Question

determine if triangle ijk and triangle lmn are or are not similar, and, if they are, state how you know. (note that figures are not necessarily drawn to scale.)
triangle ijk: side ij = 14, side ik = 14, side jk = 18, angle at i is 82 degrees.
triangle lmn: side ln = 70, side lm = 70, angle at l is 82 degrees.
answer attempt 1 out of 2
the triangles ~ similar.

Explanation:

Step1: Identify sides around equal angles

For $\triangle IJK$: sides $\overline{IK}=14$, $\overline{IJ}=14$, included $\angle I=82^\circ$
For $\triangle LMN$: sides $\overline{LM}=70$, $\overline{LN}=70$, included $\angle L=82^\circ$

Step2: Calculate side ratios

Ratio of corresponding sides:
$\frac{IK}{LM} = \frac{14}{70} = \frac{1}{5}$
$\frac{IJ}{LN} = \frac{14}{70} = \frac{1}{5}$

Step3: Apply SAS similarity rule

Included angles are equal ($82^\circ$), and the pairs of surrounding sides are proportional. By the Side-Angle-Side (SAS) Similarity Criterion, the triangles are similar.

Answer:

The triangles are similar. They satisfy the SAS Similarity Criterion: the included angles are equal ($82^\circ$), and the ratios of the corresponding sides forming these angles are equal ($\frac{1}{5}$).