Geometry: Properties of Similar Triangles

Properties of Similar Triangles

When we talk about two things being similar, we are trying to convey that our two objects are a lot alike. They don't have to be interchangeable or identical, but they do have to have enough in common. You might think that all triangles are similar, because they have the same number of sides and the same number of angles. But similarity is a special relationship between only certain triangles. In order for two triangles to be declared similar, they must satisfy certain angle and length criteria.

Two triangles are similar if all pairs of corresponding angles are congruent and all pairs of corresponding sides are proportional. If we look at the two triangles in Figure 13.2, we can specify the corresponding angles and the various proportionalities that must be satisfied.

Figure 13.2Two similar triangles.

The angle congruencies are: ∠A ~= ∠R , ∠B ~= ∠S and ∠C ~= ∠T. The proportionalities involved are: ^AB/_RS = ^BC/_ST = ^AC/_RT and all possible rearrangements using the various properties of proportions.

I will use the symbol ~ to indicate that two triangles are similar. If you run through the list, you will see that ~ is an equivalence relation, so it will have reflexive, symmetric, and transitive properties.

If you know that two triangles are similar, you can use the extended proportionalities to learn more about the triangles.

• Example 5: If ΔABC ~ ΔRST as shown in Figure 13.3, use the indicated measures to find the measures of the remaining sides and angles of each of the triangles.

Figure 13.3ΔABC ~ ΔRST.

• Solution: You have quite a list of things to find: the measure of the third angle in ΔABC ; all three angles of ΔRST, and the lengths AC and ST. Because the interior angles of a triangle add up to 180º, you can find m∠A fairly easily:
• m∠A + m∠B + m∠C = 180º
• m∠A + 90º + 37º = 180º
• m∠A = 53º
• Because corresponding angles are congruent, you know that m∠R = 53º , m∠S = 90º and m∠R = 37º. Using your proportionalities, you have
• ^AB/_RS = ^BC/_ST = ^AC/_RT
• You can substitute the known values into the proportionality to find the value of ST:
• ³/₆ = ⁴/_ST
• 3(ST) = 24
• ST = 8
• To determine AC, you will use the other half of the proportionality:
• ³/₆ = ^AC/₁₀
• 6(AC) = 30
• AC = 5

You'll be using the idea that corresponding sides of similar triangles are proportional, so you might as well familiarize yourself with an abbreviation for that phrase now—it's CPSSAT.