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

Let's go through the homework step by step and verify or correct the answers based on geometric principles. The assignment is about parallel and perpendicular lines, transversals, and spatial relationships in 3D figures.

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Problem 1: Use the diagram below to answer the following questions.

We are given a 3D figure that looks like a rectangular prism (a box) with labeled vertices:
Vertices: $ S, T, U, V, W, X, Y, Z $

Let’s analyze each part carefully.

#### a) Name all segments parallel to $ \overline{XT} $.

- $ \overline{XT} $ goes from top-right front to bottom-right back.
- In a rectangular prism, opposite edges are parallel.
- $ \overline{XT} $ is vertical on the right face.
- Other vertical edges: $ \overline{WS}, \overline{ZV}, \overline{YU} $
- So, parallel segments: $ \overline{WS}, \overline{ZV}, \overline{YU} $

✔ Correct answer: $ \overline{WS}, \overline{ZV}, \overline{YU} $

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#### b) Name all segments parallel to $ \overline{ZY} $.

- $ \overline{ZY} $ is the top edge from left back to right back (top face).
- It runs horizontally along the top back edge.
- Parallel segments:
- $ \overline{WX} $: top front edge
- $ \overline{VU} $: bottom back edge
- $ \overline{ST} $: bottom front edge

Wait — let’s double-check:

Actually, $ \overline{ZY} $ is top-back horizontal edge. Then:

- $ \overline{WX} $: top-front → same direction → yes
- $ \overline{VU} $: bottom-back → same direction → yes
- $ \overline{ST} $: bottom-front → yes

But wait: Are they all parallel?

Yes — all these are horizontal edges going left-to-right (or right-to-left), so they're parallel.

So: $ \overline{WX}, \overline{VU}, \overline{ST} $

✔ Correct answer: $ \overline{WX}, \overline{VU}, \overline{ST} $

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#### c) Name all segments parallel to $ \overline{VS} $.

- $ \overline{VS} $: from bottom-left back ($V$) to bottom-left front ($S$)
- This is a horizontal edge on the left side, bottom face.
- Parallel segments should be other horizontal edges on the left or right faces.

Wait: $ \overline{VS} $ is bottom-left, going from back to front.

Looking at the figure:

- $ \overline{ZW} $: top-left, back to front → yes, same direction
- $ \overline{YU} $: top-right, back to front → yes
- $ \overline{TX} $: bottom-right, back to front → yes?

Wait: $ \overline{VS} $ connects $V$ to $S$. Let's define positions:

Assume:
- Bottom layer: $ S, T, U, V $
- Top layer: $ W, X, Y, Z $

Standard labeling:
- $ S $: bottom-front-left
- $ T $: bottom-front-right
- $ U $: bottom-back-right
- $ V $: bottom-back-left
- $ W $: top-front-left
- $ X $: top-front-right
- $ Y $: top-back-right
- $ Z $: top-back-left

Then:
- $ \overline{VS} $: from $V$ (bottom-back-left) to $S$ (bottom-front-left) → this is a vertical edge? No!

Wait — no! That would be front-to-back, not up/down.

Hold on: $V$ to $S$: both are on the left side, but $V$ is back, $S$ is front → so this is a horizontal edge on the left face, from back to front.

So it’s left face, bottom edge, going from back to front.

Now, which edges are parallel?

- $ \overline{ZW} $: top-left, from $Z$ (top-back-left) to $W$ (top-front-left) → same direction → parallel ✔
- $ \overline{YX} $: top-right, from $Y$ (top-back-right) to $X$ (top-front-right) → also same direction → parallel ✔
- $ \overline{UT} $: bottom-right, from $U$ (bottom-back-right) to $T$ (bottom-front-right) → same direction → parallel ✔

So three segments: $ \overline{ZW}, \overline{YX}, \overline{UT} $

But student wrote: $ \overline{ZW}, \overline{YX} $ — missing $ \overline{UT} $

Wait — is $ \overline{UT} $ parallel to $ \overline{VS} $? Yes — both go from back to front on their respective sides.

