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 midpoint 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: Recall postulate definitions

The Two - Point Postulate states that through any two points, there is exactly one line. In the given diagram, for points $D$ and $E$, there is a line passing through them.

Step2: Recall Line - Point Postulate

The Line - Point Postulate: A line contains at least two points. For example, line $\overleftrightarrow{EF}$ contains points $E$ and $F$.

Step3: Recall Line Intersection Postulate

The Line Intersection Postulate: If two lines intersect, then their intersection is exactly one point. In the diagram, lines (if we consider the lines formed by the edges of the planes' intersections) may intersect at a single point like point $J$.

Step4: Recall Three - Point Postulate

The Three - Point Postulate: Through any three non - collinear points, there is exactly one plane. Points $D$, $E$, and $F$ (assuming they are non - collinear) determine a plane.

Step5: Recall Plane - Line Postulate

The Plane - Line Postulate: If two points lie in a plane, then the line containing them lies in the plane. If points $A$ and $B$ lie in plane $A$, then line $\overleftrightarrow{AB}$ lies in plane $A$.

Step6: Recall Plane Intersection Postulate

The Plane Intersection Postulate: If two planes intersect, then their intersection is a line. Planes in the diagram may intersect along a line such as the line formed by the intersection of the two large planes.

Step7: Sketch for exercise 7

Draw three lines $\overline{AB}$, $\overline{CD}$, and $\overline{BD}$ such that two of the lines intersect at one point and one of the lines intersects another at a different point. For example, have $\overline{AB}$ and $\overline{BD}$ intersect at point $B$ and $\overline{CD}$ intersect $\overline{AB}$ at a non - $B$ point.

Step8: Sketch for exercise 8

Draw two planes $S$ and $T$ intersecting at a right - angle. Draw a line $\overline{AB}$ that lies on both planes (it is on the intersection line of the two planes) and mark point $C$ as the mid - point of $\overline{AB}$.

Step9: Analyze statement 9

For planes $W$ and $V$, if the diagram shows that their intersection is the line $\overline{TU}$, then we can assume the statement. If the intersection line in the diagram is $\overline{TU}$, the answer is yes.

Step10: Analyze statement 10

Points $T$, $U$, and $R$: If we can see from the diagram that they all lie on the same plane (a flat surface that can be extended), then they are coplanar. If they appear to be on the same surface, the answer is yes.

Step11: Analyze statement 11

$\angle TZX$ and $\angle UZY$: Vertical angles are formed by two intersecting lines. If the lines forming these angles intersect at a point and the angles are opposite each other, then they are vertical angles. If the diagram shows this intersection property, the answer is yes.

Step12: Analyze statement 12

For $\overline{TU}$ and plane $W$, if the line $\overline{TU}$ is drawn such that all its points lie within the boundaries of plane $W$ (or can be extended to lie within the plane), then $\overline{TU}$ lies in plane $W$. If the diagram indicates this, the answer is yes.

Step13: Answer question 13

Yes, it is possible. Consider three pages of a book that meet at the spine. The spine represents the line of intersection of the three planes (the pages).

Step14: Answer question 14

No, the friend is incorrect. The Plane - Line Postulate states that if two points of a line lie in a plane, then the line lies in the plane. Just because a line passes through a single point in a plane does not mean the entire line lies in the plane. A line can pass through a plane at a single point and extend out of the plane.

Step15: Answer question 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 the lines, it could be a non - coplanar line that just intersects the bisected line at the mid - point. If $\overline{EF}$ bisects both $\overline{AB}$ and $\overline{CD}$, since $\overline{AB}$ and $\overline{CD}$ lie in plane $Z$ and the mid - points of $\overline{AB}$ and $\overline{CD}$ are in plane $Z$, and two points (the mid - points) of $\overline{EF}$ lie in plane $Z$, then $\overline{EF}$ lies in plane $Z$ according to the Plane - Line Postulate.

Answer:

1. For example, for points $D$ and $E$, there is a line passing through them.
2. Line $\overleftrightarrow{EF}$ contains points $E$ and $F$.
3. Lines may intersect at a single point like point $J$.
4. Points $D$, $E$, and $F$ (assuming non - collinear) determine a plane.
5. If points $A$ and $B$ lie in plane $A$, then line $\overleftrightarrow{AB}$ lies in plane $A$.
6. Planes may intersect along a line.
7. Sketch three lines intersecting at two points as described.
8. Sketch two planes intersecting at a right - angle with the given line and mid - point.
9. Yes/No depending on the diagram.
10. Yes/No depending on the diagram.
11. Yes/No depending on the diagram.
12. Yes/No depending on the diagram.
13. Yes, example: three pages meeting at the spine.
14. No, explanation: a line through a single point in a plane may not be in the plane.
15. If bisects one, not always; if bisects both, yes.