Question

two right Triangles are congruent if the hypotenuse and one side Of one triangle are respectively equal to the hypotenuse and one side of the Other Triangle​

Answer 1

Given:-

• Two right triangles ABC and DEF in which ∠B = ∠E = 90°, AC = DF, BC = EF

To Prove:-

• ∆ABC ≅ ∆ DEF

Construction:-

produce DE to G so that EG = AB. join GF.

Proof:-

In ∆s ABC and GEF , we have

AB = GE [By construction]

∠B = ∠FEG = 90°

BC = EF [Given]

So, by SAS criterion of congruence, we obtain

∆ ABC ≅ ∆ GEF

➝ ∠A = ∠G ....i

AC = GF [c.p.c.t] ...ii

Now,

AC = GF

AC = DF [Given]

∴ DF = GF

⇒ ∠D = ∠G [∠ opposite to the equal sides in ∆ DGF are equal ....iii]

from (i) and (iii) , we get

∠A = ∠D

Thus , in ∆s ABC and DEF , we have

∠A = ∠D

∠B = ∠E

⇒ ∠A + ∠B = ∠D + ∠E [ .·.∠A + ∠B

+ ∠C = 180° & ∠D +

∠E + ∠F = 180°]

⇒ 180° - ∠C = 180° - ∠F

⇒ ∠C = ∠F ....v

Now, in in ∆s ABC and DEF , we have

BC = EF

∠C = ∠F

AC = DF

So, by SAS criterion of congruence, we obtain

∆ABC ≅ ∆ DEF

Hence proved ✓✓

Answer 2

Step-by-step explanation:

If in two right triangles the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle, then prove that two triangles are congruent.

Hypotenuse is equal in both the two right triangles. AC=DF. One side of the triangle is equal to the other side of the triangle.