Geometry: The Big Three

The Big Three

In order to prove that two triangles are similar, you would need to verify that all three corresponding angles are congruent and that the required proportionality relationships hold between all corresponding sides. When you were working with congruent triangles you had some postulates and theorems to help you prove congruence. I'll give you some postulates and theorems to help you with similarity problems. Unfortunately, some of your similarity theorems have the same initials as the congruent triangle postulates. It's important to pay attention to whether you are trying to show that two triangles are similar or congruent. I'll throw the word €œsimilarity€ into any postulates or theorems just so you are clear on which one I'm using.

The AAA Similarity Postulate

Let me introduce you to your first shortcut involving the similarity of two triangles. It's a postulate, so it's something you can't prove. You will just have to believe in it and use it to your heart's content.

• Postulate 13.1: AAA Similarity Postulate. If the three angles of one triangle are congruent to the three angles of a second triangle, then the two triangles are similar.

This postulate lets you prove similarity without messing with the proportionalities. You only have to check the angle relationships. But you can even do better than that! If two angles of one triangle are congruent to two angles of another triangle, then the third angles must also be congruent. So if you want to show that two triangles are similar, all you have to do is show that two angles of one triangle are congruent to two angles of the other triangle.

• Theorem 13.1: AA Similarity Theorem. If two angles of one triangle are congruent to two angles of a second triangle, then the two triangles are similar.

This theorem is easier to apply than the AAA Similarity Postulate (because you only have to check two angles instead of three). There's not much to the proof of Theorem 13.1. It relies mainly on fact that the measures of the interior angles of a triangle addup to 180º. Let's use it to prove the similarity of some triangles.

• Example 7: If ¯AB €Œ €Œ ¯DE as shown in Figure 13.4, write a two-column proof that shows ”ABC ~ ”EDC.

Figure 13.4The segments ¯AB and ¯DE are parallel.

• Solution: In order to write this proof, you need a game plan. The figures are getting a bit more complicated, and you have to use more and more of your previous results in order to write out proofs. Because ¯AB €Œ €Œ ¯DE , you can look at ¯AB and ¯DE as two parallel lines cut by a transversal ¯AE. In this case, ˆ BAE and ˆ DEA are alternate interior angles, so they are congruent. Because ˆ ACB and ˆ DCE are vertical angles, they are congruent. So by the AA Similarity Theorem, you see that ”ABC ~ ”EDC.

One technique for estimating the height of an object (like a tree, or a pyramid) uses the ideas of similar triangles. This technique assumes that you know your own height and can measure the lengths of shadows. In order for this technique to work, both you and the object you are trying to measure must cast a shadow.

Suppose that the sun is shining, and you want to determine the height of a nearby tree. In order for this technique to work, the sun can't be shining directly overhead€”otherwise neither you nor the tree will cast a measurable shadow. Figure 13.5 shows the role that similar triangles play in this technique. Suppose you are 6 feet tall, and you cast a shadow of length 8 feet. You don't know how tall the tree is, but its shadow is 36 feet long. If you assume that both you and the tree have good posture and stand perpendicular to the ground, both you and the tree form two triangles. Because the sun is very far away, you can assume that ˆ A and ˆ D are congruent. Using your AA Similarity Theorem, you can show that ”ABC ~ ”EDC.

Using the idea that CSSTAP, we see that

• ^AC/_BC = ^DF/_EF or ⁶/₈ = ^h/₃₆.

Cross-multiply and we see that

• h = ^{6 — 36}/₈ = 27.

So the tree is roughly 27 feet tall. Thales used this method to estimate the height of the pyramids, and he was accurate enough to have amazed his friends and impressed some pharaohs.

Figure 13.5The stick figure and the tree cast a shadow. Two similar triangles are formed, and because the sides are proportional, we can determine the height of the tree.

The SAS and SSS Similarity Theorems

There are other theorems that can help show that two triangles are similar. I will just state two other theorems that can be useful. I won't take the time to prove these theorems, because there's so much to discuss and I'm running out of space.

• Theorem 13.2: The SAS Similarity Theorem. If an angle of one triangle is congruent to an angle of a second triangle and the including sides of that angle are proportional, then the triangles are similar.
• Theorem 13.3: The SSS Similarity Theorem. If three sides of one triangle are proportional to the three corresponding sides of another triangle, then the triangles are similar.
• Example 8: Given the two triangles shown in Figure 13.6, find RS.

Figure 13.6”ABC and ´RST.

• Solution: We see that ˆ C ~= ˆ T and that the pairs of including sides are proportional:
• ⁶/₁₀ = ¹²/₂₀
• Don't be fooled into thinking that the two sides are not proportional because
• ⁶/₁₀ ‰  ²⁰/₁₂.
• You have to try various combinations to determine how the sides correspond. In this case, ¯AC corresponds to ¯ST, ¯BC corresponds to ¯RT, and ¯AB corresponds to ¯RS. So ”ABC ~ ”RST by the SAS Similarity Theorem, and you can use the fact that CSSTAP to determine RS:
• ^RS/_ST = ^AB/_AC
• ^RS/₂₀ = ¹⁴/₁₀
• RS = 28

Here's your chance to shine. Remember that I am with you in spirit and have provided the answers to these questions in Answer Key.

• 1. Use the Means-Extremes Property to solve for x: ^{x - 2}/₃ = ^{x + 1}/₄.
• 2. Use the Means-Extremes Property to solve for x: ⁶/_x = ^x/₂₄.
• 3. Suppose that the measures of two supplementary angles are in the ratio of 2 to 7. Find the measure of each angle.
• 4. If ˆ A ~= ˆ D as shown in Figure 13.7, write a two-column proof to show that ^BC/_AB = ^CE/_DE.

Figure 13.7

ˆ A ~= ˆ D.

• 5. The distance across an alligator-infested lake is to be measured safely by using similar triangles, as shown in Figure 13.8. If XY = 160 feet, YW = 40 feet, TY = 120 feet, and WZ = 50 feet, find XT.

Figure 13.8Measuring the length of a lake using similar triangles.