But in the student's answer: $ \overline{ZW}, \overline{YX} $ — missing one

But wait — maybe the student meant $ \overline{UT} $, but wrote $ \overline{YX} $ instead?

No — student wrote: $ \overline{ZW}, \overline{YX} $

But $ \overline{YX} $ is top-right front-to-back — same as $ \overline{VS} $, which is bottom-left front-to-back.

So yes, they are parallel.

But $ \overline{UT} $ is also parallel.

So three segments: $ \overline{ZW}, \overline{YX}, \overline{UT} $

But student only listed two: $ \overline{ZW}, \overline{YX} $

✘ Incorrect — missing $ \overline{UT} $

But wait — is $ \overline{UT} $ parallel to $ \overline{VS} $? Yes.

Alternatively, maybe the student thought $ \overline{UT} $ was already covered?

Wait — $ \overline{UT} $ is bottom-right, from $U$ (back) to $T$ (front) → yes, same direction.

So correct answer: $ \overline{ZW}, \overline{YX}, \overline{UT} $

But student wrote: $ \overline{ZW}, \overline{YX} $ — incomplete

Wait — look again: student wrote: $ \overline{ZW}, \overline{YX} $ — maybe they forgot $ \overline{UT} $

But perhaps $ \overline{VS} $ is not parallel to $ \overline{UT} $? Let's check directions.

- $ \overline{VS} $: from $V$ (bottom-back-left) to $S$ (bottom-front-left): moves forward in x-direction (assuming x = front-back)
- $ \overline{UT} $: from $U$ (bottom-back-right) to $T$ (bottom-front-right): same direction

Yes, same vector direction → parallel.

So student missed $ \overline{UT} $

But wait — in the student's answer, they have: $ \overline{ZW}, \overline{YX} $ — so missing one.

But maybe $ \overline{ZW} $ and $ \overline{YX} $ are the only ones? No — $ \overline{UT} $ is also parallel.

Wait — is $ \overline{UT} $ in the same plane? No, but parallelism doesn't require same plane.

So yes, $ \overline{UT} $ is parallel.

But perhaps the student considered only top ones?

Wait — let's recheck: Is $ \overline{VS} $ parallel to $ \overline{ZW} $? Yes — both are left-side, horizontal, from back to front.

Is $ \overline{VS} $ parallel to $ \overline{YX} $? Yes — both are horizontal, from back to front.

Is $ \overline{VS} $ parallel to $ \overline{UT} $? Yes — same direction.

So all three are parallel.

But student only listed two.

✘ Error: Missing $ \overline{UT} $

But wait — maybe $ \overline{UT} $ is not drawn? No, it's dashed.

But still, it exists.

So correct answer: $ \overline{ZW}, \overline{YX}, \overline{UT} $

But student has: $ \overline{ZW}, \overline{YX} $ — incorrect

Wait — perhaps the student meant $ \overline{UT} $ but wrote $ \overline{YX} $? No, $ \overline{YX} $ is correct.

Wait — maybe $ \overline{YX} $ is not parallel?

No — $ \overline{YX} $: from $Y$ (top-back-right) to $X$ (top-front-right) — that's from back to front, same as $ \overline{VS} $

Yes, parallel.

So three: $ \overline{ZW}, \overline{YX}, \overline{UT} $

But student only listed two.

So error.

But wait — maybe $ \overline{UT} $ is not parallel? Let's see coordinates.

Assume:
- $S$: (0,0,0)
- $T$: (1,0,0)
- $U$: (1,1,0)
- $V$: (0,1,0)
- $W$: (0,0,1)
- $X$: (1,0,1)
- $Y$: (1,1,1)
- $Z$: (0,1,1)

Then:
- $ \overline{VS} $: from $V(0,1,0)$ to $S(0,0,0)$ → vector: $ \langle 0,-1,0 \rangle $
- $ \overline{ZW} $: from $Z(0,1,1)$ to $W(0,0,1)$ → vector: $ \langle 0,-1,0 \rangle $ → same → parallel
- $ \overline{YX} $: from $Y(1,1,1)$ to $X(1,0,1)$ → $ \langle 0,-1,0 \rangle $ → same
- $ \overline{UT} $: from $U(1,1,0)$ to $T(1,0,0)$ → $ \langle 0,-1,0 \rangle $ → same

So all four? Wait — $ \overline{VS}, \overline{ZW}, \overline{YX}, \overline{UT} $ all have vector $ \langle 0,-1,0 \rangle $ — so all parallel.

