Question

Given ∠ABE = 45° and ∠EAB = 63° in ΔABE and∠MNP= 72° and ∠NMP = 63° in ΔMNP. Are the two triangles, ΔABE and ΔMPN similar? If so, by what criterion?

Answer 1

Yes ,we can prove the two triangles are similar by angle angle test.

Explanation:

Given:

∠ABE = 45°

∠EAB = 63° and

∠MNP= 72°

∠NMP = 63°

To Prove:

ΔABE ~ ΔMPN

Proof:

In a Triangle sum of the angles of a triangle is 180°

In ΔMPN

∴ ∠MNP + ∠NMP + ∠MPN = 180°

Substituting the given values we get,

`72+63+\angle MPN = 180\\135 + \angle MPN = 180\\\angle MPN = 180-135\\\angle MPN = 45`

∠MPN = 45° ..........................( 1 )

Now,for triangles to be similar

1. minimum two angles should be congruent i.e AA test.
2. all the three sides should be proportional i.e SSS test

In Δ ABE and Δ MPN

∠ ABE ≅ ∠ MPN = 45° ……….{From ( 1 ) and Given}

∠ EAB ≅ ∠ NMP = 63° ………...{Given}

Δ ABE ~ Δ MPN ….{Angle-Angle test}

..........Proved