2. use the diagram to the right to name the following.
a) a line containing point f. ____
b) another name for line k. ____
c) a plane containing point a. ____
d) an example of three non - collinear points. ____
e) the intersection of plane m and line k. ____

Question

Accepted Answer

### Part (a)
# Explanation:
## Step1: Identify line with F
Point $ F $ is on line $ j $ (or the line through $ D $, $ E $, $ F $). So a line containing $ F $ is line $ j $ (or $ \overleftrightarrow{DF} $, $ \overleftrightarrow{EF} $, etc.).
# Answer: Line $ j $ (or $ \overleftrightarrow{DF} $, $ \overleftrightarrow{EF} $, etc.)

### Part (b)
# Explanation:
## Step1: Rename line $ k $
Line $ k $ passes through $ B $, $ E $, $ C $. So another name is $ \overleftrightarrow{BC} $ (or $ \overleftrightarrow{BE} $, $ \overleftrightarrow{EC} $).
# Answer: $ \overleftrightarrow{BC} $ (or $ \overleftrightarrow{BE} $, $ \overleftrightarrow{EC} $)

### Part (c)
# Explanation:
## Step1: Plane with point $ A $
Point $ A $ is inside plane $ M $. So the plane containing $ A $ is plane $ M $.
# Answer: Plane $ M $

### Part (d)
# Explanation:
## Step1: Find non - collinear points
Non - collinear points do not lie on the same line. Points $ A $, $ B $, $ D $ (or $ A $, $ C $, $ F $, etc.) are non - collinear as they don't lie on a single line.
# Answer: $ A $, $ B $, $ D $ (or other valid triplet like $ A $, $ C $, $ F $)

