prove that two triangles are congruent if they have 2 equal adjacent angles and the side inbetween them is also the same.
A side and the two angles at either end completely define the point where the two other sides intersect. Thus all sides and angles are reproduced for both triangles and the triangles are congruent.
That is not the way they want you to wrte geometric proofs, but I hope you can fill in the gaps.
You might try this geometry link:
http://www.themathpage.com/abooki/propI-26.htm
ASA
To prove that two triangles are congruent, we need to demonstrate that all corresponding angles and sides are equal. In this case, you mentioned that two angles and the side between them are equal.
To begin, let's label the triangles. Suppose we have triangle ABC and triangle DEF, where angle A is congruent to angle D, angle B is congruent to angle E, and side AC is congruent to side DF.
To prove the congruence of the triangles, we will use the Angle-Side-Angle (ASA) congruence criterion. This states that if two angles and the side between them in one triangle are equal to corresponding angles and the corresponding side in another triangle, then the triangles are congruent.
1. Prove that angle C is congruent to angle F:
Since angle A is congruent to angle D, and angle B is congruent to angle E, it follows that angle C is congruent to angle F. This is because the sum of the angles in a triangle is always 180 degrees, and if two angles of one triangle are equal to two angles of another triangle, the third angles must also be equal.
2. Prove that side AB is congruent to side DE:
Given that side AC is congruent to side DF (as mentioned), we can add side BC to both sides of the respective triangle. This gives us side AB congruent to side DE, because when two sides of a triangle are equal to the corresponding sides of another triangle, the third side must also be equal.
By proving that both angle C is congruent to angle F and side AB is congruent to side DE, we have satisfied the ASA congruence criterion. Thus, we can conclude that triangle ABC is congruent to triangle DEF.