Question

nancee knight
triangle congruence theorems
sss (side-side-side)
ab≅de (s)
bc≅ef (s)
ac≅df (s)
sas (side-angle-side)
ab≅de
∠b≅∠e
bc≅ef
asa (angle-side-angle)

hl (hypotenuse-leg)

aas (angle-angle-side)

multiple-choice question
which of the following is the included sides shown in the asa image?
ab≅de
bc≅ef
ac≅df

Explanation:

Step1: Recall ASA (Angle - Side - Angle) criteria

ASA states that if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the triangles are congruent.

Step2: Analyze the ASA image

In the ASA (Angle - Side - Angle) image, the side is between the two angles. Looking at the given options:

• \(AB\cong DE\) is part of SSS (where all three sides \(AB\cong DE\), \(BC\cong EF\), \(AC\cong DF\) are used)
• \(BC\cong EF\) is part of SAS (where two sides and the included angle are used, e.g., if we have two sides \(AB\cong DE\), \(BC\cong EF\) and the included angle \(\angle B=\angle E\))
• \(AC\cong DF\) is not relevant for ASA. In ASA, we focus on the side that is between the two angles. The key for ASA is the side that is between the two angles. In the ASA congruence, the included side is the one that is common to the two angles.

Answer:

None of the options \(AB\cong DE\), \(BC\cong EF\), \(AC\cong DF\) are the included sides for ASA. If we assume there was a mis - labeling and we go by the general ASA concept (side between two angles), but based on the given options, there is an error. However, if we consider the standard ASA (where the side is between the two angles in the congruence statement for triangles), and if we assume the triangles \(\triangle ABC\) and \(\triangle DEF\) with angles \(\angle A=\angle D\), \(\angle B=\angle E\), the included side is \(AB\) (for \(\triangle ABC\)) and \(DE\) (for \(\triangle DEF\)) when we write \(\triangle ABC\cong\triangle DEF\) by ASA (\(\angle A=\angle D\), \(AB = DE\), \(\angle B=\angle E\)). But among the given options \(AB\cong DE\) is the closest (assuming the angles adjacent to these sides are the ones used in ASA). So, if we have to choose from the given options, the answer is \(AB\cong DE\)