signs in the figure, $overleftrightarrow{dg}$ and $overleftrightarrow{dp}$ are in plane $j$ and $h$ lies on $overleftrightarrow{dg}$. state the postulate that can be used to show that each statement is true. 3. points $g$ and $p$ are collinear. 4. points $d$, $h$, and $p$ are coplanar.

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Accepted Answer

### Question 3: Points $ G $ and $ P $ are collinear.
# Explanation:
## Step 1: Recall the collinear postulate
The postulate for collinear points is: *If two points lie on the same line, then they are collinear*.
We know that $ \overrightarrow{DG} $ and $ \overrightarrow{DP} $ are given, and from the diagram (or context), points $ D $, $ G $, and $ P $ lie on the same line (since $ \overrightarrow{DG} $ and $ \overrightarrow{DP} $ share the line $ DG $ or $ DP $). Specifically, $ G $ lies on $ \overrightarrow{DG} $ and $ P $ lies on $ \overrightarrow{DP} $, and these rays are part of the same line.

## Step 2: Apply the postulate
Since points $ G $ and $ P $ lie on the same line (containing $ D $, $ G $, $ P $), by the collinear postulate, they are collinear.

### Question 4: Points $ D $, $ H $, and $ P $ are coplanar.
# Explanation:
## Step 1: Recall the coplanar postulate
The postulate for coplanar points is: *If points lie in the same plane, then they are coplanar*. Also, another relevant postulate: *If two points lie in a plane, then the line containing them lies in the plane*, and *points on that line are also in the plane*.

We know $ \overrightarrow{DG} $ and $ \overrightarrow{DP} $ are in plane $ J $, and $ H $ lies on $ \overrightarrow{DG} $. So, $ D $ and $ H $ are in plane $ J $, and $ D $ and $ P $ are in plane $ J $ (since $ \overrightarrow{DP} $ is in plane $ J $).

## Step 2: Apply the postulate
Since $ D $, $ H $, and $ P $ all lie in plane $ J $ ( $ D $ and $ H $ are in plane $ J $ via $ \overrightarrow{DG} $, $ D $ and $ P $ are in plane $ J $ via $ \overrightarrow{DP} $, and the line through $ D $, $ H $, $ P $ is contained in plane $ J $), by the coplanar postulate, they are coplanar.

# Answers:
3. The postulate "If two points lie on the same line, then they are collinear" shows $ G $ and $ P $ are collinear (since $ G $ and $ P $ lie on the same line $ DP $ or $ DG $).
4. The postulate "If points lie in the same plane, then they are coplanar" (or related plane - line postulates) shows $ D $, $ H $, $ P $ are coplanar (since $ D $, $ H $, $ P $ lie in plane $ J $).

(For concise answers, you can state the postulate and conclusion directly:
3. Postulate: Through any two points, there is exactly one line (or "If two points lie on a line, they are collinear"). Conclusion: $ G $ and $ P $ are collinear.
4. Postulate: If two points lie in a plane, the line containing them lies in the plane (or "Points in the same plane are coplanar"). Conclusion: $ D $, $ H $, $ P $ are coplanar.)

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QUESTION IMAGE

Question

Explanation:

Question 3: Points \( G \) and \( P \) are collinear.

Step 1: Recall the collinear postulate

Step 2: Apply the postulate

Question 4: Points \( D \), \( H \), and \( P \) are coplanar.

Step 1: Recall the coplanar postulate

Step 2: Apply the postulate

Answer: