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.

 The Angle-Side-Angle Similarity Theorem states that if two triangles have two pairs of sides are of the same proportions and their included angles are congruent, then these two triangles are similar.  To be similar triangles can be different sizes, but all angles must be congruent.  If two triangles have a pair of congruent angles, then we know their opposite side of that angle must be in proportion to each other.  If we also have two pairs of sides that are of the same proportions then these triangles would be similar.  This theorem guarantees these conditions are met.