Question

Point E is the midpoint of Line segment A B and point F is the midpoint of Line segment C D .

3 intersecting lines are shown. A line has points C, E, F, D. Another line has points A, E, and B and intersects the other line at point E. Another line goes from point A to point F.
Which statements about the figure must be true? Select three options.

Line segment A B is bisected by Line segment C D .
Line segment C D is bisected by Line segment A B .
AE = One-halfAB
EF = One-halfED
CE + EF = FDWhat is the midpoint of Line segment F B?

A number line going from negative 9 to positive 9. A closed circle appears at negative 6 and is labeled A. A closed circle appears at negative 2 and is labeled F. A closed circle appears at positive 1 and is labeled G. A closed circle appears at positive 2 and is labeled H. A closed circle appears at positive 3 and is labeled L. A closed circle appears at positive 8 and is labeled B.

point A
point G
point H
point LWhat is the midpoint of Line segment A B ?

A number line going from negative 9 to positive 9. A closed circle appears at negative 6 and is labeled A. A closed circle appears at negative 2 and is labeled F. A closed circle appears at positive 1 and is labeled G. A closed circle appears at positive 2 and is labeled H. A closed circle appears at positive 3 and is labeled L. A closed circle appears at positive 8 and is labeled B.

point F
point G
point H
point IIf JM = 5x – 8 and LM = 2x – 6, which expression represents JL?

A line with the following points labeled left to right: J, K, L, M.

3x – 2
3x – 14
7x – 2
7x – 14Using the segment addition postulate, which is true?

A number line going from negative 9 to positive 9. A closed circle appears at negative 6 and is labeled A. A closed circle appears at negative 1 and is labeled B. A closed circle appears at positive 2 and is labeled C. A closed circle appears at positive 8 and is labeled D.

AB + BC = AD
AB + BC = CD
BC + CD = AD
BC + CD = BDRaj correctly determined that ray LH is the bisector of AngleGLI.

A line contains points K, L, M. 4 lines extend from point L. One line extends to point F, another to G, another to H, and another to I.
Which information could he have used to determine this?

AngleGLH Is-congruent-to AngleILM
mAngleKLM = 5mAngleILM
mAngleGLI = 2mAngleGLH
mAngleGLI = mAngleGLH + mAngleHLIPoint M lies between points L and N on Line segment L N .

The space between L and M is 10 x + 8. The space between M and N is 5 x minus 4.
If LN = 12x + 16, what is the length of Line segment L N in units?

16 units
40 units
48 units
64 units
Given that mAngleKLH = 120° and mAngleKLM = 180°, which statement about the figure must be true?

AngleHLM is bisected by Ray L J .
AngleGLJ is bisected by Ray L H .
mAngleKLG = mAngleHLJ
mAngleHLI = mAngleILMPoint V lies between points U and W on Line segment U W.

A line has points U, V, W. The length between U V is 2 x minus 4. The length between V W is 4 x + 10.
If UW = 9x – 9, what is UW in units?

5 units
6 units
30 units
36 unitsRay UW is the angle bisector of AngleVUT.

Three lines extend from point U. They are lines U V, U W, and U T.
If mAngleVUW = (4x + 6)° and mAngleWUT = (6x – 10)°, what is the measure of AngleWUT?

32°
38°
48°
76°

Answer 1

I'll answer each part in order with short justifications.

1) Which three statements must be true (E midpoint of AB; F midpoint of CD; points C, E, F, D are collinear)?
- True: Line segment AB is bisected by line segment CD. (E is midpoint of AB and lies on CD.)
- True: AE = 1/2 AB. (Definition of midpoint.)
- True: CE + EF = FD. (CE + EF = CF, and CF = FD because F is midpoint of CD.)
- The other two statements are not necessarily true.

2) Midpoint of segment FB (points on number line: A = -6, F = -2, G = 1, H = 2, L = 3, B = 8)
- Midpoint = (−2 + 8)/2 = 3 → point L.

3) Midpoint of segment AB (A = -6, B = 8)
- Midpoint = (−6 + 8)/2 = 1 → point G.

4) If JM = 5x – 8 and LM = 2x – 6, find JL (points left to right J, K, L, M)
- JL = JM − LM = (5x − 8) − (2x − 6) = 3x − 2.

5) Using the segment addition postulate, which is true? (A = −6, B = −1, C = 2, D = 8)
- BC + CD = BD (3 + 6 = 9).

6) Raj determined ray LH bisects ∠GLI. Which info could show that?
- m∠GLI = 2·m∠GLH. (If the whole angle equals twice one of the parts, that part is half the whole → LH bisects ∠GLI.)

7) Point M between L and N: LM = 10x + 8, MN = 5x − 4, LN = 12x + 16. Find LN.
- Solve (10x+8)+(5x−4)=12x+16 → 15x+4 = 12x+16 → x = 4. Then LN = 12(4)+16 = 64 units.

8) Given m∠KLH = 120° and m∠KLM = 180°, which statement must be true?
- m∠KLG = m∠HLJ. (This is the statement that follows from the given configuration.)

9) Point V between U and W: UV = 2x − 4, VW = 4x + 10, UW = 9x − 9. Find UW.
- (2x−4)+(4x+10)=9x−9 → 6x+6 = 9x−9 → x = 5. UW = 9(5)−9 = 36 units.

10) Ray UW bisects ∠VUT. If m∠VUW = (4x+6)° and m∠WUT = (6x−10)°, find m∠WUT.
- Set 4x+6 = 6x−10 → x = 8. Then m∠WUT = 6(8) − 10 = 38°.

If you want, I can show algebra steps or re-evaluate the ambiguous angle-question (8) with the exact diagram labels to give a clearer justification.