To determine which theorem cannot prove triangle congruency, first recall what it means to be "congruent". Congruent means equal or in more mathematical terms, both corresponding angles and sides of the two triangles are congruent.

There are five theorems used in geometry to prove whether two triangles are congruent.

1. Side, Side, Side

2. Side, Angle, Side

3. Angle, Side, Angle

4. Angle, Angle, Side

5, Hypotenuse and One Leg from Right Triangle

Of the answers below, Angle, Angle, Angle (AAA) is not among one of the theorems to prove triangle congruency. The reason AAA does not prove triangle congruency because two triangles can have the same angles but have different side lengths thus, the triangles would not be congruent.

For this particular problem there are two geometric theorems that can prove that the triangles are similar.

The geometric theorems that could be used:

1. Angle, Side, Angle (ASA)

2. Angle, Angle, Side (AAS)

For triangle C the AAS is the most evident to use. Since ASA is not an option in the answer selections, AAS is the correct answer.

For this particular problem there are two geometric theorems that could potentially be used to prove that the triangles are similar.

The geometric theorems that could be used:

1. Hypotenuse and One Leg of a Right Triangle (HL)

2. Side, Side, Side (SSS)

To use SSS or HL, the Pythagorean theorem will need to be used to calculate the missing side.

For these triangles the HL is the most evident to use and since SSS is not an option in the answer selections, HL is the correct answer.

For this particular problem there are two geometric theorems that could potentially be used to prove that the triangles are similar.

The geometric theorems that could be used:

1. Hypotenuse and One Leg of a Right Triangle (HL)

2. Side, Side, Side (SSS)

To use SSS or HL, the Pythagorean theorem will need to be used to calculate the missing side.

For these triangles the HL is the most evident to use but is not an option in the answer selections, therefore, SSS is the correct answer.

For this particular problem there are two geometric theorems that can prove that the triangles are similar.

The geometric theorems that could be used:

1. Angle, Side, Angle (ASA)

2. Angle, Angle, Side (AAS)

For triangle C the AAS is the most evident to use. Since AAS is not an option in the answer selections, ASA is the correct answer.

Recall that a rigid motion preserves the distance from points within a shape in the plane. Since the statement clearly states that the two triangles are congruent, that means that the distances between the points within the shape are preserved. Also, since the corresponding points between the two triangles are separated by five spaces this is describing a translation which when combined with the preserved shape, describes a rigid motion. Therefore, the statement is true.

To determine whether this statement is true or false, first recall what it means to be "congruent". Congruent means equal or in more mathematical terms, both corresponding angles and sides of the two triangles are congruent.

The statement says that is an equilateral triangle which means all sides and angles are the same. After the triangle goes through a transformation, it becomes an isosceles triangle. To be isosceles means that two of the sides and angles of the triangle are the same. Since, not all of the angles are the same in it is not congruent to .

Therefore, the statement, " is congruent to ." is false.

The statement, "The __________ geometric theorem deals with proving congruency among right triangles; specifically when the length of one leg and the length of the hypotenuse are known." is describing the geometric theorem known as the Hypotenuse and One Leg theorem. When abbreviated this is seen as, HL.

Therefore, the missing term is, Hypotenuse and One Leg (HL)

Recall that a rigid motion preserves the distance from points within a shape in the plane. Therefore, if has had a rigid motion applied to it to result in that means that all corresponding sides and angles of these two triangle are congruent.

Therefore, the statement "both corresponding angles and sides are congruent" is the correct solution to this particular question.

The statement, "The __________ geometric theorem can be used to identify whether triangles that each have three known side lengths are congruent." is describing the geometric theorem known as the Side, Side, Side theorem. When abbreviated this is seen as, SSS.

Therefore, the missing term is, Side, Side, Side (SSS).