = ∠ 2$ [ CPCT ]
Also ,
$∠ 3 = ∠ 4$ [ CPCT ]
Hence ,
$B D$ bisects both the angles $A B C$ and $A D E$