Similarity in Other Polygons Examples

Determine whether these parallelograms are similar.

To be similar, they must have proportional side lengths and congruent angles. We'll check the sides first.

Yes, those ratios are equivalent, so check on the side length requirement.

As for the angles, we know the polygons are parallelograms, which means that interior adjacent angles are supplementary, so they'll add up to 180°. In the first parallelogram, the missing measures of the acute angles must be 180° – 115° = 65°. That's the given measurement of the acute angle in the second parallelogram, so we know the angles are congruent.

Proportional sides and congruent angles. These parallelograms are similar.

Determine whether these polygons are similar.

If they are similar, then both the side lengths and diagonals will be proportional. We don't know anything about the angles, so our only answer choices here are "no" or "not enough information."

The first two (the side length ratios) are proportional, but the ratio of diagonal lengths is not equivalent. So, no, they're not similar.

Determine whether these polygons are similar.

The side lengths are definitely proportional (⁵⁄₈ = ⁵⁄₈ = ⁵⁄₈ = …), but we have no idea about the angles. There's not enough information given to determine similarity.

These rhombi are similar. Find the value of y.

Proportion, proportion, proportion!

3y + 2 = 20
3y = 18
y = 6