Geometry: The ASA Postulate
The ASA Postulate
You can prove that two triangles are congruent if you know about all three sides of the triangles, or if you know about two sides and the included angle. This next postulate will enable you to prove that two triangles are congruent based on two angles and the included side.
Eureka!
You might be reaching abbreviation overload. I would advise you to always write the abbreviations so that the middle letter is the included item (two sides and the included angle for SAS, and two angles and the included side for ASA).
- Postulate 12.3: The ASA Postulate. If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the triangles are congruent.
Let's see how this postulate can be used.
- Example 3: If ¯AC ~= ¯DC and ∠A ~= ∠D , as shown in Figure 12.5, write a two-column proof to show that ΔACE ~= ΔDCB.
Figure 12.5¯AC ~= ¯DC and ∠A ~= ∠D.
- Solution: The figures are starting to get complicated. Whenever this happens, break things apart to get a grip on each piece of the puzzle. Figure 12.6 might help you sort through the mess.
Figure 12.6The triangles in Figure 12.5 pulled apart. You still have that ¯AC ~= ¯DC and ∠A ~+ ∠D.
Notice that ∠C ~= ∠C (by the reflexive property of ~=). Because you are given that ∠A ~= ∠D, you have information about two angles. You just need to know about the included sides. The included side for ΔACE is ¯AC , and the included side for Δ is ¯. You are given that those two sides are congruent! That enables you to use the ASA Postulate! Write out the details.
| Statements | Reasons | |
|---|---|---|
| 1. | ¯AC ~= ¯DC and ∠A ~= ∠D | Given |
| 2. | ∠C ~= ∠C | Reflexive property of ~= |
| 3. | ΔACE ~= ΔDCB | ASA Postulate |
Notice that the proof is fairly short. The hard part is pulling the picture apart to see how the pieces relate. That's not unusual. The hardest part about writing a proof usually involves coming up with the game plan.
Excerpted from The Complete Idiot's Guide to Geometry © 2004 by Denise Szecsei, Ph.D.. All rights reserved 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.
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