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 similarity into any postulates or theorems just so you are clear on which one I'm using. The AAA Similarity PostulateLet 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.
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.
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.
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 overheadotherwise 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 A and D are congruent. Using your AA Similarity Theorem, you can show that ABC ~ EDC. Using the idea that CSSTAP, we see that
Cross-multiply and we see that 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. The SAS and SSS Similarity TheoremsThere 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.
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.
Excerpted from The Complete Idiot's Guide to Geometry © 2004 by Denise Szecsei, Ph.D.. All rights resrved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc. To order this book direct from the publisher, visit the Penguin USA website or call 1-800-253-6476. You can also purchase this book at Amazon.com and Barnes & Noble.
If two sides of a triangle are congruent , then the angles opposite to these sides are congruent.
∠ P ≅ ∠ Q Proof: Let S be the midpoint of P Q ¯ . Join R and S .
Since S is the midpoint of P Q ¯ , P S ¯ ≅ Q S ¯ . By Reflexive Property , R S ¯ ≅ R S ¯ It is given that P R ¯ ≅ R Q ¯ Therefore, by SSS , Δ P R S ≅ Δ Q R S Since corresponding parts of congruent triangles are congruent, ∠ P ≅ ∠ Q The converse of the Isosceles Triangle Theorem is also true. If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
If ∠ A ≅ ∠ B , then A C ¯ ≅ B C ¯ . |