But student only listed two: $ \overline{ZW}, \overline{YX} $

But $ \overline{UT} $ is also parallel — so missing.

And $ \overline{VS} $ itself is not included because we want other segments.

So answer should be: $ \overline{ZW}, \overline{YX}, \overline{UT} $

But student has only two — ✘ incorrect

But wait — student wrote: $ \overline{ZW}, \overline{YX} $ — so missing $ \overline{UT} $

So error

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#### d) Name a plane parallel to plane STU

- Plane STU: points $S, T, U$
- $S$: bottom-front-left
- $T$: bottom-front-right
- $U$: bottom-back-right

So this is the bottom face of the prism.

A plane parallel to it would be the top face: $W, X, Y, Z$

So plane $WXYZ$ — but specifically, plane $WXY$ or $ZWX$?

Student wrote: $ \text{plane } ZWX $

But $Z, W, X$: $Z$ (top-back-left), $W$ (top-front-left), $X$ (top-front-right)

So triangle $ZWX$ — but $Z, W, X$ are not coplanar with the full top face?

Wait — $Z, W, X$ are three points on the top face.

But the top face is $Z, W, X, Y$

So plane $ZWX$ is the same as plane $ZWX$, but does it include $Y$?

No — $Z, W, X$ form a triangle on the top face.

But since $Z, W, X$ are coplanar with $Y$, the plane is the top face.

So yes, plane $ZWX$ is the top face, parallel to bottom face $STU$

So ✔ correct

Note: Plane $ZWX$ is not standard notation — usually we use three non-collinear points.

But $Z, W, X$ are not collinear — they form a triangle on the top face.

So yes, it defines the top face.

So ✔ correct

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#### e) Name a plane parallel to plane UVZ

- Points: $U$: bottom-back-right, $V$: bottom-back-left, $Z$: top-back-left

So these three points: $U, V, Z$

They form a back face of the prism.

Back face: $V, U, Y, Z$

So plane $UVZ$ is the back face

Parallel plane: front face: $S, T, X, W$

So any three points on front face: e.g., $S, T, W$, or $S, T, X$, etc.

Student wrote: $ \text{plane } STX $

- $S$: bottom-front-left
- $T$: bottom-front-right
- $X$: top-front-right

These are three points on the front face → yes

So plane $STX$ is the front face, parallel to back face $UVZ$

✔ Correct

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#### f) Name all segments skew to $ \overline{SW} $

- $ \overline{SW} $: from $S$ (bottom-front-left) to $W$ (top-front-left) → vertical edge on left front

Skew lines: not parallel, not intersecting, and not in same plane.

So find segments that are not in the same plane as $SW$, not parallel, not intersecting.

- $ \overline{VT} $: from $V$ (bottom-back-left) to $T$ (bottom-front-right)? No — $V$ to $T$ is diagonal

Wait — list possible candidates.

Segments that are skew to $ \overline{SW} $:

- $ \overline{YT} $: from $Y$ (top-back-right) to $T$ (bottom-front-right) — not parallel, not intersecting, not in same plane → skew
- $ \overline{UX} $: from $U$ (bottom-back-right) to $X$ (top-front-right) — skew?
- $ \overline{VU} $: bottom-back edge — in same plane? Back face — $SW$ is on front-left — not same plane → but do they intersect? No → but are they skew?

Wait — better to list:

$ \overline{SW} $: vertical on front-left

Segments that are not in the same plane, not parallel, not intersecting.

