Answer:
Sure, I can help you with that. To prove that two triangles are congruent, you need to show that three pairs of corresponding sides or angles are equal. There are five ways to do this: SSS, SAS, ASA, AAS, and HL (for right triangles).
In your problem, you are given that angle R and angle V are right angles. That means that triangle RST and triangle VST are both right triangles. You can use the **HL (hypotenuse-leg) theorem** to prove that they are congruent. This theorem states that if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent.
To use this theorem, you need to show that the hypotenuses and one pair of legs are congruent. The hypotenuses are sides 1 and 4 in your diagram. The legs are sides 2 and 3. You can use the **parallel lines theorem** to show that these sides are congruent. This theorem states that if two parallel lines are cut by a transversal, then the corresponding angles are congruent.
In your diagram, sides AB and DC are parallel lines, and side AC is a transversal. That means that angle BAC is congruent to angle DCA, and angle ACB is congruent to angle DAC. These are corresponding angles in the triangles RST and VST. By the **congruent supplements theorem**, if two angles are supplementary to the same angle (or to congruent angles), then they are congruent. Angle R and angle V are both supplementary to angle BAC (or DCA), so they are congruent. Angle T is also supplementary to angle ACB (or DAC), so it is congruent to itself.
Now we have shown that angle R is congruent to angle V, and angle T is congruent to angle T. By the **side-angle-side (SAS) theorem**, if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. Therefore, side 1 is congruent to side 4, and side 2 is congruent to side 3.
We have all the information we need to use the HL theorem. Here is a possible table for your proof:
| Statements | Reasons |
| --- | --- |
| 1. AB || DC | Given |
| 2. Angle R and angle V are right angles | Given |
| 3. Angle BAC = Angle DCA | Parallel lines theorem |
| 4. Angle R = Angle V | Congruent supplements theorem |
| 5. Angle T = Angle T | Reflexive property of equality |
| 6. Side 1 = Side 4 | SAS theorem |
| 7. Side 2 = Side 3 | SAS theorem |
| 8. Triangle RST = Triangle VST | HL theorem |
Explanation: