name ____________ date ____________
3-1 practice
parallel lines and transversals
refer to the figure at the right to identify each of the following.
1. all planes that intersect plane stx
2. all segments that intersect $\overline{qu}$
3. all segments that are parallel to $\overline{xy}$
4. all segments that are skew to $\overline{vw}$
classify the relationship between each pair of angles as alternate interior,

Question

Accepted Answer

### 1. All planes that intersect plane $ STX $
# Explanation:
## Step1: Analyze the figure structure
The figure is a prism - like shape. Plane $ STX $ is a face. Planes that share an edge with plane $ STX $ will intersect it.
- Plane $ STU $: shares edge $ ST $ with plane $ STX $.
- Plane $ STW $ (or plane $ STY $? Wait, looking at the labels: $ Z, V, T, U $ and $ X, W, T, S $, $ Y $ is at the bottom. Wait, the planes that intersect plane $ STX $: plane $ STU $ (contains $ ST $ and $ TU $), plane $ STW $ (contains $ ST $ and $ SW $)? No, more accurately, from the vertices: plane $ STU $ (vertices $ S, T, U, Q $? Wait, the figure has two hexagons? No, it's a prism with two pentagons? Wait, the labels are $ Q, R, S, X, Y, Z, U, V, W, T $. Let's re - examine:
Plane $ STX $ has points $ S, T, X $. Planes that intersect it:
- Plane $ STU $: contains $ S, T, U $ (shares line $ ST $ with plane $ STX $).
- Plane $ STW $: Wait, no, $ W $ is connected to $ X $ and $ S $? Wait, maybe the correct planes are plane $ STU $, plane $ STW $ (no, $ W $ is on the other side), plane $ XYZ $? No, better approach: In a 3 - D figure, two planes intersect if they share a line. Plane $ STX $ and plane $ STU $ share line $ ST $, plane $ STX $ and plane $ XYW $ (no, $ Y $ is at the bottom). Wait, looking at the vertical edges: $ TU $ and $ VW $ and $ XY $ are vertical? Wait, the figure is a prism with two congruent pentagons (top: $ Q, R, S, W, V, U $? No, the top is $ Q, R, S $ and bottom is $ X, Y, Z $, with vertical edges $ QZ, RU, SV, TW, UX $? Wait, maybe the planes intersecting plane $ STX $ are plane $ STU $, plane $ STW $ (no), plane $ XYZ $ (no). Wait, the correct planes should be plane $ STU $ (contains $ S, T, U $), plane $ XYW $ (no), plane $ ZV T $ (no). Wait, maybe the answer is plane $ STU $, plane $ XYW $ (no), let's think again. The plane $ STX $ and plane $ STU $ intersect along $ ST $, plane $ STX $ and plane $ XYZ $ (no, $ XYZ $ is bottom). Wait, the plane $ STX $ and plane $ TUZ $ (no). Maybe the intended answer is plane $ STU $, plane $ XYW $ (no), I think the correct planes are plane $ STU $, plane $ STW $ (no), plane $ ZV T $ (no). Wait, perhaps the figure is a pentagonal prism, with top face $ Q, R, S, W, V, U $ (no, pentagon) and bottom face $ X, Y, Z, V, U $? No, the labels are $ Q, R, S, X, Y, Z, U, V, W, T $. So plane $ STX $ intersects plane $ STU $ (through $ ST $), plane $ XYW $ (through $ X $), plane $ ZV T $ (through $ T $)? No, I think the correct answer is plane $ STU $, plane $ XYW $ (no), maybe the answer is plane $ STU $, plane $ XYZ $ (no). Wait, maybe the figure is such that plane $ STX $ intersects plane $ STU $, plane $ XYW $, and plane $ ZVT $. But more likely, from the given labels, the planes intersecting plane $ STX $ are plane $ STU $, plane $ XYW $, and plane $ ZVT $. But I think the standard answer for such a figure (pentagonal prism) would be plane $ STU $, plane $ XYW $, and plane $ ZVT $ (or plane $ STU $, plane $ XYZ $, plane $ ZVT $). Wait, maybe I made a mistake. Let's assume the figure is a prism with vertical edges $ TU, VW, XY $. Then plane $ STX $ (with $ S, T, X $) intersects plane $ STU $ ( $ S, T, U $ ), plane $ XYW $ ( $ X, Y, W $ ), and plane $ ZVT $ ( $ Z, V, T $ ).

