1.1 Naming Points, Lines. planes, and angles! - Karlee Pins ... - Free Printable

Problem Analysis and Solution

The task involves analyzing geometric diagrams to answer questions about points, lines, planes, and their intersections. Let's solve each part step by step.

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#### Problem 1:
Diagram Description: A parallelogram with labeled points \( V, W, X, Y, Z \) and lines \( a \) and \( b \).

##### Questions:
1. a) How many points appear in the figure?
- Points: \( V, W, X, Y, Z \)
- Answer: 5

2. b) How many lines appear in the figure?
- Lines: \( a, b, VW, WX, XY, YZ, VZ, VX \)
- Answer: 8

3. c) How many planes appear in the figure?
- The entire parallelogram forms one plane.
- Answer: 1

4. d) Name a line containing point \( V \).
- Line \( a \) contains point \( V \).
- Answer: \( a \)

5. e) Name the intersection of lines \( a \) and \( b \).
- Lines \( a \) and \( b \) intersect at point \( W \).
- Answer: \( W \)

6. f) Give another name for line \( b \).
- Line \( b \) can also be named as \( WZ \).
- Answer: \( WZ \)

7. g) Name three non-collinear points.
- Non-collinear points are points that do not lie on the same line. For example, \( V, X, Y \).
- Answer: \( V, X, Y \)

8. h) Give another name for plane \( D \).
- Plane \( D \) can also be named using any three non-collinear points on it, such as \( V, W, X \).
- Answer: \( VWX \)

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#### Problem 2:
Diagram Description: A 3D figure with two parallel planes \( K \) and \( L \), and several points and lines.

##### Questions:
1. a) How many points appear in the figure?
- Points: \( P, Q, R, S, T, U, M, N, O \)
- Answer: 9

2. b) How many lines appear in the figure?
- Lines: \( e, c, d, PQ, QR, RS, ST, TU, UP, MO, NO, MN \)
- Answer: 12

3. c) How many planes appear in the figure?
- Planes: \( K \) and \( L \)
- Answer: 2

4. d) Name three collinear points.
- Collinear points lie on the same line. For example, \( M, N, O \) are collinear on line \( e \).
- Answer: \( M, N, O \)

5. e) Name four non-coplanar points.
- Non-coplanar points do not lie in the same plane. For example, \( P, Q, S, T \) (one from each plane and not aligned in a single plane).
- Answer: \( P, Q, S, T \)

6. f) Give another name for line \( e \).
- Line \( e \) can also be named as \( MN \).
- Answer: \( MN \)

7. g) Name the intersection of \( \overleftrightarrow{PQ} \) and \( \overleftrightarrow{MO} \).
- The lines \( \overleftrightarrow{PQ} \) and \( \overleftrightarrow{MO} \) intersect at point \( Q \).
- Answer: \( Q \)

8. h) Name the intersection of plane \( K \) and line \( c \).
- Line \( c \) intersects plane \( K \) at point \( R \).
- Answer: \( R \)

9. i) Give another name for plane \( L \).
- Plane \( L \) can also be named using any three non-collinear points on it, such as \( M, N, O \).
- Answer: \( MNO \)

10. j) Give another name for \( \overleftrightarrow{PQ} \).
- \( \overleftrightarrow{PQ} \) can also be named as \( \overleftrightarrow{QP} \).
- Answer: \( \overleftrightarrow{QP} \)

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#### Problem 3:
Diagram Description: A triangular prism with vertices \( A, B, C, D, E, F, G, H \).

##### Questions:
1. a) How many points appear in the figure?
- Points: \( A, B, C, D, E, F, G, H \)
- Answer: 8

2. b) How many lines appear in the figure?
- Lines: \( AB, BC, CA, AD, DE, EF, FA, DG, GH, HC, BE, CF \)
- Answer: 12

3. c) How many planes appear in the figure?
- Planes: \( ABC, ABE, BCH, DEF, DFG, EGH \)
- Answer: 6

4. d) Name three collinear points.
- Collinear points lie on the same line. For example, \( A, B, C \) are collinear on the base.
- Answer: \( A, B, C \)

5. e) Name four coplanar points.
- Coplanar points lie in the same plane. For example, \( A, B, E, D \) are coplanar in plane \( ABE \).
- Answer: \( A, B, E, D \)

6. f) Name the intersection of planes \( ABC \) and \( ABE \).
- The intersection of planes \( ABC \) and \( ABE \) is line \( AB \).
- Answer: \( AB \)

7. g) Name the intersection of planes \( BCH \) and \( DEF \).
- The intersection of planes \( BCH \) and \( DEF \) is line \( EH \).
- Answer: \( EH \)

8. h) Name the intersection of \( \overline{AD} \) and \( \overline{DF} \).
- The intersection of segments \( \overline{AD} \) and \( \overline{DF} \) is point \( D \).
- Answer: \( D \)

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Final Answers:

1.
- a) 5
- b) 8
- c) 1
- d) \( a \)
- e) \( W \)
- f) \( WZ \)
- g) \( V, X, Y \)
- h) \( VWX \)

2.
- a) 9
- b) 12
- c) 2
- d) \( M, N, O \)
- e) \( P, Q, S, T \)
- f) \( MN \)
- g) \( Q \)
- h) \( R \)
- i) \( MNO \)
- j) \( \overleftrightarrow{QP} \)

3.
- a) 8
- b) 12
- c) 6
- d) \( A, B, C \)
- e) \( A, B, E, D \)
- f) \( AB \)
- g) \( EH \)
- h) \( D \)

Boxed Final Answer:
\[
\boxed{
\begin{array}{l}
\text{Problem 1: } 5, 8, 1, a, W, WZ, VXY, VWX \\
\text{Problem 2: } 9, 12, 2, MNO, PQST, MN, Q, R, MNO, QP \\
\text{Problem 3: } 8, 12, 6, ABC, ABED, AB, EH, D
\end{array}
}
\]