Problem 2 Instructions: State what additional information is required in order to know that the triangles are congruent for the given reason. Given: AAS
The Correct Answer and Explanation is:
Correct Answer:
QP ≅ HG
Explanation
The problem requires us to identify the additional piece of information needed to prove that the two triangles, ΔQRP and ΔHIG, are congruent using the Angle-Angle-Side (AAS) Congruence Postulate.
The AAS postulate states that if two angles and a non-included side of one triangle are congruent to the corresponding two angles and non-included side of another triangle, then the two triangles are congruent. A non-included side is any side that is not directly between the two angles.
First, let’s analyze the information already provided in the diagram. The angle markings indicate the following congruencies:
- Angle (A): The single arc in ∠R and ∠I shows that ∠R ≅ ∠I.
- Angle (A): The double arcs in ∠Q and ∠H show that ∠Q ≅ ∠H.
We now have the two “Angle” components required for the AAS postulate. The next step is to find the “Side” component. According to the AAS rule, this must be a non-included side.
In ΔQRP, the given angles are ∠Q and ∠R. The side located between these two angles is side QR. This is the included side. Therefore, the non-included sides are the other two sides: QP and RP.
In the second triangle, ΔHIG, the given angles are ∠H and ∠I. The side located between them is HI, which is the included side. The non-included sides are HG and IG.
To use AAS, we must show that a pair of corresponding non-included sides are congruent. Based on the congruent angles, we can determine the corresponding vertices: Q corresponds to H, and R corresponds to I. This implies that the remaining vertices, P and G, also correspond.
Therefore, the corresponding non-included sides are:
- Side QP corresponds to side HG.
- Side RP corresponds to side IG.
We need one of these pairs to be congruent. Thus, the additional information required could be either QP ≅ HG or RP ≅ IG. Selecting the first pair, we find that if we are given QP ≅ HG, we can prove the triangles are congruent by AAS.
