In quadrilateral $PQRS$,$PQ = PS$ and $RQ = RS$. Prove that $\angle PQR = \angle PSR$.
(N/A) $1$. Consider the two triangles $\triangle PQR$ and $\triangle PSR$.
$2$. In these triangles,we are given that $PQ = PS$ (Side).
$3$. We are also given that $RQ = RS$ (Side).
$4$. The side $PR$ is common to both triangles,so $PR = PR$ (Side).
$5$. By the $SSS$ (Side-Side-Side) congruence criterion,$\triangle PQR \cong \triangle PSR$.
$6$. Since the triangles are congruent,their corresponding parts are equal ($CPCT$ - Corresponding Parts of Congruent Triangles).
$7$. Therefore,$\angle PQR = \angle PSR$.
$2$. In these triangles,we are given that $PQ = PS$ (Side).
$3$. We are also given that $RQ = RS$ (Side).
$4$. The side $PR$ is common to both triangles,so $PR = PR$ (Side).
$5$. By the $SSS$ (Side-Side-Side) congruence criterion,$\triangle PQR \cong \triangle PSR$.
$6$. Since the triangles are congruent,their corresponding parts are equal ($CPCT$ - Corresponding Parts of Congruent Triangles).
$7$. Therefore,$\angle PQR = \angle PSR$.
Explore More
Similar Questions
In triangles $ABC$ and $PQR$,$\angle A = \angle Q$ and $\angle B = \angle R$. Which side of $\triangle PQR$ should be equal to side $BC$ of $\triangle ABC$ so that the two triangles are congruent? Give reason for your answer.
Medium
View Solution$PL, QM$ and $RN$ are the altitudes of $\Delta PQR$. If $PL = QM = RN$,then by $AAS$ criterion of congruence,prove that $\Delta PQR$ is an equilateral triangle.
Medium
View SolutionIn the given figure,$l \parallel m$ and $M$ is the mid-point of a line segment $AB$. Show that $M$ is also the mid-point of any line segment $CD,$ having its end points on $l$ and $m,$ respectively.
Medium
View Solution$ABCD$ is a quadrilateral in which $AB = BC$ and $AD = CD$. Show that $BD$ bisects both the angles $ABC$ and $ADC$.
Medium
View SolutionIn triangles $ABC$ and $PQR$,$\angle A = \angle Q$ and $\angle B = \angle R$. Which side of $\triangle PQR$ should be equal to side $AB$ of $\triangle ABC$ so that the two triangles are congruent? Give reason for your answer.
Easy
View SolutionVedclass Products
For Students
Vedclass Test Series
Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.
Start Free TrialFor Teachers
Exam Paper Generator
Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.
Try FreeFor Institutes
Online Exam Module
Live online exams with unlimited students, 360° analytics & white-label branding.
See Demo