# congruent triangle question

There are two triangles ABC and DEF with corresponding sides abc and def. Given that angle D = angle E, and angle A = angle C, and side b = side e, determine if the triangles are congruent.

The answer is yes based on the theorem that two angles and a side of one triangle is equal to two angles and a side of another triangle.

My question is it only gives that two angles are equal in each triangle, it never says angle A = angle D. How then can the triangles be congruent?

• What does "corresponding sides" mean? Corresponding... to what ? For example, is $\;a\;$ the length of $\;AC,\,AB\,$ or of $\;BC\;$ ? I'd say the last one but I cannot be sure, and this is very important in these cases. Nov 21, 2016 at 11:20
• a is the side opposite of angle A, b is the side opposite of angle B ...etc. Nov 21, 2016 at 11:22
• Ok, now that's clearer...yet there is no theorem as the one you mention. What exists is the theorem: if two angles in one triangle are equal to two angles in other triangle, and if both sides between the equal angles are equal in both triangles, then the triangles are congruent. This is not what you have here, as according to what you wrote, $\;b\;$ is the length of side $\;AC\;$ in $\;ABC\;$ (between the equal angles...!) , whereas $\;e\;$ is the length of $\;DF\;$ , not between the equal angles... Nov 21, 2016 at 11:27

Let's take triangle with angles $A,C,B$ $30^\circ, 30^\circ, 120^\circ$ and side $b=1$, and another one with angles $D,E,F$ $45^\circ, 45^\circ, 90^\circ$ and side $e=1$. These fulfill the conditions given, but obviously are not congruent.