Which Shows Two Triangles That Are Congruent by AAS?
In geometry, two triangles are congruent if they have the same shape and size. This means that all three sides and all three angles of each triangle are equal in length and measure. There are several different congruence theorems that can be used to prove that two triangles are congruent. One of these theorems is the Angle-Angle-Side (AAS) congruence theorem.
The AAS congruence theorem states that if two angles and the included side of one triangle are congruent to the corresponding two angles and included side of another triangle, then the two triangles are congruent. In other words, if two triangles have the same two angles and the side that is between those two angles is also the same length, then the two triangles are congruent.
Which of the following shows two triangles that are congruent by AAS?
A. Two right triangles are connected at one side.
B. The triangles have 1 congruent side and 2 congruent angles.
C. The second triangle is a reflection of the first triangle.
D. Two triangles are connected at one side. The triangles have 2 congruent sides and 1 congruent angle.
E. Two triangles are connected at one side. All sides are congruent.
The correct answer is (D). The triangles in this option have two congruent sides (AB and AC) and one congruent angle (∠A). The side that is between these two angles is also the same length (BC). Therefore, the two triangles are congruent by AAS.
The other options are not correct because they do not meet the requirements of the AAS congruence theorem. Option (A) shows two right triangles that are connected at one side. However, the triangles do not have two congruent angles. Option (B) shows two triangles that have one congruent side and two congruent angles. However, the side that is between these two angles is not the same length. Option (C) shows two triangles that are reflections of each other. However, reflections are not congruent to their originals. Option (E) shows two triangles that are connected at one side and have all sides congruent. However, this does not meet the requirements of the AAS congruence theorem.
Here are some additional questions that can be used to assess students’ understanding of the AAS congruence theorem:
- Can two triangles be congruent if they have only one congruent angle?
- Can two triangles be congruent if they have only one congruent side?
- Can two triangles be congruent if they have two congruent angles and two congruent sides that are not adjacent?
These questions can be used to help students develop a deeper understanding of the AAS congruence theorem and its applications.