Question

s.6 angle - angle criterion for similar triangles
△mkl and △ijk are shown below.
which statement is true?
△mkl is similar to △ijk.
△mkl is not similar to △ijk.
there is not enough information to determine whether the triangles are similar.

Explanation:

Step1: Analyze $\triangle MKL$

Since $\triangle MKL$ has two equal sides (marked with red ticks), it is an isosceles triangle. Using the angle - sum property of a triangle ($\angle K+\angle M+\angle L = 180^{\circ}$) and $\angle K = 56^{\circ}$, and $\angle M=\angle L$ (isosceles triangle property), we get $\angle M=\angle L=\frac{180 - 56}{2}=62^{\circ}$.

Step2: Analyze $\triangle IJK$

Since $\triangle IJK$ has two equal sides (marked with red ticks), it is an isosceles triangle. Using the angle - sum property of a triangle ($\angle I+\angle J+\angle K = 180^{\circ}$) and $\angle I = 56^{\circ}$, and $\angle J=\angle K$ (isosceles triangle property), we get $\angle J=\angle K=\frac{180 - 56}{2}=62^{\circ}$.

Step3: Check for similarity

For $\triangle MKL$ and $\triangle IJK$, we have $\angle K_{MKL}=\angle I_{IJK} = 56^{\circ}$ and $\angle L_{MKL}=\angle K_{IJK}=62^{\circ}$. By the AA (angle - angle) similarity criterion (if two angles of one triangle are equal to two angles of another triangle, the triangles are similar), $\triangle MKL\sim\triangle IJK$.

Answer:

$\triangle MKL$ is similar to $\triangle IJK$.