Mastering geometry proofs

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A good strategy for completing a proof in geometry is to work backwards. If you are having trouble determining where to start, try going to the end first. Look at the statement you are trying to prove and ask yourself, “What do I need to know in order to prove this statement?” In other words, what would the last line of the proof be? Then figure out what you need in order to prove the next to last line of the proof, and so on.

Here’s an example to show how this works:

Given: \overline{AE} and \overline{BD} bisect each other.
Prove: \overline{AB} \cong \overline{DE}.










How do you approach this? The first question to ask yourself is “what do I need to know to prove the two segments are congruent?” The answer: Show that the two triangles are congruent—then the segments will be congruent by CPCTC.

So next, the question becomes “how do I show that the triangles are congruent?” Well, you can show triangles are congruent with SSS, SAS or ASA.

Continuing to work backwards, the next question is “can I prove SSS?” Well, no, because you’re trying to prove two sides are congruent. You can’t use what you’re trying to prove in a congruence proof! So ask the next question: “can I prove SAS?” And the answer is yes!

Because \overline{AE} and \overline{BD} bisect each other, \overline{AC} \cong \overline{CE} and \overline{BC} \cong \overline{CD}.    And \angle{ACB} \cong \angle{DCE}  because they are vertical angles.

You have simplified the proof to finding two congruent sides and a congruent angle. This leads to SAS which leads to two congruent triangles which leads to the two congruent segments you were trying to prove.

Triangle Congruence Proofs and CPCTC

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The triangle congruence theorems (SSS, SAS, ASA and HL) can only be used to show that two triangles are congruent to each other. You can’t use them to show that sides or angles are congruent.

However, CPCTC is used to show that parts of triangles are congruent to each other. CPCTC cannot be used to show that triangles are congruent.

When you construct a proof that asks you to show that two segments or two angles in a triangle are congruent, most of the time you will first show the triangles are congruent, and then use CPCTC to show the corresponding parts are congruent.

Remember that CPCTC can’t be used until you have already shown that the triangles are congruent.

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