Question

1. write and solve an equation to find $m\angle7$. show all work.$\angle4 = 9x - 15$ and $\angle7 = 3x + 33$10. if $m\angle2 = 40$, find the measure of all the other labeled angles.$m\angle1 = \\_\\_\\_\\_\\_$$m\angle5 = \\_\\_\\_\\_\\_$$m\angle2 = \\_\\_\\_\\_\\_$$m\angle6 = \\_\\_\\_\\_\\_$$m\angle3 = \\_\\_\\_\\_\\_$$m\angle7 = \\_\\_\\_\\_\\_$$m\angle4 = \\_\\_\\_\\_\\_$$m\angle8 = \\_\\_\\_\\_\\_$11. given the transformation below of $abcde$ to its image $abcde$.a. which of the following rules expresses the transformation?i. $f(x,y) \to f(x,-y)$ii. $f(x,y) \to f(-x,-y)$iii. $f(x,y) \to f(-y,x)$iv. $f(x,y) \to f(-y,-x)$b. select all the statements that must be true.i. segment $dd$ is parallel to $ee$ii. segment $dd$ is the same length as $ee$iii. the segment from $e$ to $(0,0)$ is parallel to the segment from $e$ to $(0,0)$iv. the segment from $e$ to $(0,0)$ is the same length as the segment from $e$ to $(0,0)$v. the segment from $e$ to $(0,0)$ is perpendicular to the line from $e$ to $(0,0)$

Explanation:

Problem 9

Step1: Identify angle relationship

$\angle 4 + \angle 7 = 180^\circ$

Step2: Substitute angle expressions

$(9x - 15) + (3x + 33) = 180$

Step3: Simplify and solve for $x$

$12x + 18 = 180$
$12x = 162$
$x = \frac{162}{12} = 13.5$

Step4: Calculate $m\angle7$

$m\angle7 = 3(13.5) + 33$

Step1: Use linear pair for $\angle1$

$\angle1 + \angle2 = 180^\circ$
$\angle1 = 180 - 40 = 140^\circ$

Step2: Vertical angle for $\angle3$

$\angle3 = \angle1 = 140^\circ$

Step3: Vertical angle for $\angle4$

$\angle4 = \angle2 = 40^\circ$

Step4: Corresponding angles for right set

$\angle5 = \angle1 = 140^\circ$, $\angle6 = \angle2 = 40^\circ$, $\angle7 = \angle3 = 140^\circ$, $\angle8 = \angle4 = 40^\circ$

Take a vertex of $ABCDE$, e.g., $A(2,1)$. Its image $A'(1,-2)$. Test the rules:

• i. $(2,-1)

eq (1,-2)$

• ii. $(-2,-1)

eq (1,-2)$

• iii. $(-1,2)

eq (1,-2)$

• iv. $(-1,-2) = (1,-2)$ (matches). This rule represents a 270° clockwise rotation (or 90° counterclockwise rotation) about the origin.

Answer:

$m\angle7 = 73.5^\circ$

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Problem 10