Answer:
AB≅AD; ΔADC≅ΔEBC; SAS only; VY; AB≅CE
Step-by-step explanation:
For HL, we must have the hypotenuse and a leg of each triangle congruent to the corresponding hypotenuse and leg of the other triangle.
We can see from the diagram that AC≅AC by the reflexive property. AC is a leg in each right triangle. This means we need the hypotenuse of each triangle congruent to the corresponding hypotenuse of the other triangle; this means we need AB≅AD.
ASA means "angle side angle." It states that if two angles and the included side of one triangle is congruent to the corresponding two angles and the included side of the other triangle, then the triangles are congruent.
We can see that BC≅CD from the diagram. This is the side; this means we need two angles.
We know that ACD≅ECB by the diagram. This is one angle and the side; we need one more angle.
Since m∠ADE = 90°, this means that m∠ADC must be 90° as well. Since m∠EBA = 90°, this means that m∠EBC must be 90° as well. This gives us the last corresponding pieces for the two triangles to be congruent using ASA.
Since ∠CBA≅∠DEA, and ∠BAC≅∠EAD, this means the third pair of angles, ∠ACB and ∠ADE, must be congruent as well. This is because all triangles have a sum of 180°.
This means we have two sides and the angle between them; this is SAS.
ΔUTW is similar to ΔVTX. This means the angle measures are the same for both triangles. We can also see that ∠VXW ≅ ∠UWT.
∠UWX and ∠UWT make a linear pair; this makes them supplementary. Since ∠VXW≅∠UWT, this means that ∠UWX and ∠UWT are supplementary as well.
This means that ∠UYX, which is opposite ∠UWX, will be congruent to it. This makes VYU≅VXT.
This gives us a pair of congruent angles at the bottom of ΔVYU, which makes it isosceles. This means that the sides adjacent to the base angles are congruent, which means that VU≅VY.
We are given that ΔABC≅ΔECD. This means corresponding pieces are congruent. From the similarity statement, we can see that AB corresponds to EC.
