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In order to prove that triangles are congruent, all the angles and sides have to be congruent. What if we aren't given any angles? We can use the SSS postulate (which has no A's—unlike your geometry tests). If all the sides are congruent, then the two triangles are congruent.
So are all the sides congruent? We're given that AB ≅ BC ≅ AD ≅ CD. Two of the sides match, but that isn't enough to prove congruence. What about the third side, AC? It's a side for both ∆ABC and ∆ADC, and it' congruent to itself, for sure. That means all three sides are congruent to one another, meaning ∆ABC ≅ ∆ADC.
Is it true that ∆HMR ≅ ∆ATP?
To prove that triangles are congruent, we can use either the SSS postulate or the SAS postulate (or we can find all the angles and sides, but why waste time?). We know that HM ≅ AT and MR ≅ TP.
Two sides are good, but not good enough. We need either another side or the included angles, and we have neither. The answer is that ∆HMR and ∆ATP might be congruent or not congruent, but we have no way of being sure unless we're given more information.
If X bisects VZ and WY, is it true that ∆VWX ≅ ∆ZYX?
A bisector splits something into two equal halves, so it's useful to know that X bisects VZ and WY. That means WX ≅ XY and VX ≅ XZ. Two sides of the triangles are congruent to one another, but that's not enough to prove that the triangles are congruent. Yet.
We'll move on to angles. Can we say anything about ∠VXW and ∠YXZ? They aren't just vertical-facing angles; they're actually vertical angles. Since vertical angles are congruent, we can say that ∠VXW ≅ ∠YXZ.
Two congruent sides and congruent included angles? That's a recipe just reeking with sass. Or maybe just SAS. In that case, ∆VWX ≅ ∆ZYX.