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Answer: Choice D) SAS</h3>
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Explanation:
LA stands for Leg Angle. More specifically, the angle is an acute angle. This theorem only works for right triangles. We see that ED = LK which is one pair of congruent leg segments. So that takes care of the L part of LA. However, we don't know anything about the acute angles. We don't know if angle E = angle L, or if angle F = angle M, or something along those lines. We need angle markers to tell us or not.
In short, we only have half the info needed for LA, so we must cross choice A off the list.
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HA stands for hypotenuse angle, or hypotenuse acute angle, and this rule only applies for right triangles. We don't know anything about the hypotenuse lengths if they are the same or not.
Similar to choice A, we don't know anything about the acute angles either.
Those two facts mean we must cross off choice B.
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AAS stands for Angle Angle Side. The order is important because the side is not between the angles. The diagram only shows one pair of angles that are congruent to one another. That pair being the 90 degree angles indicated with the square marker. We don't have another pair of angles, so we cannot use AAS.
Cross choice C off the list.
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We can use SAS because we have two pairs of congruent sides and one pair of congruent angles.
The congruent sides are
- ED = LK (double tickmarks)
- DF = KM (single tickmarks)
That takes care of the two "S"s of "SAS". The "A" would be those two right angles which are congruent to one another. The angles are between the congruent sides which is important. Recall that SSA is not a valid congruence theorem.
So that's why choice D is the only answer.
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Side note: We could use the LL theorem, where L stands for leg, since we are given two pairs of congruent legs for each triangle. Like LA and HA, this only applies to right triangles. LL is a special case of SAS. However, LL isn't listed so we'll just be sticking with SAS as the only answer.