## Step2: Confirm the intersection
Two planes intersect if they have a non - empty intersection (a line). Plane $ STX $ and plane $ STU $ share the line $ ST $, plane $ STX $ and plane $ XYW $ share the line $ XW $ (no, $ X $ is in both), plane $ STX $ and plane $ ZVT $ share the line $ VT $ (no, $ T $ is in both). Wait, maybe the correct planes are plane $ STU $, plane $ XYZ $, and plane $ ZV U $. I think the intended answer is plane $ STU $, plane $ XYW $, and plane $ ZVT $ (or plane $ STU $, plane $ XYZ $, plane $ ZV U $). But to be precise, from the figure (as per the labels $ Q, R, S, X, Y, Z, U, V, W, T $), the planes that intersect plane $ STX $ are plane $ STU $, plane $ XYW $, and plane $ ZVT $.

# Answer:
Plane $ STU $, Plane $ XYW $, Plane $ ZVT $ (or other correct planes based on the actual figure, but likely Plane $ STU $, Plane $ XYZ $, Plane $ ZV U $ - the answer may vary slightly based on the exact figure, but the key is planes sharing an edge with $ STX $)

### 2. All segments that intersect $ \overline{QU} $
# Explanation:
## Step1: Analyze the segment $ \overline{QU} $
Segment $ \overline{QU} $ is a side of the top (or one of the) face. Segments that meet $ \overline{QU} $ at a point will intersect it.
- $ \overline{Q R} $: meets $ \overline{QU} $ at $ Q $.
- $ \overline{Q Z} $ (if $ QZ $ is a vertical edge): meets $ \overline{QU} $ at $ Q $. Wait, looking at the labels, $ U $ is connected to $ T $ and $ Z $? So $ \overline{UT} $: meets $ \overline{QU} $ at $ U $.
- $ \overline{Q R} $: at $ Q $, $ \overline{UT} $: at $ U $, $ \overline{Q Z} $ (if exists): at $ Q $, $ \overline{QU} $ also intersects with $ \overline{QV} $? No, the figure labels: $ Q, R, S, X, Y, Z, U, V, W, T $. So segments intersecting $ \overline{QU} $ are $ \overline{Q R} $, $ \overline{Q Z} $ (if $ QZ $ is an edge), $ \overline{UT} $, and $ \overline{QV} $ (no, $ V $ is below $ U $). Wait, more accurately, at point $ Q $: $ \overline{Q R} $ and $ \overline{Q Z} $ (assuming $ QZ $ is a vertical edge) intersect $ \overline{QU} $, and at point $ U $: $ \overline{UT} $ and $ \overline{UZ} $ (if $ UZ $ is an edge) intersect $ \overline{QU} $. But from the given figure (prism - like), the segments intersecting $ \overline{QU} $ are $ \overline{Q R} $, $ \overline{Q Z} $, $ \overline{UT} $, $ \overline{UZ} $ (if exists). But likely, the answer is $ \overline{Q R} $, $ \overline{UT} $, $ \overline{Q Z} $ (or $ \overline{Q R} $, $ \overline{UT} $, $ \overline{QV} $ - no, $ V $ is not connected to $ Q $).

## Step2: List the intersecting segments
Segments that share a common point with $ \overline{QU} $ are the ones that intersect it. So $ \overline{Q R} $ (intersects at $ Q $), $ \overline{UT} $ (intersects at $ U $), $ \overline{Q Z} $ (intersects at $ Q $), $ \overline{UZ} $ (intersects at $ U $) (depending on the figure). But the most probable are $ \overline{Q R} $, $ \overline{UT} $, $ \overline{Q Z} $.

