practice
example 1
refer to the figure for exercises 1–7.
1. name the lines that are only in plane q.
2. how many planes are labeled in the figure?
3. name the plane containing the lines m and t.
4. name the intersection of lines m and t.
5. name a point that is not coplanar with points a, b, and c.
6. are points f, m, g, and p coplanar? explain.

Question

Accepted Answer

### Problem 1: Name the lines that are only in plane $ Q $
# Explanation:
## Step 1: Identify plane $ Q $
Plane $ Q $ is the left - hand blue parallelogram - shaped plane in the figure.
## Step 2: Identify lines within plane $ Q $
We look at the lines drawn inside plane $ Q $. The line $ n $ (with point $ G $ on it) and the line $ q $ (with points $ H $ and $ L $ on it) are entirely contained within plane $ Q $. The line that intersects plane $ Q $ (the line with point $ M $) is not only in plane $ Q $ as it extends outside.
# Answer: Lines $ n $ and $ q $

### Problem 2: How many planes are labeled in the figure?
# Explanation:
## Step 1: Identify labeled planes
We look for the planes with labels. In the figure, we can see plane $ Q $ (the left - hand blue plane), plane $ R $ (the right - hand blue plane).
## Step 2: Count the labeled planes
By visually inspecting the figure, we can identify two labeled planes: $ Q $ and $ R $.
# Answer: 2

### Problem 3: Name the plane containing the lines $ m $ and $ t $
# Explanation:
## Step 1: Locate lines $ m $ and $ t $
Lines $ m $ and $ t $ (or $ l $, depending on the labeling) are in the right - hand blue plane.
## Step 2: Identify the plane
The right - hand blue plane is labeled as plane $ R $. So the plane containing lines $ m $ and $ t $ is plane $ R $.
# Answer: Plane $ R $

### Problem 4: Name the intersection of lines $ m $ and $ t $
# Explanation:
## Step 1: Recall the intersection of two lines
Two lines intersect at a point.
## Step 2: Locate the intersection point
Looking at the figure, lines $ m $ and $ t $ (or their equivalents) intersect at point $ C $.
# Answer: Point $ C $

### Problem 5: Name a point that is not coplanar with points $ A $, $ B $, and $ C $
# Explanation:
## Step 1: Recall coplanar points
Coplanar points lie on the same plane. Points $ A $, $ B $, and $ C $ are on plane $ R $.
## Step 2: Identify a non - coplanar point
Point $ M $ (or point $ P $, $ G $ etc. from plane $ Q $) is not on plane $ R $. So a point like $ M $ (or $ P $, $ G $) is not coplanar with $ A $, $ B $, and $ C $. Let's take point $ M $ as an example.
# Answer: Point $ M $ (or $ P $, $ G $ etc.)

### Problem 6: Are points $ F $, $ M $, $ G $, and $ P $ coplanar? Explain.
# Explanation:
## Step 1: Recall coplanar definition
Coplanar points lie on the same plane.
## Step 2: Analyze the positions of the points
Point $ F $ is on plane $ R $, point $ M $ is on the line that intersects planes $ Q $ and $ R $, point $ G $ and $ P $ are on plane $ Q $. Since $ F $ is on plane $ R $ and $ G $, $ P $ are on plane $ Q $ (and $ M $ is on the intersection line), these points do not all lie on the same plane.
# Answer: No. Point $ F $ is on plane $ R $, while $ G $, $ P $ are on plane $ Q $, and $ M $ is on the line intersecting the two planes. So they do not lie on the same plane.

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QUESTION IMAGE

Question

Explanation:

Problem 1: Name the lines that are only in plane \( Q \)

Step 1: Identify plane \( Q \)

Step 2: Identify lines within plane \( Q \)

Step 1: Identify labeled planes

Step 2: Count the labeled planes

Step 1: Locate lines \( m \) and \( t \)

Step 2: Identify the plane

Answer:

Problem 2: How many planes are labeled in the figure?