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] $\therefore\ \triangle\text{ADC}\cong\triangle\text{ABC}$ [byASA 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] $\therefore\ \triangle\text{ADC}\cong\triangle\text{ABC}$ [byASA 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
In the adjoining figure, CE || AD and CF || BA. Prove that $\text{ar}(\triangle\text{CBG})=\text{ar}(\triangle\text{AFG}).$
→Factorize the following expressions : $8a^3 - b^3 - 4ax + 2bx$
→A class consists of 50 students out of which 30 are girls. The mean of marks scored by girls in a test is 73 (out of 100) and that of boys is 71. Determine the mean score of the whole class.
The diameter of a sphere is $6\ cm.$ It is melted and drawn into a wire of diameter $0.2\ cm.$ Find the length of the wire.
→Factorize: $a^3 - 3a^2b + 3ab^2 - b^3 + 8$
→In figure, if $\angle\text{ACB} = 40^\circ, \angle\text{DPB} = 120^\circ,$ find $\angle\text{CBD}.$
→Factorize: $a^2 + 4b^2 - 4ab - 4c^2$
→If a and b are different positive primes such that
$(\text{a}+\text{b})^{-1}(\text{a}^{-1}+\text{b}^{-1})=\text{a}^\text{x}\text{b}^\text{y},$ find $\text{x}+\text{y}+2.$
→$(\text{a}+\text{b})^{-1}(\text{a}^{-1}+\text{b}^{-1})=\text{a}^\text{x}\text{b}^\text{y},$ find $\text{x}+\text{y}+2.$
A corn cob $($see Fig.$),$ shaped somewhat like a cone, has the radius of its broadest end as $2.1 \ cm$ and length as $20 \ cm.$ If each $1 \ cm^2$ of the surface of the cob carries an average of four grains, find how many grains you would find on the entire cob?
→If $x + 1$ is a factor of $ax^3 + x^2 - 2x + 4a - 9,$ find the value of $a.$
→