Question

question 2
bcpss_geometry_module5
which additional statements are required to prove △abc ≅ △ikm using the sas postulate, if ∠i ≅ ∠a? select all that apply
a) ∠b ≅ ∠k
b) ∠c ≅ ∠m
c) (overline{ab}) ≅ (overline{ik})
d) (overline{bc}) ≅ (overline{km})
e) (overline{ac}) ≅ (overline{im})

Explanation:

Step1: Recall the SAS (Side - Angle - Side) postulate

The SAS postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Step2: Analyze each option

• Option A ($\angle B\cong\angle K$): This is an angle - angle - side (AAS) or angle - side - angle (ASA) related condition, not SAS.
• Option B ($\angle C\cong\angle M$): This is also an angle - related condition not relevant to SAS.
• Option C ($\overline{AB}\cong\overline{LK}$): If we consider the given $\angle I\cong\angle A$ (assuming it's the included angle), and if $\overline{AB}\cong\overline{LK}$ and we can find the other side (say $\overline{AC}\cong\overline{LM}$), but by itself, if we assume the included angle is $\angle A$ (or $\angle I$) between $\overline{AB}$ and $\overline{AC}$ (and corresponding sides in the other triangle), $\overline{AB}\cong\overline{LK}$ is a side.
• Option D ($\overline{BC}\cong\overline{KM}$): This side is not adjacent to the given congruent angle ($\angle A\cong\angle I$) in the context of SAS.
• Option E ($\overline{AC}\cong\overline{LM}$): If we have $\angle A\cong\angle I$ (included angle), $\overline{AB}\cong\overline{LK}$ (one side) and $\overline{AC}\cong\overline{LM}$ (the other side adjacent to the included angle), then by SAS $\triangle ABC\cong\triangle LKM$.

Answer:

C. $\overline{AB}\cong\overline{LK}$, E. $\overline{AC}\cong\overline{LM}$