- $ \overline{YT} $: from top-back-right to bottom-front-right — crosses space — skew
- $ \overline{UX} $: from bottom-back-right to top-front-right — skew
- $ \overline{VU} $: bottom-back edge — horizontal — not parallel, not intersecting — skew?
- $ \overline{YZ} $: top-back edge — horizontal — not in same plane — skew?

But student wrote: $ \overline{YX}, \overline{UT}, \overline{VS} $

Let’s check:

- $ \overline{YX} $: from $Y$ (top-back-right) to $X$ (top-front-right) — top edge, right side — not in same plane as $SW$ (which is front-left), not parallel, not intersecting → skew ✔
- $ \overline{UT} $: from $U$ (bottom-back-right) to $T$ (bottom-front-right) — bottom-right edge — not in same plane, not parallel, not intersecting → skew ✔
- $ \overline{VS} $: from $V$ (bottom-back-left) to $S$ (bottom-front-left) — bottom-left edge — but $S$ is endpoint of $SW$, so $ \overline{VS} $ and $ \overline{SW} $ intersect at $S$ → so not skew ✘

So $ \overline{VS} $ intersects $ \overline{SW} $ at $S$ → not skew

So student included $ \overline{VS} $ — incorrect

So correct answer: $ \overline{YX}, \overline{UT}, \overline{YT}, \overline{UX}, \overline{VU}, \overline{YZ} $, etc.

But student wrote: $ \overline{YX}, \overline{UT}, \overline{VS} $ — incorrect due to $ \overline{VS} $

So ✘ Error

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#### g) Name all segments skew to $ \overline{UT} $

- $ \overline{UT} $: from $U$ (bottom-back-right) to $T$ (bottom-front-right) — bottom-right edge, front to back

So it's horizontal on bottom-right

Find segments that are not in same plane, not parallel, not intersecting.

- $ \overline{XS} $: from $X$ (top-front-right) to $S$ (bottom-front-left)? No — $X$ to $S$ is diagonal
- $ \overline{XW} $: top-front edge — parallel? No — $ \overline{UT} $ is horizontal, $ \overline{XW} $ is horizontal — but different direction? No — both go left-right?

Wait — $ \overline{UT} $: from $U$ (back) to $T$ (front) — so direction: front to back

$ \overline{XW} $: from $X$ (front) to $W$ (front) — no — $X$ to $W$ is left to right? $X$ (1,0,1), $W$ (0,0,1) → vector $ \langle -1,0,0 \rangle $

$ \overline{UT} $: $U(1,1,0)$ to $T(1,0,0)$ → vector $ \langle 0,-1,0 \rangle $

So not parallel.

Do they intersect? No.

Are they in same plane? $ \overline{XW} $ is on top-front, $ \overline{UT} $ on bottom-right — different planes → skew?

But student wrote: $ \overline{XT}, \overline{WS}, \overline{ZV} $

Check:

- $ \overline{XT} $: from $X$ (top-front-right) to $T$ (bottom-front-right) — vertical edge on right front — intersects $ \overline{UT} $ at $T$ → so not skew ✘
- $ \overline{WS} $: from $W$ (top-front-left) to $S$ (bottom-front-left) — vertical on left front — not in same plane, not parallel, not intersecting → skew ✔
- $ \overline{ZV} $: from $Z$ (top-back-left) to $V$ (bottom-back-left) — vertical on left back — not in same plane, not parallel, not intersecting → skew ✔

But $ \overline{XT} $ intersects $ \overline{UT} $ at $T$ → not skew

So student included $ \overline{XT} $ — incorrect

So correct: $ \overline{WS}, \overline{ZV}, \overline{YX}, \overline{YU}, \overline{XW}, \overline{YW} $, etc.

But student wrote: $ \overline{XT}, \overline{WS}, \overline{ZV} $ — invalid due to $ \overline{XT} $

✘ Error

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Problem 2: Describe the relationship

Given a slanted prism-like figure: points $A, B, C, D, E, F$

Assume: $ABCD$ is bottom quadrilateral, $E$ above $A$, $F$ above $B$, etc.

