prove that two triangles are congruent if they have 2 equal adjacent angles and the side inbetween them is also the same.

To prove that two triangles are congruent if they have two equal adjacent angles and the side in between them is also the same, we can use the ASA (Angle-Side-Angle) congruence criterion. ASA states that if two triangles have two pairs of corresponding angles equal, and the included side (the side between those angles) is also equal, then the triangles are congruent.

Here's how you can prove it step by step:

Step 1: Given that two triangles have ∠A = ∠X, ∠B = ∠Y, and segment AB = XY.

Step 2: To prove that the two triangles are congruent, we need to show that the remaining angles and sides are also equal.

Step 3: Since triangles have a total of 180 degrees, the third angle of triangle ABC (let's call it ∠C) can be found by subtracting the sum of ∠A and ∠B from 180 degrees: ∠C = 180° - ∠A - ∠B.

Step 4: Similarly, the third angle of triangle XYZ (let's call it ∠Z) can be found using the same formula: ∠Z = 180° - ∠X - ∠Y.

Step 5: Since ∠A = ∠X and ∠B = ∠Y (given), we can substitute the values obtained in step 3 and step 4: ∠C = ∠Z.

Step 6: Now, we can conclude that two corresponding angles of triangles ABC and XYZ are equal, namely ∠A = ∠X, ∠B = ∠Y, and ∠C = ∠Z.

Step 7: We also know that the included side, segment AB and XY, is equal: AB = XY (given).

Step 8: By using the ASA congruence criterion, we can state that triangles ABC and XYZ are congruent.

Therefore, we have proven that if two triangles have two equal adjacent angles and the side in between them is also the same, then the triangles are congruent using the ASA congruence criterion.