Question

in exercises 1 - 6, use the diagram to write an example of the postulate.

1. two - point postulate
2. line - point postulate
3. line intersection postulate
4. three point postulate
5. plane - line postulate
6. plane intersection postulate

in exercises 7 and 8, sketch a diagram of the description.

1. \\(\overline{ab},\overline{cd},\\) and \\(\overline{bd}\\) that intersect at exactly two points
2. planes \\(s\\) and \\(t\\) intersecting at a right angle, \\(\overline{ab}\\) on plane \\(s\\) and plane \\(t\\), and point \\(c\\) is the mid - point of \\(\overline{ab}\\)

in exercises 9 - 12, use the diagram to determine whether you can assume the statement.

1. planes \\(w\\) and \\(v\\) intersect at \\(\overline{tu}\\).
2. points \\(t,u,\\) and \\(r\\) are coplanar.
3. \\(\angle tzx\\) and \\(\angle uzy\\) are vertical angles.
4. \\(\overline{tu}\\) lies in plane \\(w\\).
5. is it possible for three planes to intersect along the same line? explain your reasoning.
6. your friend claims that if the plane - line postulate is true, then all lines that pass through a point in a plane must also be in that same plane. is your friend correct? explain your reasoning.
7. \\(\overline{ab}\\) and \\(\overline{cd}\\) lie in plane \\(z\\), where points \\(a\\) and \\(b\\) are not collinear with points \\(c\\) and \\(d\\). if \\(\overline{ef}\\) bisects either \\(\overline{ab}\\) or \\(\overline{cd}\\), will \\(\overline{ef}\\) always lie in plane \\(z\\)? if \\(\overline{ef}\\) bisects both \\(\overline{ab}\\) and \\(\overline{cd}\\), will \\(\overline{ef}\\) always lie in plane \\(z\\)? explain your reasoning.

Explanation:

Step1: Define postulates

Two - Point Postulate: Through any two points, there is exactly one line. Example from the diagram: Through points \(A\) and \(B\), there is exactly one line \(\overleftrightarrow{AB}\).

Step2: Line - Point Postulate

A line contains at least two points. In the diagram, line \(\overleftrightarrow{TU}\) contains points \(T\) and \(U\).

Step3: Line Intersection Postulate

If two lines intersect, then their intersection is exactly one point. In the diagram, if we consider two lines (not shown explicitly in a way that they intersect in the given elements, but conceptually), for example, if we had two lines in the planes, their intersection would be a single point.

Step4: Three - Point Postulate

Through any three non - collinear points, there is exactly one plane. In the diagram, points \(A\), \(B\), and \(D\) (assuming non - collinear) determine a plane.

Step5: Plane - Line Postulate

If a line intersects a plane not containing it, then the intersection is exactly one point. If we consider a line like \(\overleftrightarrow{TU}\) intersecting a plane (say plane \(A\)), the intersection is a point.

Step6: Plane Intersection Postulate

If two planes intersect, then their intersection is a line. In the diagram, the intersection of the two planes shown is a line (not labeled explicitly but conceptually exists).

For Exercises 7:
We can draw three lines \(\overline{AB}\), \(\overline{CD}\), and \(\overline{BD}\) such that \(\overline{AB}\) and \(\overline{CD}\) intersect at one point, and \(\overline{BD}\) intersects one of them at a different point.

For Exercises 8:
Draw two planes \(S\) and \(T\) intersecting at a right - angle. Draw a line segment \(\overline{AB}\) that lies on both planes \(S\) and \(T\), and mark point \(C\) as the mid - point of \(\overline{AB}\).

For Exercises 9 - 12:
9. Planes \(W\) and \(V\) intersect at \(\overline{TU}\). So the statement is True.
10. Points \(T\), \(U\), and \(R\) are not coplanar. We can see that \(R\) is not in the same plane as \(T\) and \(U\) (assuming the standard interpretation of the 3 - D diagram), so the statement is False.
11. \(\angle TZX\) and \(\angle UZY\) are vertical angles. Vertical angles are formed by two intersecting lines. Here, if we consider the lines that form these angles (implicit in the intersection of elements in the diagram), they are vertical angles, so the statement is True.
12. \(\overline{TU}\) lies in plane \(W\). From the diagram, we can see that \(\overline{TU}\) is part of the intersection of planes and does not lie entirely in plane \(W\), so the statement is False.

13. Yes, it is possible for three planes to intersect along the same line. Imagine three pages of a book intersecting along the spine. The spine represents the line of intersection of the three planes (the pages).
14. Your friend is incorrect. The Plane - Line Postulate only states that if a line intersects a plane not containing it, the intersection is a point. A line that passes through a point in a plane can pass through the plane and extend out of it. For example, a flagpole passing through a point on the ground (the plane) extends into the air and is not entirely in the ground - plane.
15. If \(\overline{EF}\) bisects either \(\overline{AB}\) or \(\overline{CD}\), \(\overline{EF}\) does not always lie in plane \(Z\). If \(\overline{EF}\) bisects only one of them, it can be in a different plane. If \(\overline{EF}\) bisects both \(\overline{AB}\) and \(\overline{CD}\), and \(\overline{AB}\) and \(\overline{CD}\) lie in plane \(Z\), and the mid - points of \(\overline{AB}\) and \(\overline{CD}\) are used to define \(\overline{EF}\), then \(\overline{EF}\) lies in plane \(Z\) because two points (the mid - points) in plane \(Z\) determine a line in plane \(Z\).

Answer:

1. Through points \(A\) and \(B\), there is exactly one line \(\overleftrightarrow{AB}\).
2. Line \(\overleftrightarrow{TU}\) contains points \(T\) and \(U\).
3. Conceptually, if two lines in the planes intersect, their intersection is a single point.
4. Points \(A\), \(B\), and \(D\) (assuming non - collinear) determine a plane.
5. If a line like \(\overleftrightarrow{TU}\) intersects a plane (say plane \(A\)), the intersection is a point.
6. The intersection of the two planes shown is a line.
7. Sketch three lines intersecting at two points as described.
8. Sketch two planes intersecting at a right - angle with a line segment on both planes and its mid - point marked.
9. True
10. False
11. True
12. False
13. Yes, like the spine of a three - page book.
14. No, a line can pass through a point in a plane and extend out of it.
15. If it bisects only one, no; if it bisects both, yes.