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Problem 8 Medium Difficulty

Write a two-column proof. Given: $\overline{F H}$ is a median of $\triangle D F G$
$$\mathrm{m} \angle D H F>\mathrm{m} \angle G H F$$Prove: $D F>G F$
(FIGURE CANNOT COPY)

Answer

$D F>G F$

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Video Transcript

at a two column proof that proves segment DF is greater than segment GF. We're starting with Triangle D F G, and we've been given an interior segment f h and we're told that that segment is a median. So we're going to start by re stating that segment FH excuse me is the median of triangle D F g. And this is given a lot of students make mistake and immediately right all of the givens. But what does segment F h being a median? Tell us. Well, it tells us that Point H is the midpoint of segment D. G. And this is based on the definition of median. A median starts at the midpoint of a segment and ends at the Vertex opposite of that side of the triangle opposite of that segment. And now that we know H is a midpoint, then we know that segment D H and segment G h are congruent to each other so we can say segment D. H is congruent to segment GH and this is definition of midpoint. We also know that segment H f is congrats to itself by reflexive property of congruence. So by hinge here, um We now have two sides that air corresponding and groups of two triangles. Next, I want to go to our other. Given we are told that the measurement of angle D h f is not congrats it but greater than the measurement of angle G h f. And if you look at where those angles are, D H f is on the left side of our median G h f is on the right side of the median and they're not congruity It so hinged there, then tells us if these air the included angles of correspondent congrats sides, then we can also compare the sides opposite of the angles. So if the measurement of D H F is the larger measurement, then we know that segment d f has to be the larger segment and the smaller angle measurement of angle G H f is opposite of the smaller side of the two sides. So that gives us d. F is greater than GF and that was what we were supposed to prove. And this was by hinge here. Um and I apologize. I didn't write the reason here. This was given

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