Question

which similarity theorems should you try: similar or not similar: δlmn ~ drag these tiles and drop them in the correct blank above

Explanation:

Step1: Identify Vertical Angles

In the diagram, $\angle MLN$ and $\angle KLI$ are vertical angles, so they are equal. Let's denote $\angle MLN = \angle KLI = \theta$.

Step2: Check Side Ratios

Calculate the ratios of the sides around the equal angles. For $\triangle LMN$ and $\triangle LKI$ (assuming the other triangle is $\triangle LKI$), the sides adjacent to $\theta$:

• For $\triangle LMN$: The sides are $LN = 28 + 21 = 49$ (wait, no, looking at the diagram, maybe the segments are $ML = 28$, $LK = 21$, and $LN = 36$, $LI = 27$? Wait, let's re - examine. If we consider the two triangles formed by the intersecting lines, let's say $\triangle LMN$ and $\triangle LIK$. The sides:
• $\frac{ML}{KL}=\frac{28}{21}=\frac{4}{3}$
• $\frac{LN}{LI}=\frac{36}{27}=\frac{4}{3}$

Since the included angle (vertical angles) is equal and the ratios of the two pairs of corresponding sides are equal, by the Side - Angle - Side (SAS) similarity criterion, the triangles are similar. So $\triangle LMN \sim \triangle LIK$ (or the other triangle with the correct labeling) and they are similar.

Answer:

The similarity theorem is SAS (Side - Angle - Side) similarity, and $\triangle LMN$ is similar to the other triangle (e.g., $\triangle LIK$) with the conclusion "Similar" based on SAS similarity (since two sides are in proportion $\frac{28}{21}=\frac{36}{27}=\frac{4}{3}$ and included angles are equal as vertical angles).