To prove that triangles ∆ KML and ∆ UST are congruent using the ASA (Angle-Side-Angle) congruence theorem, we need the following information:
- Two angles must be congruent (e.g., Angle K ≅ Angle U and Angle M ≅ Angle S).
- The side between those two angles must be congruent (e.g., side KL must be congruent to side UT).
From the information provided, you mentioned congruencies for sides (KL ~ UT, MK ~ SU, ML ~ ST) and angles (Angle M ~ Angle S).
To use the ASA theorem specifically, we need:
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Two pairs of corresponding angles to be congruent:
- Let's say we already have one pair: Angle M ~ Angle S.
- We need information about another corresponding angle (for example: Angle K ~ Angle U).
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The side between the two congruent angles (which could be either KL, MK, or ML, depending on which angles you choose).
With this in mind, to complete the ASA condition, we would need:
- One more pair of angles: either Angle K ≅ Angle U or any other corresponding angle that is already known, along with either KL ≅ UT (the side between the angles) or a verification that it fits the structure of the triangles as per ASA.
So, ultimately:
You need to know one more angle (Angle K ≅ Angle U) and the side KL ~ UT to prove ∆ KML ≅ ∆ UST by ASA.