$ABC$ is an isosceles triangle with $AB = AC$. Draw $AP \perp BC$ to show that $\angle B = \angle C$.
(N/A) Given: $ABC$ is an isosceles triangle with $AB = AC$. $AP \perp BC$.
To prove: $\angle B = \angle C$.
Proof:
In $\Delta ABP$ and $\Delta ACP$:
$1$. $\angle APB = \angle APC = 90^\circ$ (Since $AP \perp BC$)
$2$. $AB = AC$ (Given)
$3$. $AP = AP$ (Common side)
Therefore,by $RHS$ congruence criterion,$\Delta ABP \cong \Delta ACP$.
Since the triangles are congruent,their corresponding parts are equal $(CPCT)$.
Thus,$\angle B = \angle C$.
To prove: $\angle B = \angle C$.
Proof:
In $\Delta ABP$ and $\Delta ACP$:
$1$. $\angle APB = \angle APC = 90^\circ$ (Since $AP \perp BC$)
$2$. $AB = AC$ (Given)
$3$. $AP = AP$ (Common side)
Therefore,by $RHS$ congruence criterion,$\Delta ABP \cong \Delta ACP$.
Since the triangles are congruent,their corresponding parts are equal $(CPCT)$.
Thus,$\angle B = \angle C$.
Explore More
Similar Questions
In quadrilateral $ACBD$,$AC = AD$ and $AB$ bisects $\angle A$ (see figure). Show that $\Delta ABC \cong \Delta ABD$. What can you say about $BC$ and $BD$?
Medium
View SolutionLine $l$ is the bisector of an angle $\angle A$ and $B$ is any point on $l$. $BP$ and $BQ$ are perpendiculars from $B$ to the arms of $\angle A$ (see Fig). Show that :
$(i)$ $\Delta APB \cong \Delta AQB$
$(ii)$ $BP = BQ$ or $B$ is equidistant from the arms of $\angle A$.
$(i)$ $\Delta APB \cong \Delta AQB$
$(ii)$ $BP = BQ$ or $B$ is equidistant from the arms of $\angle A$.
Medium
View SolutionIn an isosceles triangle $ABC$,with $AB = AC$,the bisectors of $\angle B$ and $\angle C$ intersect each other at $O$. Join $A$ to $O$. Show that :
$(i)$ $OB = OC$
$(ii)$ $AO$ bisects $\angle A$
$(i)$ $OB = OC$
$(ii)$ $AO$ bisects $\angle A$
Medium
View Solution$ABC$ is a right-angled triangle in which $\angle A = 90^o$ and $AB = AC$. Find $\angle B$ and $\angle C$.
Medium
View Solution$\Delta ABC$ is an isosceles triangle in which altitudes $BE$ and $CF$ are drawn to equal sides $AC$ and $AB$ respectively. Show that these altitudes are equal.
Medium
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