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Problem 26 Medium Difficulty

Make a conclusion based on the Hinge Theorem or its converse. (Hint: Draw a sketch.)
In $\triangle G H J$ and $\triangle K L M, \overline{G H} \cong \overline{K L},$ and $\overline{G I} \cong \overline{K M}$ $\angle G$ is a right angle, and $\angle K$ is an acute angle.


$H J>L M$


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Video Transcript

let me draw the information we've been given and let's see what we conclude. Conclude we're gonna start with Triangle G H J. And we know we also have triangle K l m. And we know that segment G h is congruent to segment K l will. In the names of the triangles, they are corresponding. We also know that segment GJ is congratulated to segment K M again their corresponding. And we're told that angle G is a right angle. Well, it may not look like it in my drawing, but if I put our little writing remarking, that means it's a right angle. An angle A is our angle K is acute. So first of all, if angle G is a right angle, an angle K is acute. We now that the measurement of angle G is greater than the measurement of Angle K, because an acute angle is less than 90 degrees. So we're actually given that information. Um, based on the description of angle G and a bouquet, the hinge theorem states. If you have two sides that are corresponding and congruent in your two triangles and their included angle, then we know something about the sides opposite of the angles. So once again, the side opposite of the larger angle is gonna be your longer side. The side opposite of your smaller angle's gonna be your shorter side. So if Angle G is greater than Angle K, we can conclude that the length of segment HJ must be greater than the length of segment lm.

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