# Answer:
$ \overline{Q R} $, $ \overline{UT} $, $ \overline{Q Z} $ (or other correct segments based on the actual figure, but these are the likely ones)

### 3. All segments that are parallel to $ \overline{XY} $
# Explanation:
## Step1: Analyze the direction of $ \overline{XY} $
In a prism, the lateral edges (vertical edges, if $ XY $ is a lateral edge) are parallel. So if $ \overline{XY} $ is a vertical edge (connecting bottom face to top face), then other vertical edges will be parallel to it.
- $ \overline{VW} $: if $ VW $ is a vertical edge (same direction as $ XY $).
- $ \overline{TU} $: if $ TU $ is a vertical edge (same direction as $ XY $).
- $ \overline{RS} $? No, $ RS $ is a top face edge. Wait, $ XY $ is a bottom face edge? No, in the figure, $ Y $ is at the bottom, $ X $ is at the bottom, $ Z $ is at the bottom. So $ XY $ is a bottom face edge? No, maybe $ XY $ is a lateral edge (connecting bottom $ Y $ to top $ T $? Wait, the labels: $ Y $ is at the bottom, $ T $ is in the middle, $ U $ is at the top. So $ XY $ is a lateral edge (from $ X $ (bottom) to $ T $ (middle)? No, the figure has $ X, Y, Z $ at the bottom, $ V, W $ in the middle, $ U, T, S $ at the top, $ Q, R $ at the top. So if $ XY $ is a vertical edge (from $ X $ to $ S $? No, $ S $ is at the top. Wait, maybe $ XY $ is parallel to $ VW $ and $ TU $ because they are all lateral edges (same direction, same length, and non - intersecting, so parallel).

## Step2: Identify parallel segments
Segments that are parallel to $ \overline{XY} $ are those that have the same direction and do not intersect. So $ \overline{VW} $ and $ \overline{TU} $ (assuming they are lateral edges like $ XY $).

# Answer:
$ \overline{VW} $, $ \overline{TU} $ (or other correct segments based on the actual figure, but these are the likely parallel segments)

### 4. All segments that are skew to $ \overline{VW} $
# Explanation:
## Step1: Define skew segments
Skew segments are non - parallel and non - intersecting, and lie in different planes.
- $ \overline{QR} $: lies in the top face, $ \overline{VW} $ is a lateral edge. They are non - parallel and non - intersecting.
- $ \overline{RS} $: top face edge, non - parallel and non - intersecting with $ \overline{VW} $.
- $ \overline{XY} $: bottom face edge, non - parallel and non - intersecting with $ \overline{VW} $ (if $ XY $ is not parallel). Wait, $ \overline{VW} $ is a lateral edge. Segments like $ \overline{QR} $, $ \overline{RS} $, $ \overline{XY} $, $ \overline{YZ} $, $ \overline{ZQ} $ (if exists) are skew to $ \overline{VW} $ because they are in different planes and do not intersect or are not parallel.

## Step2: List skew segments
Segments that are not parallel to $ \overline{VW} $ and do not intersect it. So $ \overline{QR} $, $ \overline{RS} $, $ \overline{XY} $, $ \overline{YZ} $, $ \overline{ZQ} $ (depending on the figure). But the most probable are $ \overline{QR} $, $ \overline{RS} $, $ \overline{XY} $, $ \overline{YZ} $.

# Answer:
$ \overline{QR} $, $ \overline{RS} $, $ \overline{XY} $, $ \overline{YZ} $ (or other correct segments based on the actual figure, but these are the likely skew segments)

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QUESTION IMAGE

Question

Explanation:

1. All planes that intersect plane \( STX \)

Step1: Analyze the figure structure

Step2: Confirm the intersection

Step1: Analyze the segment \( \overline{QU} \)

Step2: List the intersecting segments

Step1: Analyze the direction of \( \overline{XY} \)

Step2: Identify parallel segments

Answer:

2. All segments that intersect \( \overline{QU} \)