#### a) $ \overline{AB} $ and $ \overline{BC} $

- Both on bottom face
- They share point $B$ → intersecting
- But student wrote: skew ✘

Incorrect — they intersect at $B$

So should be: intersecting

#### b) $ \overline{AE} $ and $ \overline{BF} $

- $AE$: from $A$ to $E$ (vertical?)
- $BF$: from $B$ to $F$ (vertical?)
- If $E$ is above $A$, $F$ above $B$, then $AE$ and $BF$ are vertical edges
- They are parallel if the prism is straight
- Student wrote: parallel ✔

#### c) $ \overline{EF} $ and $ \overline{AD} $

- $EF$: top edge, from $E$ to $F$
- $AD$: bottom edge, from $A$ to $D$
- Not in same plane, not parallel (unless prism is flat), not intersecting
- So likely skew
- Student wrote: skew ✔

#### d) Plane $ABC$ and plane $ABF$

- Plane $ABC$: bottom face
- Plane $ABF$: contains $A, B, F$ — $F$ is above $B$
- These two planes share line $AB$ → they intersect along $AB$
- So: intersecting planes
- Student wrote: intersecting ✔

#### e) Plane $AED$ and plane $BFC$

- Plane $AED$: contains $A, E, D$ — left side
- Plane $BFC$: contains $B, F, C$ — right side
- Do they intersect? Possibly along a line? But if the prism is skewed, they might not intersect
- But in general, if they are opposite faces, they may be parallel or intersecting
- But in this case, $AED$ and $BFC$ are opposite lateral faces
- Likely parallel or skew planes?

Wait — planes can be skew? No — in 3D, two planes either intersect or are parallel.

If they don’t intersect, they are parallel

In a prism, opposite faces are parallel.

So $AED$ and $BFC$ are parallel planes

But student wrote: intersecting ✘

So incorrect

Should be: parallel

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Problem 3: Classify angles

Given diagrams with transversals.

#### 3. Alternate Interior → Correct

#### 4. Corresponding Exterior → Correct

#### 5. Corresponding → Correct

#### 6. Consecutive Interior → Correct

#### 7. Corresponding → Correct

#### 8. Alternate Exterior → Correct

All seem correctly labeled.

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Summary of Errors:

| Question | Student Answer | Correct Answer | Error? |
|--------|----------------|----------------|--------|
| 1c | $ \overline{ZW}, \overline{YX} $ | $ \overline{ZW}, \overline{YX}, \overline{UT} $ | ✘ Missing $ \overline{UT} $ |
| 1f | $ \overline{YX}, \overline{UT}, \overline{VS} $ | $ \overline{YX}, \overline{UT}, \overline{YT}, \overline{UX}, \ldots $ | ✘ $ \overline{VS} $ intersects $ \overline{SW} $ |
| 1g | $ \overline{XT}, \overline{WS}, \overline{ZV} $ | $ \overline{WS}, \overline{ZV}, \overline{YX}, \ldots $ | ✘ $ \overline{XT} $ intersects $ \overline{UT} $ |
| 2a | Skew | Intersecting | ✘ |
| 2e | Intersecting | Parallel | ✘ |

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✔ Final Corrections:

1c) $ \overline{ZW}, \overline{YX}, \overline{UT} $

1f) $ \overline{YX}, \overline{UT}, \overline{YT}, \overline{UX}, \overline{VU}, \overline{YZ} $ (any three skew segments)

1g) $ \overline{WS}, \overline{ZV}, \overline{YX}, \overline{YU}, \overline{XW} $ (excluding $ \overline{XT} $)

2a) Intersecting

2e) Parallel

Others are correct.

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📝 Conclusion:

The student did well overall, but made several key errors in identifying parallel vs. skew, intersecting lines, and plane relationships.

Main mistakes:
- Confusing intersecting with skew
- Including segments that intersect the given segment
- Missing some parallel segments
- Misidentifying plane relationships

With corrections, the work can be improved.