### Part (e)
# Explanation:
## Step1: Intersection of plane and line
The intersection of plane $ M $ and line $ k $ is the set of points common to both. From the diagram, line $ k $ intersects plane $ M $ at point $ E $ and also contains $ B $ and $ C $ but the intersection as a line segment (or line) within the plane: the intersection is $ \overleftrightarrow{BE} $ (or $ \overleftrightarrow{EC} $, $ \overleftrightarrow{BC} $) within plane $ M $, but more precisely, the intersection of plane $ M $ and line $ k $ is the line segment (or line) $ \overleftrightarrow{BC} $ (since line $ k $ intersects plane $ M $ at $ E $, $ B $, $ C $ are on line $ k $ and in plane $ M $). Wait, actually, the intersection of a plane and a line is either a point (if line is skew or parallel) or the line itself (if line lies on the plane). Here line $ k $ intersects plane $ M $ at $ E $, $ B $ is on plane $ M $ and line $ k $, $ C $ is on line $ k $ and plane $ M $? Wait, looking at the diagram, line $ k $ has points $ B $, $ E $, $ C $, and plane $ M $ contains $ B $, $ E $, $ C $? Wait, no, the dashed line is $ M $'s edge? Wait, the plane $ M $ is a quadrilateral? Wait, the diagram shows plane $ M $ with line $ j $ (through $ D $, $ E $, $ F $) and line $ k $ (through $ B $, $ E $, $ C $). So the intersection of plane $ M $ and line $ k $ is the set of points that are in both. So points $ B $, $ E $, $ C $ are in both? Wait, no, maybe $ E $ is the intersection? Wait, no, the line $ k $ passes through $ B $ (on plane $ M $), $ E $ (on plane $ M $), and $ C $ (on plane $ M $)? Wait, maybe the plane $ M $ contains line $ k $ at least at $ B $, $ E $, $ C $. So the intersection of plane $ M $ and line $ k $ is line $ k $ (or $ \overleftrightarrow{BC} $) within plane $ M $, but more accurately, the intersection is the line segment (or line) $ \overleftrightarrow{BE} $ or $ \overleftrightarrow{BC} $. But from the diagram, the intersection of plane $ M $ and line $ k $ is point $ E $? No, wait, line $ k $ goes through $ B $ (on plane $ M $), $ E $ (on plane $ M $), and $ C $ (outside? Wait, the diagram has $ C $ outside the plane? Wait, the plane $ M $ is a quadrilateral with vertices? Wait, the diagram shows plane $ M $ with line $ j $ (with $ D $, $ E $, $ F $) and line $ k $ with $ B $, $ E $, $ C $, where $ B $ and $ F $ are inside the plane, $ C $ is outside? Wait, no, the arrow for $ k $ is on both sides. So line $ k $ intersects plane $ M $ at $ E $ and $ B $? Wait, maybe the intersection of plane $ M $ and line $ k $ is $ \overleftrightarrow{BE} $ (the part of line $ k $ that is in plane $ M $). But the standard answer for intersection of a plane and a line is either a point (if line is not on the plane) or the line (if line is on the plane). If line $ k $ is on plane $ M $ at $ B $, $ E $, then the intersection is line $ k $ (or $ \overleftrightarrow{BC} $) but within the plane, the intersection is the set of points common to both, so $ \overleftrightarrow{BE} $ (or $ \overleftrightarrow{BC} $ if $ C $ is in the plane). Wait, maybe the answer is point $ E $? No, that doesn't make sense. Wait, re - examining: the plane $ M $ and line $ k $ intersect at the set of points that are in both. From the diagram, line $ k $ passes through $ B $ (in plane $ M $), $ E $ (in plane $ M $), and $ C $ (maybe in plane $ M $ or not). But the most likely answer is that the intersection of plane $ M $ and line $ k $ is $ \overleftrightarrow{BE} $ (or $ \overleftrightarrow{BC} $) but actually, the correct answer is that the intersection is the line segment (or line) $ \overleftrightarrow{BC} $ (or $ \overleftrightarrow{BE} $) because line $ k $ intersects plane $ M $ at those points. But the standard answer for such a problem is that the intersection of plane $ M $ and line $ k $ is $ \overleftrightarrow{BE} $ (or $ \overleftrightarrow{BC} $), but more precisely, the intersection is point $ E $? No, I think I made a mistake. Wait, the line $ k $ and plane $ M $: if line $ k $ passes through $ B $ (in plane $ M $) and $ C $ (outside), and $ E $ is the intersection point? No, the diagram shows $ B $ and $ C $ on line $ k $, with $ B $ inside the plane and $ C $ outside? Wait, the plane $ M $ is a quadrilateral, so $ B $ is on the edge, $ E $ is inside, $ C $ is outside. So line $ k $ intersects plane $ M $ at $ B $ and $ E $? No, the intersection of a line and a plane is either a single point (if the line is not on the plane) or the entire line (if the line is on the plane). If line $ k $ is not on the plane, it intersects at one point. But in the diagram, $ B $ is on the plane and on line $ k $, $ E $ is on the plane and on line $ k $, so line $ k $ is on the plane? Then the intersection is line $ k $ (or $ \overleftrightarrow{BC} $). But the problem says "the intersection of plane $ M $ and line $ k $". So the answer is $ \overleftrightarrow{BE} $ (or $ \overleftrightarrow{BC} $, or line $ k $ within the plane). But the most appropriate answer is $ \overleftrightarrow{BE} $ (or $ \overleftrightarrow{BC} $). Wait, maybe the answer is point $ E $? No, that's not right. I think the correct answer is that the intersection of plane $ M $ and line $ k $ is $ \overleftrightarrow{BE} $ (or $ \overleftrightarrow{BC} $), but to be precise, looking at the diagram, line $ k $ intersects plane $ M $ at $ E $ and $ B $, so the intersection is the line segment $ \overleftrightarrow{BE} $ (or the line $ \overleftrightarrow{BC} $ if $ C $ is in the plane). But the standard answer for this type of problem is that the intersection of plane $ M $ and line $ k $ is $ \overleftrightarrow{BE} $ (or $ \overleftrightarrow{BC} $). However, if we consider that line $ k $ and plane $ M $ intersect at point $ E $, that would be wrong. So the correct answer is $ \overleftrightarrow{BE} $ (or $ \overleftrightarrow{BC} $).
# Answer: $ \overleftrightarrow{BE} $ (or $ \overleftrightarrow{BC} $, or line $ k $ within plane $ M $)

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QUESTION IMAGE

Question

Explanation:

Part (a)

Step1: Identify line with F

Step1: Rename line \( k \)

Step1: Plane with point \( A \)

Answer:

Part (b)