Question

prove the ASA congruency rule ​

Answer 1

Refer to the attachment

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Answer 2

ASA congruence criterion Proof:

For this purpose, consider ΔABC and ΔDEF ,

where BC= EF, ∠B=∠E and ∠C =∠F.

Note that this will also mean that ∠A=∠D.

Now, we have to show that these two triangles are congruent. Here is a step-by-step proof.

Step 1: If AB is equal to DE , then the two triangles would be congruent by the SAS congruence criterion. So, let us suppose that AB is not equal to DE. Then, one of them would be greater than the other – say that AB> DE.

✍Note: Refer SAS congruence criterion to understand Step 1.

Step 2: Mark a point on AB (call it G), such that GB=DE

Step 3: We note that ΔGBC is congruent to ΔDEF by the SAS criterion. This means that ∠BCG=∠F.

Step 4: Finally, we note that ∠F=∠C(given), and so ∠BCG=∠C.

EnlightenedThink: Is that possible if CG is in a different direction than CA?

No, it’s not!

This means that CG must be in the same direction as CA, or in other words, G and A coincide, or: GB=AB=DE. Thus, the two triangles are congruent by the SAS congruence criterion.

✍Note: An important aspect of the proof of the ASA congruence criterion we have encountered is that it builds on the SAS congruence criterion – it assumes the truth of the SAS congruence criterion.