Question
ABC is a right-angled triangle in which $\angle A =90^{\circ}$ and $AB = AC$. Find $\angle B$ and $\angle C$.
✓
Answer
$ABC$ is a right triangle in which,
$AB = AC$
$\Rightarrow \angle C =\angle B$ [ opposite angles of equal side are equal]$ .....(i)$
We know that, in $ABC , \angle A +\angle B +\angle C =180^{\circ}$ [ Angle sum property of triangle]
$\Rightarrow 90^{\circ}+\angle B+\angle B=180^{\circ}\left[\angle A=90^{\circ} \text { (given) and } \angle B=C \text { (from eq. (i) }\right]$
$\Rightarrow 2 \angle B=180^{\circ}-90^{\circ}$
$\Rightarrow 2 \angle B=90^{\circ}$
$\Rightarrow \angle B=45^{\circ}$
Also $\angle C =45^{\circ}[$ As $\angle B =\angle C ]$
$AB = AC$
$\Rightarrow \angle C =\angle B$ [ opposite angles of equal side are equal]$ .....(i)$
We know that, in $ABC , \angle A +\angle B +\angle C =180^{\circ}$ [ Angle sum property of triangle]
$\Rightarrow 90^{\circ}+\angle B+\angle B=180^{\circ}\left[\angle A=90^{\circ} \text { (given) and } \angle B=C \text { (from eq. (i) }\right]$
$\Rightarrow 2 \angle B=180^{\circ}-90^{\circ}$
$\Rightarrow 2 \angle B=90^{\circ}$
$\Rightarrow \angle B=45^{\circ}$
Also $\angle C =45^{\circ}[$ As $\angle B =\angle C ]$
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
Multiply: $\sqrt{10}$ by $\sqrt{40}$
→How many planes can be made to pass through a line and a point not on the line?
Examine whether the following number are rational or irrational:
$\big(5-\sqrt{5}\big)\big(5+\sqrt{5}\big)$
→$\big(5-\sqrt{5}\big)\big(5+\sqrt{5}\big)$
Factorise:
$6 x^2-11 x-35$
→$6 x^2-11 x-35$
The measure of an angle is twice the measure of its supplementary angle. Find its measure.
Factorise: $2\text{x}^2-\text{x}+\frac{1}{8}$
→If $a^2+b^2+c^2=250$ and $a b+b c+c a=3$, find $a+b+c$.
→Two circles intersect at two points $B$ and $C.$ Through $B,$ two line segments $\ce{ABD}$ and $\ce{PBQ}$ are drawn to intersect the circles at $\ce{A, D, P, Q}$ respectively $($see figure$).$ Prove that $\ce{ACP=QCD}$.
→In the we have $AB = BC, BX = BY$. Show that $AX = CY.$
→The perimeter of an isosceles triangle is $42 \ cm$ and its base is $1 \frac{1}{2}$ times each of the equal sides. Find the height of the triangle. $($Given, $\sqrt{7}=2.64)$.
→