1. d) JX = JY
2. b) mâ FEG = mâ DEF
1. For the question "If line JK is perpendicular to line XY at its midpoint M, which statement is true?", let's look at the available options and see why or why not they're correct.
a) JX = KY
* There no mention of point M also being a midpoint of JK. So we don't know if JM = KM. And because of that, even though XM = YM, we don't have a similar triangle to make any statement about the equality of JX and KY.
b) JX = KX
* Once again, we don't know of JM = KM, so we don't have a similar triangle and can't make any statements about the equality between JX and KX.
c) JM = KM
* Since the problem doesn't mention M being a midpoint of JK, we don't know about their being equal or not.
d) JX = JY
* Finally. We have 2 right triangles. XM = YM because M is the midpoint of XY, and JM = JM because it's equal to itself. So we have two congruent right triangles. Therefore JX = JY, making this the correct answer.
2. Let's see what the options are and why the option is correct or incorrect.
a) mâ DEF = mâ DEG
* Bad choice. This is asking for the angle made by the bisector to be equal to the angle being bisected. And that doesn't make sense.
b) mâ FEG = mâ DEF
* Good choice. We want the two angles created by the bisector to be equal to each other and that's what this condition is checking. So this is the correct answer.
c) mâ GED = mâ GEF
* This is just like option "a" above. Checking an angle created by the bisector against the entire angle. So it's also wrong.
d) mâ DEF = mâ EFG
* The angle â EFG has all of the correct vertexes, but in the wrong order. So the wrong angle is being mentioned, and this is a wrong answer.
