Question
$ABCD$ is a quadrilateral such that diagonal $AC$ bisects the angles $A$ and $C$. Prove that $AB = AD$ and $CB = CD$.
✓
Answer
Given in a quadrilateral $ABCD$, diagonal $AC$ bisects the angles $A$ and $C.$
To prove $\text{AB}=\text{CD}\ \text{ and }\text{CB}=\text{CD}$ Proof in $\triangle\ \text{ADC}\ \text{and}\ \triangle\ \text{ABC},$
$\angle\ \text{DAC}=\angle\ \text{BAC}$
$\big[\because$ AC is the bisector of $\angle\ \text{A}\ \text{and}\ \angle\ \text{C}\big]$
$\angle\ \text{DCA}=\angle\ \text{BCA}$
$\big[\because$ AC is the bisector of $\angle\ \text{A}\ \text{and}\ \angle\text{C}\big]$
$\text{and}\ \text{AC}=\text{AC}$ [common side]
$\therefore\ \triangle\text{ADC}\cong\triangle\text{ABC}$ [by$ASA$ congruence rule]
$\text{AD}=\text{AB}$ [by CPCT] $\text{and}\ \text{CD}=\text{CB}$ [by $CPCT$] Hence proved.
To prove $\text{AB}=\text{CD}\ \text{ and }\text{CB}=\text{CD}$ Proof in $\triangle\ \text{ADC}\ \text{and}\ \triangle\ \text{ABC},$
$\angle\ \text{DAC}=\angle\ \text{BAC}$
$\big[\because$ AC is the bisector of $\angle\ \text{A}\ \text{and}\ \angle\ \text{C}\big]$
$\angle\ \text{DCA}=\angle\ \text{BCA}$
$\big[\because$ AC is the bisector of $\angle\ \text{A}\ \text{and}\ \angle\text{C}\big]$
$\text{and}\ \text{AC}=\text{AC}$ [common side]
$\therefore\ \triangle\text{ADC}\cong\triangle\text{ABC}$ [by$ASA$ congruence rule]
$\text{AD}=\text{AB}$ [by CPCT] $\text{and}\ \text{CD}=\text{CB}$ [by $CPCT$] Hence proved.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
→
Solve the following equations:
$4^{2\text{x}}=(\sqrt[3]{16})^{\frac{-6}{\text{y}}}=(\sqrt{8})^2$
→$4^{2\text{x}}=(\sqrt[3]{16})^{\frac{-6}{\text{y}}}=(\sqrt{8})^2$
In the given figure, $O$ is the canter of the circle and $\angle\text{AOB}=70^\circ.$ Calculate the values of
$i. \angle\text{OCA}$
$ii. \angle\text{OAC}$
→$i. \angle\text{OCA}$
$ii. \angle\text{OAC}$
Find the median of the data: $15, 6, 16, 8, 22, 21, 9, 18, 25$
→If $2^\text{x}=3^\text{y}=12^\text{z},$ show that $\frac{1}{\text{z}}=\frac{1}{\text{y}}+\frac{2}{\text{x}}.$
→The approximate velocities of some vehicles are given below:
Draw a bar graph to represent the above data.
→|
Name of vehicle
|
Bicycle
|
Scooter
|
Car
|
Bus
|
Train
|
|
Velocity (in km/hr)
|
$27$
|
$45$
|
$90$
|
$72$
|
$63$
|
If the temperature of a liquid can be measured in kelvin units as $x^\circ K$ or in fahrenheit units as $y^\circ F$, the relation between the two systems of measurement of temperature is given by the linear equation. $\text{y}=\frac{9}{5}(\text{x}-273)+32$
$i.$ find the temperature of the liquid in fahrenheit, if the temperature of the liquid is $313K.$
$ii.$ If the temperature is $158^\circ F$, then find the temperature in kelvin.
→$i.$ find the temperature of the liquid in fahrenheit, if the temperature of the liquid is $313K.$
$ii.$ If the temperature is $158^\circ F$, then find the temperature in kelvin.
Find the area of an equilateral triangle having each side $4\ cm$.
→In figure, $PQ\ ||\ AB$ and $PR\ ||\ BC$. If $\angle\text{QPR}=102^\circ,$ determine $\angle\text{ABC}.$ Give reasons. 
→Find the median of the data: $25, 34, 31, 23, 22, 26, 35, 29, 20, 32$
→