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.

Question

Accepted Answer

### Question 3:
# Brief Explanations:
To determine if points $ G $ and $ P $ are collinear, we use the postulate: *If two points lie on a line, then they are collinear*. From the figure, $ D $ is on $ \overleftrightarrow{DG} $ and $ \overleftrightarrow{DP} $, but $ G $ is on $ \overleftrightarrow{DG} $ and $ P $ is on $ \overleftrightarrow{DP} $. Wait, actually, looking at the diagram (the T - intersection), $ D $ is the intersection point, $ \overleftrightarrow{DG} $ is horizontal, $ \overleftrightarrow{DP} $ is vertical. Wait, no, maybe I misread. Wait, the problem says "Points $ G $ and $ P $ are collinear". Wait, no, actually, maybe the line $ \overleftrightarrow{DG} $ and $ \overleftrightarrow{DP} $ intersect at $ D $, but $ G $ is on $ \overleftrightarrow{DG} $, $ P $ is on $ \overleftrightarrow{DP} $. Wait, no, maybe the postulate here is: *Through any two points, there is exactly one line*? No, collinear means they lie on the same line. Wait, maybe the diagram shows that $ D $, $ G $, and another point? Wait, no, the first part: "Points $ G $ and $ P $ are collinear". Wait, no, maybe I made a mistake. Wait, the problem says "State the postulate that can be used to show that each statement is true." For "Points $ G $ and $ P $ are collinear" – no, actually, looking at the diagram (the T - sign), $ D $ is the center, $ \overleftrightarrow{DG} $ is horizontal (left - right), $ \overleftrightarrow{DP} $ is vertical (up - down)? Wait, no, the arrows: $ \overleftrightarrow{DG} $ has arrows both ways, $ \overleftrightarrow{DP} $ has an arrow down (towards $ P $). Wait, maybe $ D $, $ G $ are on a horizontal line, $ D $, $ P $ are on a vertical line. So $ G $ and $ P $ are not on the same line. Wait, maybe the question is miswritten? Wait, no, maybe I misread the points. Wait, the first sub - question: "3. Points $ G $ and $ P $ are collinear." Wait, no, maybe it's $ D $, $ G $, and another? Wait, no, let's re - examine. The postulate for collinear points is: *If two points lie on a line, then they are collinear* (or *Through any two points, there is exactly one line, and points on the same line are collinear*). But in this case, if $ G $ is on $ \overleftrightarrow{DG} $ and $ P $ is on $ \overleftrightarrow{DP} $, but $ \overleftrightarrow{DG} $ and $ \overleftrightarrow{DP} $ intersect at $ D $, so $ G $, $ D $, $ P $ – no, $ G $ and $ P $ are not on the same line. Wait, maybe the question is "Points $ D $, $ G $ and $ P $"? No, the question says "Points $ G $ and $ P $ are collinear." Wait, maybe the diagram is different. Alternatively, maybe the postulate is: *If two lines intersect, then their intersection is a point* – no, that's for intersection. Wait, maybe I made a mistake. Let's move to question 4.

### Question 4:
# Brief Explanations:
The postulate for coplanar points (points lying in the same plane) is: *If two lines intersect, then they lie in the same plane (and all points on those lines are coplanar with the intersection point)*. We know that $ \overleftrightarrow{DG} $ and $ \overleftrightarrow{DP} $ intersect at $ D $, so they lie in a plane (plane $ J $, as given: " $ \overleftrightarrow{DG} $ and $ \overleftrightarrow{DP} $ are in plane $ J $"). $ H $ lies on $ \overleftrightarrow{DG} $, so $ H $ is in plane $ J $. $ P $ lies on $ \overleftrightarrow{DP} $, so $ P $ is in plane $ J $. $ D $ is the intersection point, so $ D $, $ H $ (on $ \overleftrightarrow{DG} $ in plane $ J $), and $ P $ (on $ \overleftrightarrow{DP} $ in plane $ J $) are all in plane $ J $, hence coplanar. The postulate used is: *If two lines intersect, then they lie in a plane, and all points on those lines are coplanar with the lines' intersection point* (or more simply, *Points on intersecting lines are coplanar*).

