Prove that if in two triangles two angles and the included side of one triangle are equal to two angles and the included side of the other triangle, then the two triangles are congruent.
Solution:
Consider two triangles ABC and DEF
∠B = ∠E and ∠C = ∠F
BC = EF
We have to prove that the triangles ABC and DEF are congruent.
Case 1 : Let AB = DE
In triangles ABC and DEF,
AB = DE
∠B = ∠E
Given, BC = EF
By SAS criterion, the triangles ABC and DEF are congruent.
Case 2 : AB > DE
Take a point P on AB such that PB = DE
In triangles PBC and DEF,
PB = DE
Given, ∠B = ∠E
Given, BE = EF
By SAS criterion, the triangles PBC and DEF are congruent.
By CPCTC,
∠PCB = ∠DFE
Given, ∠ACB = ∠DFE
So, ∠PCB = ∠ACB
The condition is satisfied only when P coincides with A.
AB = DE
By case 1, the triangles ABC and DEF are congruent.
Case 3 : If AB < DE
Take a point M on DE such that AB = ME
In triangles ABC and MEF,
AB = ME
Given, ∠B = ∠E
Given, BE = EF
By SAS criterion, the triangles ABC and MEF are congruent.
By CPCTC,
∠ACB = ∠MFE
Given, ∠ACB = ∠DFE
So, ∠MFE = ∠DFE
The condition is satisfied only when M coincides with D.
AB = DE
By case 1, the triangles ABC and DEF are congruent.
Therefore, it is proved that the two triangles are congruent.
✦ Try This: In a ΔABC, AD is the bisector of ∠A, meeting side BC at D. If AB = 10 cm, AC = 6 cm and BC = 12 cm, find BD and DC.
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 7
NCERT Exemplar Class 9 Maths Exercise 7.4 Sample Problem 2
Prove that if in two triangles two angles and the included side of one triangle are equal to two angles and the included side of the other triangle, then the two triangles are congruent
Summary:
It is proven that if in two triangles two angles and the included side of one triangle are equal to two angles and the included side of the other triangle, then the two triangles are congruent by ASA criterion
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