Question

sc 2: i can use angle addition and line - segment addition to analyze line segments, adjacent angles, and linear pairs of angles.
2a) every point in this diagram is on the same line. $overline{vz}$ is 52 units long. $overline{xz}$ is 20 units long. $overline{wx}$, $overline{xy}$, and $overline{yz}$ have the same length. what are the lengths of these three line segments? $overline{wx}$, $overline{vw}$, and $overline{vy}$
2b) the length of $overline{tg}$ is $4x + 8$. find the length of $overline{jg}$. (your answer will not have “x” in it.)
2c) find the measure of $angle abc$. (your answer will not have “x” in it.)
2d) m is the mid - point of $overline{jl}$. what is the length of $overline{jm}$? (your answer will not have “x” in it.)
sc 3: i can find mid - points of line segments and use mid - points when analyzing geometric figures.
3a) point g is (-4, 4); point h is (6, 4). what is the mid - point of $overline{gh}$?
3b) the mid - point of $overline{er}$ is (1, 4). one endpoint of $overline{er}$ is (5, 1). what is the other endpoint of $overline{er}$?

1. not online. you can create 3 line segments from 3 points on a plane. you can create 6 line segments from 3 points on a plane. fill in this table. attach scratch paper to show evidence of your thinking.

Explanation:

2a)

Step1: Find length of $\overline{WZ}$

Since $\overline{VZ}=52$ units and $\overline{XZ} = 20$ units, then $\overline{WZ}=\overline{VZ}-\overline{XZ}$. So $\overline{WZ}=52 - 20=32$ units.

Step2: Find length of $\overline{WX},\overline{XY},\overline{YZ}$

Since $\overline{WX},\overline{XY},\overline{YZ}$ have the same length and $\overline{WZ}=\overline{WX}+\overline{XY}+\overline{YZ}$, let the length of each be $l$. Then $3l=\overline{WZ}=32$, so $l=\frac{32}{3}$ units.

Step3: Find length of $\overline{VW}$

$\overline{VW}=\overline{VZ}-\overline{WZ}=52 - 32 = 20$ units.

Step4: Find length of $\overline{VY}$

$\overline{VY}=\overline{VW}+\overline{WY}$, and $\overline{WY}=2\times\frac{32}{3}=\frac{64}{3}$ units, so $\overline{VY}=20+\frac{64}{3}=\frac{60 + 64}{3}=\frac{124}{3}$ units.
$\overline{WX}=\frac{32}{3}$ units, $\overline{VW}=20$ units, $\overline{VY}=\frac{124}{3}$ units

2b)

Step1: Set up an equation using segment - addition

Since $\overline{TG}=\overline{TJ}+\overline{JG}$ and $\overline{TG}=4x + 8$, $\overline{TJ}=x + 2$, $\overline{JG}=6x-9$, we have $4x + 8=(x + 2)+(6x-9)$.

Step2: Solve the equation for $x$

$4x+8=x + 2+6x-9$;
$4x+8=7x-7$;
$8 + 7=7x-4x$;
$15 = 3x$;
$x = 5$.

Step3: Find the length of $\overline{JG}$

Substitute $x = 5$ into the expression for $\overline{JG}$: $\overline{JG}=6x-9=6\times5-9=30 - 9=21$.

Step1: Use the angle - addition postulate

Since $\angle ABC$ and $\angle ABD$ form a linear pair, $\angle ABC+\angle ABD = 180^{\circ}$, and $\angle ABD=(3x)^{\circ}+(15x + 18)^{\circ}=(18x + 18)^{\circ}$.
So $18x+18=180$.

Step2: Solve for $x$

$18x=180 - 18=162$;
$x = 9$.

Step3: Find the measure of $\angle ABC$

$\angle ABC=180-(15x + 18)$; substitute $x = 9$: $\angle ABC=180-(15\times9+18)=180-(135 + 18)=180 - 153=27^{\circ}$.

Answer:

21

2c)