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Triangles [Congruency in Triangles] — MATHEMATICS STD 9 — Question

ICSE BoardEnglish MediumSTD 9MATHEMATICSTriangles [Congruency in Triangles]5 Marks
Question
In the figure, given below, $\triangle \mathrm{ABC}$ is right$-$angled at $\mathrm{B} . \mathrm{ABP}$ and $\text{ACRS}$ are squares.

Prove that:$(i) \triangle \mathrm{ACQ}$ and $\triangle \mathrm{ASB}$ are congruent.$(ii) \mathrm{CQ}=\mathrm{BS}$.
✓

Answer

Given: $A(\triangle ABC)$ is right$-$angled at $B.$
$\text{ABPQ}$ and $\text{ACRS}$ are squares
To Prove:
$(i) \triangle ACQ \cong \triangle ASB$
$(ii) CQ = BS$
Proof:
$(i)$
$\angle QAB = 90^\circ\dots ...[ \text{ABPQ}$ is a square $] \dots...(1)$
$\angle SAC = 90^\circ\dots ...[\text {ACRS}$ is a square $]\dots ...(2)$
From $(1)$ and $(2) ,$ We have
$\angle QAB = \angle SAC\dots...(3)$
Adding $\angle BAC$ to both sides of $(3),$ We have
$\angle QAB + \angle BAC = \angle SAC + \angle BAC$
$\Rightarrow \angle QAC = \angle SAB\dots...(4)$
In $\triangle ACQ$ and $\triangle ASB,$
$QA = QB\dots...[$ Sides of a square $\text{ABPQ}]$
$\angle QAC = \angle SAB\dots...[$ From $(4)]$
$AC = AS\dots...[$ sides of a square $ \text{ACRS}]$
$\therefore $ By Side$ -$Angle$-$Side criterion of congruence,
$\triangle ACQ \cong \triangle ASB\dots....(ii)$
The corresponding parts of the congruent triangles are congruent,
$\therefore CQ = BS\dots...[ \text{c.p.c.t.} ]$

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Case Study : Ms Anu Gupta teaches mathematics to class 9 in a school. One day she drew a figure on the board in the class. She provided the following clues to the students.
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