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What missing piece of information would make △ABC≅△RPQ by AAS? (1 point) Responses BC¯¯¯¯¯¯¯¯≅PQ¯¯¯¯¯¯¯¯ line segment cap b cap c is congruent to line segment cap p cap q ∠B≅∠C ∠B≅∠C AC¯¯¯¯¯¯¯¯≅RQ¯¯¯¯¯¯¯¯ line segment cap A cap c is congruent to line segment cap r cap q ∠B≅∠P ∠B≅∠P Skip to navigation

3 answers

To use the Angle-Angle-Side (AAS) congruence postulate, we need two angles and the non-included side of one triangle to be congruent to the corresponding parts of another triangle.

Since we already have triangle \(ABC\) and triangle \(RPQ\) and you are looking for the missing piece of information, we can infer that if you already have:

  • One angle from triangle \(ABC\) congruent to one angle from triangle \(RPQ\) (let's assume that's ∠B ≅ ∠P).
  • Another angle from triangle \(ABC\) (∠C) that is congruent to another angle from triangle \(RPQ\) (let's say this angle is yet to be known).

Then, to satisfy the criteria for AAS congruence, we need:

  • The side \(AC\) to be congruent to side \(RQ\).

Hence, the missing piece of information would be:

AC ≅ RQ.

Thus, if you already have ∠B ≅ ∠P and ∠C ≅ ∠Q, and knowing that \(AC\) is congruent to \(RQ\) would establish that \(△ABC ≅ △RPQ\) by AAS.

NOTE: On the bottom right corner of each question, you will see a button that says “Check Answer.” Click this to check your answer before submitting. It will not tell you the correct answer, but it will tell you whether the answer you selected is correct or not. You can use this feature ONCE per question (only for Quizzes and Sample Works). If ∠O≅∠P and ∠R≅∠R by the Reflexive Property of Congruence, what additional information is needed to prove △NOR≅△QPR by the AAS Congruence Theorem? (1 point) Responses NR¯¯¯¯¯¯¯¯≅QP¯¯¯¯¯¯¯¯ line segment cap n cap r is congruent to line segment cap q cap p OR¯¯¯¯¯¯¯¯≅PR¯¯¯¯¯¯¯¯ line segment cap o cap r is congruent to line segment cap p cap r NO¯¯¯¯¯¯¯¯≅PR¯¯¯¯¯¯¯¯ line segment cap n cap o is congruent to line segment cap p cap r NO¯¯¯¯¯¯¯¯≅QP¯¯¯¯¯¯¯¯ line segment cap n cap o is congruent to line segment cap q cap p Skip to navigation

To use the Angle-Angle-Side (AAS) congruence theorem to prove that \( \triangle NOR \cong \triangle QPR \), you already have:

  • ∠O ≅ ∠P
  • ∠R ≅ ∠R (which is by the Reflexive Property of Congruence and is essentially congruence to itself).

To satisfy the AAS condition, you need an additional piece of information that includes one side from one triangle that corresponds to one of the angles in the other triangle.

Given that you need to incorporate the non-included side (not between the two angles you already have), the options can be evaluated:

  1. NR ≅ QP: This corresponds the side opposite the angle ∠O and should match the position in triangle \( QPR \) corresponding to ∠P.

  2. OR ≅ PR: This corresponds to the side opposite ∠R, but does not add the necessary side congruence to connect the other angle.

  3. NO ≅ PR: This doesn't make sense as it contradicts allowing for angle congruence because NO is opposite of another angle.

  4. NO ≅ QP: This also doesn't directly help as it adds an irrelevant congruence which does not correspond to the angles.

Given this understanding, the most appropriate choice that would provide the necessary additional information is:

NR ≅ QP.

This would allow you to conclude that \( \triangle NOR \cong \triangle QPR \) by AAS since you would then have two angles and the non-included side congruent.