Question

2 intersecting lines are shown. A line with points T, R, W intersects a line with points S, R, V at point R. Angle T R V is 95 degrees.
mAngleSRW =
°Given: mAngleORP = 80°
mAngleORN = (3x + 10)°
Prove: x = 30

3 lines are shown. A line with points P, R, N intersects a line with points M, R, O at point O. A line extends from point R to point L between angle M R P. Angle N R O is (3 x + 10) degrees and angle P R O is 80 degrees.
Which statement could be used in step 2 when proving x = 30?

A 2-column table with 5 rows. Column 1 is labeled statements with the entries measure of angle O R P = 80 degrees semicolon measure of angle O R N = (3 x + 10) degrees, blank, blank, blank, blank. Column 2 is labeled reasons with the entries given, blank, blank, blank, blank.

AngleORP and AngleORN are a linear pair
AngleORP and AngleORN are vertical angles
80 = 3x +10
x = 303 lines are shown. A line with points P, R, N intersects a line with points O, R, M at point R. A line extends from point R to point L in between angle M R P.
Marcus states that angle ORP and angle LRP are a linear pair. Which best describes his statement?

He is correct. The angles share a common vertex so they are a linear pair.
He is correct. The angles share a common ray so they are a linear pair.
He is incorrect. Angle ORP does not form a linear pair with another angle in the diagram.
He is incorrect. Ray RO and ray RL are not opposite rays.3 lines are shown. A line with points A, F, D intersects with a line with points B, F, E at point F. Another line extends from point F to point C in between angle B F D.
Which are vertical angles?

AngleAFE and AngleBFD
AngleBFC and AngleDFE
AngleAFE and AngleCFD
AngleBFC and AngleEFAGiven: mAngleEDF = 120°; mAngleADB = (3x)°; mAngleBDC = (2x)°
Prove: x = 24

3 lines are shown. A line with points E, D, C intersects a line with points A, D, F at point D. A line extends from point D to point B in between angle A D C. Angle E D F is 120 degrees, angle C D B is (2 x) degrees, and angle B D A is (3 x) degrees.
What is the missing reason in step 3?

A 2-column table has 9 rows. Column 1 is labeled statements with entries measure of angle E D F = 120 degrees measure of angle A D B = (3 x) degrees measure of angle B D C = (2 x) degrees, angle E D F and angle A D C are vertical angles, angle E D F is-congruent-to angle A D C, measure of angle A D C = measure of angle A D B + measure of angle B D C, measure of angle E D F = measure of angle A D C, measure of angle E D F = measure of angle A D B + measure of angle B D C, 120 = 3 x + 2 x, 120 = 5 x, x = 24. Column 2 is labeled Reasons with entries given, def. of vert. angles, question mark, angle add. post., definition of congruency, substitution, substitution, addition, div, prop. of equality.

vertical angles are congruent
substitution
definition of congruency
definition of equality3 lines are shown. A line with points T, R, W intersects with a line with points S, R, V at point R. Another line extends from point R to point U in between angle V R W.
Which pair of angles are vertical angles?

AngleWRU and AngleSRT
AngleWRS and AngleVRT
AngleVRU and AngleTRS
AngleVRT and AngleSRT2 intersecting lines are shown. A line with points M, H, K intersects a line with points L, H, J at point H. 4 angles are created. Clockwise, from the top, the angles are (5 m + 100) degrees, blank, blank, (2 m + 10) degrees.
What is the value of m?

10
30
70
1502 intersecting lines are shown. A line with points T, R, W intersects a line with points S, R, V at point R. 4 angles are created. Clockwise, from the top, the angles are blank, (3 x) degrees, blank, (x + 40) degrees.
What is the value of x?

20
35
60
702 lines intersect. A line with points A, B, D intersects a line with points E, B, C at point B, forming 4 angles.
In the diagram, which angles are vertical angles? Select two options.

AngleABE and AngleABC
AngleABE and AngleCBD
AngleABE and AngleEBD
AngleABC and AngleEBD
AngleABC and AngleCBDGiven: mAngleADE = 60° and mAngleCDF = (3x + 15)°
Prove: x = 15

Answer 1

I'll answer each diagram/question in order with a short justification.

1) 2 intersecting lines: TRW and SRV; m∠TRV = 95°. Find m∠SRW.
- m∠SRW = 95° (∠TRV and ∠SRW are vertical angles, so they are equal).

2) Given m∠ORP = 80° and m∠ORN = (3x + 10)°. Prove x = 30. Which statement could be used in step 2?
- "∠ORP and ∠ORN are a linear pair." (That gives m∠ORP + m∠ORN = 180°, then 80 + (3x+10) = 180 → 3x + 90 = 180 → 3x = 90 → x = 30.)

3) Marcus: "∠ORP and ∠LRP are a linear pair." Is he correct?
- He is incorrect. Ray RO and ray RL are not opposite rays, so those two angles are not a linear pair.

4) Lines A–F–D and B–F–E with ray FC between BF and DF. Which are vertical angles?
- ∠AFE and ∠BFD are vertical angles.

5) Proof with m∠EDF = 120°, m∠ADB = 3x, m∠BDC = 2x; missing reason in step 3?
- The missing reason is "vertical angles are congruent." (So from ∠EDF and ∠ADC being vertical, conclude ∠EDF ≅ ∠ADC.)

6) Which pair are vertical angles? (Lines TRW and SRV, with RU between VR and WR.)
- ∠WRS and ∠VRT are vertical angles.

7) Intersecting lines at H: clockwise from top angles are (5m + 100), ?, ?, (2m + 10). Find m.
- m = 10. (Adjacent angles are supplementary: (5m+100)+(2m+10)=180 → 7m+110=180 → 7m=70 → m=10.)

8) Intersecting lines at R: clockwise top ?, right 3x, bottom ?, left (x+40). Find x.
- x = 20. (Right and left are vertical → 3x = x + 40 → 2x = 40 → x = 20.)

9) Two lines A–B–D and E–B–C intersect at B. Which angles are vertical? (Choose two.)
- ∠ABE and ∠CBD, and ∠ABC and ∠EBD.

10) Given m∠ADE = 60° and m∠CDF = (3x + 15)°. Prove x = 15.
- If ∠ADE and ∠CDF are vertical (or otherwise congruent), then 3x + 15 = 60 → 3x = 45 → x = 15.