SOLUTION: Unit 3 Parallel & Perpendicular Lines & Transversals ... - Free Printable

Based on the provided image, which appears to be a completed homework assignment on parallel and perpendicular lines, here is a detailed explanation of the solutions for each problem.

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Problem 1: Cube Diagram Analysis

This problem uses a diagram of a cube with vertices labeled V, W, X, Y, Z, S, T, U. We need to identify parallel segments, parallel planes, and skew segments based on the properties of a cube.

a) Name all segments parallel to $\overline{XT}$.
- In a cube, opposite edges are parallel.
- $\overline{XT}$ is an edge on the top face (assuming standard orientation).
- Segments parallel to $\overline{XT}$ are those that run in the same direction and never intersect.
- These are: $\overline{WS}$ (bottom front), $\overline{ZY}$ (top back), and $\overline{VU}$ (bottom back).
- ✔ Correct Answer: $\overline{WS}, \overline{ZY}, \overline{VU}$

b) Name all segments parallel to $\overline{ZY}$.
- $\overline{ZY}$ is an edge on the top face, running from Z to Y.
- Parallel segments would be those running in the same direction.
- These are: $\overline{WX}$ (top front), $\overline{VU}$ (bottom back), and $\overline{ST}$ (bottom front).
- ✔ Correct Answer: $\overline{WX}, \overline{VU}, \overline{ST}$

c) Name all segments parallel to $\overline{VS}$.
- $\overline{VS}$ is a vertical edge on the left side of the cube.
- Parallel segments are other vertical edges: $\overline{ZW}$ (left back), $\overline{YX}$ (right back), and $\overline{UT}$ (right front).
- The student wrote $\overline{ZW}, \overline{YX}$ — this is correct but incomplete. $\overline{UT}$ should also be included.
- ⚠️ Partially Correct. Should include: $\overline{ZW}, \overline{YX}, \overline{UT}$

d) Name a plane parallel to plane STU.
- Plane STU is the bottom face of the cube (points S, T, U).
- The plane parallel to it is the top face: plane WXY or plane ZYX.
- The student wrote "ZWX" — assuming they meant plane ZWX, which is the same as plane WZX or WXY, this is acceptable if we interpret ZWX as the top face.
- ✔ Acceptable Answer: Plane ZWX (or WXY)

e) Name a plane parallel to plane UVZ.
- Plane UVZ is the left face of the cube (points U, V, Z).
- The plane parallel to it is the right face: plane TSW or plane STX.
- The student wrote "STX" — this correctly identifies the right face.
- ✔ Correct Answer: Plane STX

f) Name all segments skew to $\overline{SW}$.
- Skew lines are lines that are neither parallel nor intersecting (they lie in different planes).
- $\overline{SW}$ is a diagonal on the front face (from S to W).
- Segments skew to $\overline{SW}$ must not be parallel to it and not intersect it.
- Possible candidates: $\overline{YX}$ (top right edge), $\overline{UT}$ (bottom right edge), $\overline{VZ}$ (left vertical edge).
- The student wrote: $\overline{YX}, \overline{UT}, \overline{VZ}$ — these are all correct.
- ✔ Correct Answer: $\overline{YX}, \overline{UT}, \overline{VZ}$

g) Name all segments skew to $\overline{UT}$.
- $\overline{UT}$ is a vertical edge on the right front.
- Segments skew to it: any segment not parallel and not intersecting it.
- Examples: $\overline{XT}$ (top front — intersects at T), so not skew.
- Better choices: $\overline{VZ}$ (left vertical — parallel? No, same direction? Yes — wait, $\overline{UT}$ and $\overline{VZ}$ are both vertical, so they are parallel, not skew.
- Actually, $\overline{XT}$ and $\overline{WS}$ are horizontal, not parallel to vertical $\overline{UT}$, and don’t intersect it → they are skew.
- $\overline{ZY}$ is top back — doesn’t intersect $\overline{UT}$, not parallel → skew.
- Student wrote: $\overline{XT}, \overline{WS}, \overline{ZY}$ — these are all correct.
- ✔ Correct Answer: $\overline{XT}, \overline{WS}, \overline{ZY}$

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Problem 2: Relationship Between Lines and Planes

This problem asks to classify relationships as parallel, intersecting, or skew.

a) $\overline{AB}$ and $\overline{BC}$
- These two segments share point B → they intersect.
- But the student wrote “skew” — this is incorrect.
- ✘ Incorrect. Should be: Intersecting

b) $\overline{AE}$ and $\overline{BF}$
- In the diagram, AE and BF appear to be opposite edges of a rectangular prism or parallelepiped.
- They do not intersect and are in parallel planes → they are parallel.
- ✔ Correct Answer: Parallel

c) $\overline{EF}$ and $\overline{AD}$
- EF is on the top face, AD is on the bottom face.
- They are not parallel (different directions) and do not intersect (in different planes) → they are skew.
- ✔ Correct Answer: Skew

d) Plane ABC and plane ABF
- Both planes share line AB → they intersect along line AB.
- The student’s answer is cut off, but likely intended “intersecting”.
- ✔ Correct Answer: Intersecting

e) Plane AED and plane BFC
- These are opposite faces of the solid (like left and right faces).
- They do not intersect and are parallel → they are parallel planes.
- ✔ Correct Answer: Parallel

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Angle Classification Section

The student has classified the angle pairs correctly:

1. Alternate Interior — angles between parallel lines, on opposite sides of the transversal.
2. Corresponding Exterior — angles outside the parallel lines, in matching positions.
3. Corresponding — same relative position at each intersection.
4. Consecutive Exterior — exterior angles on the same side of the transversal.
5. Corresponding — again, matching positions.
6. Alternate Exterior — exterior angles on opposite sides of the transversal.

✔ All classifications are correct.

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## ✔ Final Summary of Corrections:

| Problem | Student's Answer | Correct Answer | Status |
|--------|------------------|----------------|--------|
| 1a | WS, ZY, VU | WS, ZY, VU | ✔ Correct |
| 1b | WX, VU, ST | WX, VU, ST | ✔ Correct |
| 1c | ZW, YX | ZW, YX, UT | ⚠️ Partial (missing UT) |
| 1d | ZWX | ZWX (or WXY) | ✔ Acceptable |
| 1e | STX | STX | ✔ Correct |
| 1f | YX, UT, VZ | YX, UT, VZ | ✔ Correct |
| 1g | XT, WS, ZY | XT, WS, ZY | ✔ Correct |
| 2a | Skew | Intersecting | ✘ Incorrect |
| 2b | Parallel | Parallel | ✔ Correct |
| 2c | Skew | Skew | ✔ Correct |
| 2d | (cut off) | Intersecting | (Assumed correct) |
| 2e | (cut off) | Parallel | (Assumed correct) |
| Angles | All labeled | All correct | ✔ Correct |

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## 📝 Overall Grade: B+ / A- (Mostly correct, with minor errors in 1c and 2a)

Great job overall! Just remember:
- In a cube, every edge has three parallel edges (not two).
- Two segments sharing a common endpoint are intersecting, not skew.

Let me know if you’d like a visual diagram or further clarification!