### Answer (Question 3 correction - maybe misread points):
Wait, maybe the first question is "Points $ D $, $ G $ and $ P $"? No, the user's question for 3 is "Points $ G $ and $ P $ are collinear." But based on the diagram (T - intersection), $ G $ is on horizontal line $ \overleftrightarrow{DG} $, $ P $ is on vertical line $ \overleftrightarrow{DP} $, so they are not collinear. Maybe a typo, and it's "Points $ D $, $ G $ and $ P $"? No, the user's question is as given. Alternatively, maybe the postulate is that "Through any two points, there is exactly one line" – but that would mean if $ G $ and $ P $ were collinear, but they aren't. Wait, maybe the diagram shows $ G $, $ D $, $ P $ – no, $ D $ is between $ G $ and $ H $ on $ \overleftrightarrow{DG} $, and $ D $ is on $ \overleftrightarrow{DP} $ towards $ P $. So $ D $, $ G $ are collinear (on $ \overleftrightarrow{DG} $), $ D $, $ P $ are collinear (on $ \overleftrightarrow{DP} $). So for "Points $ G $ and $ P $ are collinear" – this statement is false? But the problem says "show that each statement is true". So maybe I misread the points. Maybe the first question is "Points $ D $, $ G $ and $ P $"? No, the user wrote "Points $ G $ and $ P $ are collinear."

Assuming that there is a mistake and it's "Points $ D $, $ G $ and $ P $" – no, the user's question is as is. Alternatively, maybe the postulate is "If two points lie on a line, then they are collinear" – but $ G $ and $ P $ do not lie on the same line. So perhaps the first question is "Points $ D $, $ G $ are collinear" (which they are, on $ \overleftrightarrow{DG} $) or "Points $ D $, $ P $ are collinear" (on $ \overleftrightarrow{DP} $). Given the confusion, let's focus on question 4.

For question 4: "Points $ D $, $ H $, and $ P $ are coplanar." The postulate is: *If two lines intersect, then they lie in a plane, and all points on those lines are coplanar with the lines' intersection point* (or *Points that lie on intersecting lines are coplanar*). Since $ \overleftrightarrow{DG} $ and $ \overleftrightarrow{DP} $ intersect at $ D $, they lie in plane $ J $. $ H $ is on $ \overleftrightarrow{DG} $ (so in plane $ J $), $ P $ is on $ \overleftrightarrow{DP} $ (so in plane $ J $), and $ D $ is the intersection point (in plane $ J $), so $ D $, $ H $, and $ P $ are coplanar by the postulate that intersecting lines lie in a plane and all points on those lines are coplanar with the plane of the lines.

# Answer (Question 3 - assuming a typo, maybe "Points $ D $, $ G $ and $ P $" is incorrect, and the correct points for collinear are $ D $, $ G $ (on $ \overleftrightarrow{DG} $):
For "Points $ G $ and $ P $ are collinear" – this statement is false as per the diagram. But if we assume it's a mistake and the points are $ D $, $ G $ (or $ D $, $ P $):

3. **Postulate for collinear points (e.g., $ D $, $ G $)**: *If two points lie on a line, then they are collinear* (or *Through any two points, there is exactly one line, and points on that line are collinear*). Since $ D $ and $ G $ lie on $ \overleftrightarrow{DG} $, they are collinear.

4. **Postulate for coplanar points ($ D $, $ H $, $ P $)**: *If two lines intersect, then they lie in a plane, and all points on those lines are coplanar with the lines' intersection point*. Since $ \overleftrightarrow{DG} $ and $ \overleftrightarrow{DP} $ intersect at $ D $, they lie in plane $ J $. $ H $ is on $ \overleftrightarrow{DG} $ (so in plane $ J $), $ P $ is on $ \overleftrightarrow{DP} $ (so in plane $ J $), hence $ D $, $ H $, $ P $ are coplanar.

(Note: The initial confusion arises from possible typographical errors in the problem statement, but the key postulates for collinearity (points on a line) and coplanarity (points in a plane, especially from intersecting lines) are applied as above.)

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