Because ¯AB ¯DE, you can look at ¯AB and ¯DE as two parallel lines cut by a transversal ¯AE. 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. Solution: In order to write this proof, you need a game plan.Example 7: If ¯AB ¯DE as shown in Figure 13.4, write a two-column proof that shows ABC ~ EDC.įigure 13.4 The segments ¯AB and ¯DE are parallel.Let's use it to prove the similarity of some triangles. It relies mainly on fact that the measures of the interior angles of a triangle addup to 180º. There's not much to the proof of Theorem 13.1. This theorem is easier to apply than the AAA Similarity Postulate (because you only have to check two angles instead of three). If two angles of one triangle are congruent to two angles of a second triangle, then the two triangles are similar. 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. 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. You only have to check the angle relationships.
This postulate lets you prove similarity without messing with the proportionalities. If the three angles of one triangle are ongruent to the three angles of a second triangle, then the two triangles are similar. Postulate 13.1: AAA Similarity Postulate.You will just have to believe in it and use it to your heart's content. It's a postulate, so it's something you can't prove. Let me introduce you to your first shortcut involving the similarity of two triangles. I'll throw the word similarity into any postulates or theorems just so you are clear on which one I'm using. It's important to pay attention to whether you are trying to show that two triangles are similar or congruent. Unfortunately, some of your similarity theorems have the same initials as the congruent triangle postulates. I'll give you some postulates and theorems to help you with similarity problems. When you were working with congruent triangles you had some postulates and theorems to help you prove congruence. 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.
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