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HL - Problem 2
The hypotenuse-leg (HL) shortcut to determine if two triangles are congruent can be used when both triangles have right angles. The hypotenuse of a right triangle is the side of the triangle opposite the right angle. If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and leg of another right triangle, the two triangles are congruent. If the triangles are adjacent, it is already clear that they have at least one congruent side (because the side that they share is congruent to itself). Remember to name the triangles with the letters in the corresponding positions.
Geometry teachers love to give these problems where we have two triangles that share a common side. So the question is can we say this triangle BCA, which is our lower triangle, can we say that’s congruent to our upper triangle ADC? Well, we have a right angle which means we can possibly use the hypotenuse-leg short-cut.
I have two legs that are congruent and we share the hypotenuse. So I’m going to mark that with two markings because it’s the same segment so it must be congruent to itself. So yes we can say that these two are congruent by hypotenuse-leg.
So let’s figure out what corresponds angle B. Angle B has a right angle which means D must be the corresponding angle in our other triangle. Our second is angle C. Well if I look at line segment BC, we don’t have any congruence markings, so that corresponds to this line segment DA. We’ve already had D written so that means that A must correspond to C. Last what corresponds with angle A? Angle A is part of this line segment AB with one congruence marking, corresponding side is DC which means C must correspond to angle A.
Again what was the trick here? We noticed that we had a shared side and a right angle so we could use hypotenuse-leg short-cut.