Triangles LOA and LAM share side LA. Angles OLA and ALM are congruent. What additional information is needed to prove that the triangles are congruent using the AAS congruence theorem?
A) LO ≅ LM
B) OA ≅ MA
C) Angle LOA ≅ Angle LMA
D) Angle LAO ≅ Angle LAM
Answer (2)
To prove that triangles △ L O A and △ L A M are congruent using the Angle-Angle-Side (AAS) congruence theorem, we need a set of angles and a non-included side to be congruent between the triangles.
The given information tells us that ∠ O L A is congruent to ∠ A L M . Additionally, the triangles share the side L A , which means L A is congruent to itself by the Reflexive Property of congruence.
According to the AAS theorem, we need the following to prove triangle congruence:
Two angles in one triangle are congruent to two angles in another triangle.
A non-included side in one triangle is congruent to the corresponding side in the other triangle.
Currently, we have ∠ O L A ≡ ∠ A L M and L A ≡ L A .
To properly apply the AAS theorem, we need another pair of congruent angles. Analyzing the provided options:
A) L O ≡ L M : This option would not provide the needed angle congruence.
B) O A ≡ M A : This would not help since it provides another side congruence rather than an angle.
C) ∠ L O A ≡ ∠ L M A : This gives us a second angle congruence.
D) ∠ L A O ≡ ∠ L A M : This provides angle congruence too.
Since our aim is to prove using AAS, the important factor is achieving another angle congruence. Option C provides a direct solution for AAS as it gives the second required angle pair congruence.
Therefore, the correct answer is C) ∠ L O A ≡ ∠ L M A .
By having this second angle congruence ∠ L O A ≡ ∠ L M A , along with the common side L A ≡ L A and the given ∠ O L A ≡ ∠ A L M , we use the AAS theorem to establish △ L O A ≡ △ L A M .
To prove that triangles △ L O A and △ L A M are congruent using the AAS theorem, we need an additional angle congruence. The information needed is option C: ∠ L O A ≅ ∠ L M A . This provides the second angle congruence required along with the common side and the given angle congruence to establish triangle congruence.
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