Question

Are the two triangles below similar?

Yes, because there are two pairs of congruent corresponding angles
No, because there are not two pairs of congruent corresponding angles
Yes, because the corresponding sides are proportional
No, because the corresponding sides are not proportional

Answer 1

Yes, because there are two pairs of congruent corresponding angles.

Explanation:

Two triangles ABC and DEF are given .

The measure of angles in triangle ABC are:

m<A=30°,m< C=65°,

m<B= 180-(30+65)=85°(measure of sum of angles in any triangle is 180°)

In triangle DEF the measure of the angles are :

m<D=30°,m<E=85°,m<F=180-(30+85)=65°(sum of angles in a triangle add to 180°)

In triangles ABC and DEF,

< A=<D=30°

<B=<E=85°

<C=<F=65°

The two triangles are similar by AAA property.

Answer 2

Answer :- Yes the two triangles are similar because there are two pairs of congruent corresponding angles.

Explanation:-

In Δ ABC

∠A=30° , ∠C=65 °

By angle sum property of triangle

∠A + ∠B + ∠C= 180°

⇒∠B= 180°-∠A-∠C=180°-30°-65°=85°

⇒∠B=85°

Now in ΔABC and ΔDEF

∠A=∠D=30° and ∠B=∠E=85°

⇒ there are two pairs of congruent corresponding angles.

So by AA-similarity criteria

ΔABC ≈ ΔDEF