Question

quiz 1 - 1: points, lines, planes; segment addition postulate
use the figure below for questions 1 - 4.

1. give another name for line r.
2. name the intersection of lines r and s.
3. name three collinear points.
4. give another name for plane n.

use the figure below for questions 5 - 8.

1. name a point coplanar to point k.
2. name the intersection of plane r and line jl.
3. name three non - collinear points.
4. give another name for line jk.

use the figure below for questions 9 - 12.

1. if df = 61 and ef = 18, find de.
2. if de = 4x - 1, ef = 9, and df = 9x - 22, find the value of x.
3. if df = 78, de = 5x - 9, and ef = 2x + 10, find ef.
4. if de = 4x + 10, ef = 2x - 1, and df = 9x - 15, find df.

Explanation:

Step1: Recall line - naming convention

A line can be named using any two points on it. For line $r$, another name is line $AB$.

Step2: Identify intersection point

The intersection of two lines is the point where they meet. Lines $r$ and $s$ intersect at point $B$.

Step3: Recall collinear - points definition

Collinear points lie on the same line. Points $A$, $B$, and $C$ are collinear.

Step4: Recall plane - naming convention

A plane can be named using three non - collinear points on it. Another name for plane $N$ is plane $ABC$.

Step5: Recall coplanar - points definition

Coplanar points lie on the same plane. Point $M$ is coplanar to point $K$ (since they both lie on plane $R$).

Step6: Identify intersection

The intersection of a plane and a line is a point. The intersection of plane $R$ and line $JL$ is point $K$.

Step7: Recall non - collinear points

Non - collinear points do not lie on the same line. Points $J$, $K$, and $M$ are non - collinear.

Step8: Recall line - naming convention

Another name for line $JK$ is line $KJ$.

Step9: Use segment addition postulate

By the segment addition postulate $DF=DE + EF$. So, $DE=DF - EF$. Given $DF = 61$ and $EF = 18$, then $DE=61 - 18=43$.

Step10: Apply segment addition postulate

Since $DF=DE + EF$, we have $9x-22=(4x - 1)+9$.
First, simplify the right - hand side: $(4x - 1)+9=4x+8$.
Then, solve the equation $9x-22 = 4x+8$.
Subtract $4x$ from both sides: $9x-4x-22=4x-4x + 8$, which gives $5x-22 = 8$.
Add 22 to both sides: $5x-22 + 22=8 + 22$, so $5x=30$.
Divide both sides by 5: $x = 6$.

Step11: Apply segment addition postulate

Since $DF=DE + EF$, we have $78=(5x - 9)+(2x + 10)$.
Simplify the right - hand side: $(5x - 9)+(2x + 10)=7x + 1$.
Then, solve the equation $7x+1 = 78$.
Subtract 1 from both sides: $7x+1-1=78 - 1$, so $7x=77$.
Divide both sides by 7: $x = 11$.
Substitute $x = 11$ into the expression for $EF$: $EF=2x + 10=2\times11+10=22 + 10=32$.

Step12: Apply segment addition postulate

Since $DF=DE + EF$, we have $9x-15=(4x + 10)+(2x - 1)$.
Simplify the right - hand side: $(4x + 10)+(2x - 1)=6x + 9$.
Then, solve the equation $9x-15=6x + 9$.
Subtract $6x$ from both sides: $9x-6x-15=6x-6x + 9$, so $3x-15 = 9$.
Add 15 to both sides: $3x-15 + 15=9 + 15$, so $3x=24$.
Divide both sides by 3: $x = 8$.
Substitute $x = 8$ into the expression for $DF$: $DF=9x-15=9\times8-15=72-15=57$.

Answer:

1. Line $AB$
2. Point $B$
3. Points $A$, $B$, $C$
4. Plane $ABC$
5. Point $M$
6. Point $K$
7. Points $J$, $K$, $M$
8. Line $KJ$
9. $43$
10. $6$
11. $32$
12. $57$