Angles and Sides Examples

Determine if the triangles are similar, and if they are, write a similarity statement.

Since we're given the side lengths, we'll use the proportional sides part of the definition of similarity to decide if these triangles are a homerun.

Match up the shortest sides, the middle-est sides, and the longest sides in your ratios.

2.5 = 2.5 = 2.5

Each ratio can be simplified to 2.5, so they're all equivalent, meaning the side lengths are proportional, meaning the triangles are similar.

To write the similarity statement, we need to put the angles in the correct order. Match up which angle comes between each side. ∠M and ∠O are between the shortest and longest sides of their triangles, so they correspond. Let's start there and work our way around the bases, er, vertices, traveling first along the shortest sides, then the middle-est, then the longest.

∆MIL ~ ∆OTR

In ∆ATL, AT = 8, TL = 10, and LA = 12. In ∆CIN, CI = 10, IN = 12.5, and NC = 16. Are these triangles similar?

We'll figure this one out by setting up the ratios of side lengths. Match the shortest, medium-est, and longest sides.

The first two simplify to , while the last simplifies to , which are not equivalent. We don't have proportional sides, so ∆ATL is not similar to ∆CIN.

Determine if the triangles are similar, and if they are, write a similarity statement.

Let's look for congruent angles, since angle size is really the only information we can get here.

∠A ≅ ∠E, since both are equal to 85°. The question is, is ∠I ≅ ∠X or is ∠R ≅ ∠T?

Let's find the size of ∠I. Using the Triangle Angle Sum Theorem, we know that ∠I = 180° – 85° – 40° = 55°. And looky there, ∠X = 55°. This gives us two sets of congruent angles, which is enough to call it: ∆AIR ~ ∆EXT.

Determine if the triangles are similar, and if they are, write a similarity statement.

Looks like we're going to have to check for angle congruency again, since we don't know anything about side lengths.

We can automatically announce that ∠B ≅ ∠H, since all right angles are congruent.

And m∠S = 180 – 90 – 65 = 25, which is the same measure as ∠C. This gives us two pairs of congruent angles.

∆BOS ~ ∆HWC