For two triangles to be congruent they must have equal corresponding angles and sides. There are five geometric theorems that can be used to prove whether triangles are congruent or not. Since, these two triangles have two defined angles and the side between the angles are defined as well, the Angle, Side, Angle geometric theorem can be used.

Looking at the triangles the corresponding angles are not equal therefore, the triangles are not congruent.

For two triangles to be congruent, they must meet one of the five geometric theorems that prove triangles congruency. We can see that each triangle has a corresponding right angle, and each triangle has a corresponding hypotenuse of length 5. However, having one matching angle and one matching side is not enough to prove congruency.

Apply the Pythagorean Theorem, a²+b²=c², to the first triangle, to yield A²+4²=5², which simplifies to A²+16=25, which simplifies to A²=9. Solve for A by taking the square root of each side: . Therefore A = 3.

Because this is a right triangle, we can use the Hypotenuse Leg Theorem to prove congruency, since in the leftmost triangle, hypotenuse = 5 and a leg = 3, and in the rightmost triangle, the hypotenuse = 5 and a leg = 3.

After using the Pythagorean Theorem to find the missing side of the leftmost triangle, these triangles are congruent based on the Hypotenuse Leg Theorem.

From the figure, we see that there are two congruent pairs of corresponding sides, , and one congruent pair of corresponding angles, . The Side-Angle-Side Theorem (SAS) states that if two sides and the angle between those two sides of a triangle are equal to two sides and the angle between those sides of another triangle, then these two triangles are congruent.

These two triangles share three corresponding congruent sides.  The Side-Side-Side Theorem (SSS) states that if the three sides of one triangle are congruent to their corresponding sides of another triangle, then these two triangles are congruent.

The Angle-Side-Angle Theorem (ASA) states that if two angles and their included side are congruent to two angles and their included side to another triangle, then these two triangles are congruent.  Our first option cannot be correct because this figure does not give any information about the angles.  This could be proven using the SSS Theorem.  The second figure only gives information about one angle, this could be proven using the SAS Theorem.  The third option again only gives information about one angle.  The fourth figure has two angles congruent, and their included side congruent.  Clearly this is the only figure that could have congruent triangles proven through the ASA Theorem.

When we look at this figure we see that we have two pairs of congruent corresponding angles, . The Angle-Angle Theorem (AA) states that if two angles of one triangle are congruent to two angles of another triangle, then these triangles are similar.

The Side-Side-Side Similarity Theorem states that if all three sides of one triangle are proportional to another, then these triangles are similar. and  are similar by this theorem because each of their sides are proportional by a factor of 4.

Just because a triangle has two sides and one angle congruent to the two sides and angle of another triangle does not guarantee these two triangles’ congruence.  For the two triangles to be congruent, the two sides that are congruent must contain the congruent angle as well.  Below are two triangles that share congruent sides and one angle, but are not congruent.