Question

Determine if triangle J, K, LJKL and triangle M, N, OMNO are or are not similar, and, if they are, state how you know. (Note that figures are NOT necessarily drawn to scale.)

Answer 1

I don’t have the diagram or any angle/side measures, so I can’t decide for these particular triangles yet. Tell me (or upload) the picture or give the marked information. Meanwhile, here is how you can decide and how to state the result.

How to determine similarity of two triangles
- Match corresponding vertices: write the triangles in an order that pairs the likely corresponding vertices (e.g. JKL ~ MNO means J↔M, K↔N, L↔O).
- Use one of the similarity tests:
- AA (angle–angle): If two angles of one triangle equal two angles of the other, the triangles are similar. This is the most common.
- SAS (side–angle–side similarity): If two pairs of sides are proportional and the included angles are equal, the triangles are similar.
- SSS (side–side–side): If all three pairs of corresponding sides are in proportion, the triangles are similar.
- Other clues: parallel lines in the figure often give equal corresponding angles (use that to get AA). A dilation/rotation/reflection mapping one triangle onto the other also shows similarity.

How to state the result
- If they are similar, give the similarity statement with the vertices matched in corresponding order and state the reason, e.g.:
- “△JKL ~ △MNO by AA, because ∠J = ∠M and ∠K = ∠N.”
- “△JKL ~ △MNO by SAS, because JK/MN = KL/NO and ∠K = ∠N.”
- If they are not similar, show a mismatch: either find an angle in one triangle that is not equal to the corresponding angle, or show that the ratios of corresponding sides are not all equal.

If you paste the diagram or the marked angle measures/side lengths, I’ll determine whether △JKL and △MNO are similar and give the proper similarity statement and reason.