How is ∆ABC = ∆ADC not the same as ∆ABC = ∆ACD? How to prove angles ABC and ADC are the same size?

Question

Kite ABCD, for which AB = AD and BC = DC is the given information. The first two proofs show that diagonal AC creates angles BAC and DAC of the same size. 

Proof for Kite

AB = AD given

BC = DC given

AC = AC shared side

∆ABC = ∆ADC  SSS

∠BAC = ∠DAC CPCTC

In the fourth line, why would writing ∆ABC = ∆ACD have been incorrect?

Refer to the kite data above and prove that angles ABC and ADC are the same size.

Answer 1

Hello, Lorena.

That is a very good question. In fact, it is very important to name congruent triangles in the order that their sides and angles are congruent.

By saying that ΔABC ≅ ΔADC you are saying that:

— —

AB ≅ AD

— —

BC ≅ DC

— —

AC ≅ AC

In this case, this is true. However if you name the triangle ΔACD and say it is congruent to ΔABC, then you are saying that

AB ≅ AC , etc..... This is not true. That is why it is important to name the triangles in order of congruent sides/angles.

I hope that helps.