Are triangles ABC and A₁B₁C₁ similar, why, if it is known that: angle A = 17 °, angle B = 52 °

Are triangles ABC and A₁B₁C₁ similar, why, if it is known that: angle A = 17 °, angle B = 52 °, angle C₁ = 111 °, angle B₁ = 52 °?

Given triangles ABC and A₁B₁C₁, and the angles of these triangles <A = 17 °, <B = 52 °,

<C₁ = 111 °, <B₁ = 52 °. We need to prove that triangles ABC and A₁B₁C₁ are similar.

First, we need to find the 3rd corners of the triangles ABC and A₁B₁C₁. We know that the sum of the interior angles is 180 °. Using this, we find the third corner of the triangle ABC.

<A + <B + <C = 180 °;

17 ° + 52 ° + <C = 180 °;

<C = 180 ° – 69 °;

<C = 111 °.

Now we find the 3rd corners of the triangle A₁B₁C₁.

<A₁ + <B₁ + <C₁ = 180 °;

<A₁ + 52 ° + 111 ° = 180 °;

<A₁ = 17 °.

It follows that the triangles ABC and A₁B₁C₁ are similar, i.e. e.

<A₁ = <A;

<B₁ = <B;

<C₁ = <С.

Answer: Triangles ABC and A₁B₁C₁ are similar.