Ex.7.2 Q1 Congruence of Triangles  NCERT Maths Class 7
Question
Which congruence criterion do you use in the following?
(a) Given: \(\begin{align}AC &= DF\\AB &= DE\\BC &= EF\end{align}\) So, \(ΔABC ≅ ΔDEF\) 

(b) Given: \( \begin{align} ZX &= RP\\RQ &= ZY\\∠PRQ &= ∠XZY \end{align}\) So, \(ΔPQR ≅ ΔXYZ\) 

(c) Given: \(\begin{align} ∠MLN &= ∠FGH\\∠NML &= ∠GFH\\ML &= FG \end{align}\) So, \(ΔLMN ≅ ΔGFH\) 

(d) Given: \(\begin{align} EB &= DB\\AE &= BC\\∠A &= ∠C = 90^\circ \end{align}\) So, \(ΔABE ≅ ΔCDB\) 
Text Solution
(a)
What is known?
In \(ΔABC\) and \(ΔDEF\)
\[\begin{align} AC&= DF\\AB &= DE\\BC &= EF \end{align}\]
What is the unknown?
Congruence criterion by which these two triangles are congruent.
Reasoning:
Three sides of a \(ΔABC\) are equal to the corresponding three sides of the other \(ΔDEF\). So, \(SSS\) congruence criterion can be used.
Steps:
Given, \(AC\) = \(DF\), \(AB = DE\) and \(BC = EF\). The three sides of a \(ΔABC\) are equal to the corresponding three sides of the other \(ΔDEF\). So, by \(SSS\) congruence criterion, the two triangles are congruent.
(b)
What is known?
\[\begin{align} ZX &= RP\\RQ &= ZY\\∠PRQ &= ∠XZY \end{align}\]
What is unknown?
Congruence criterion by which the two triangles are congruent.
Reasoning:
The two sides and one angle of a \(ΔPRQ\) are equal to the corresponding two sides and one angle of the other \(ΔXYZ\). So, \(SSS\) congruence criterion can be used.
Steps:
Given,\( ZX = RP\), \(RQ = ZY\), \(∠PRQ = ∠XZY\). The two sides and one angle of a triangle \(ΔPRQ\) are equal to the corresponding two sides and one angle of the other triangle \(ΔXYZ\). So, by \(SAS\) congruence criterion, the two triangles are congruent.
(c)
What is known?
\[\begin{align} ∠MLN&= ∠FGH\\∠NML &= ∠GFH\\ML &= FG \end{align}\]
What is the unknown?
By which congruence criterion two triangles are congruent.
Reasoning:
The two angles and one side of a triangle \(ΔLMN\) are equal to the corresponding two sides and one angle of the other triangle \(ΔGFH\). So, by using congruency criterion based on two angels and one side \((ASA)\) of triangles can be used.
Steps:
Given, \(∠MLN = ∠FGH\), \(∠NML = ∠GFH\) and \(ML = FG\). The two sides and one angle of triangle \(ΔLMN\) are equal to the two sides and one angle of the other triangle \(ΔGFH\). So, by \(ASA\) congruence criterion, the two triangles are congruent.
(d)
What is known?
In \(ΔABE\) and \(ΔCDB\)
\[\begin{align} EB &= DB\\AE &= BC\\∠A& = ∠C = 90^\circ \end{align}\]
What is the unknown?
By which congruence criterion two triangles are congruent.
Reasoning:
In this case, the hypotenuse and one side of a rightangled triangle are respectively equal to the hypotenuse and one side of another rightangled triangle. congruency is based on of a rightangle, hypotenuse and one side \((RHS)\) criterion.
Steps:
Since, \(EB = DB\), \(AE = BC\), \(∠A = ∠C = 90^\circ\). The hypotenuse and one side of a rightangled triangle \(ΔABE\) are equal to the hypotenuse and one side of the other rightangled triangle \(ΔCDB.\) So, by \((RHS)\) congruence criterion, the two triangles are congruent.