Question

1. write and solve an equation to find $m\angle7$. show all work.$\angle4 = 6x - 15$ and $\angle7 = 3x + 33$10. if $m\angle2 = 40$, find the measure of all the other labeled angles.$m\angle1 = \underline{\quad\quad}$$m\angle2 = \underline{\quad\quad}$$m\angle3 = \underline{\quad\quad}$$m\angle4 = \underline{\quad\quad}$$m\angle5 = \underline{\quad\quad}$$m\angle6 = \underline{\quad\quad}$$m\angle7 = \underline{\quad\quad}$$m\angle8 = \underline{\quad\quad}$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. $f(x,y) \to f(-x,y)$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)$12. write and solve an equation to find $x$. show all work.13. find the values of $x$ and $y$. show all work.

Explanation:

Problem 9

Step1: Set angles as supplementary

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

Step2: Substitute angle expressions

$(6x - 15) + (3x + 33) = 180$

Step3: Simplify and solve for $x$

$9x + 18 = 180$
$9x = 162$
$x = 18$

Step4: Calculate $m\angle7$

$m\angle7 = 3(18) + 33 = 54 + 33$

Step1: Identify vertical/linear pairs

$m\angle2 = 40^\circ$, so $m\angle4 = 40^\circ$, $m\angle1 = 180-40=140^\circ$, $m\angle3=140^\circ$

Step2: Corresponding angles for parallel lines

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

a. The figure is reflected over the x-axis (flips vertically), which maps $(x,y)$ to $(x,-y)$.
b. i. Segments connecting pre-image and image points in a reflection are parallel: $DD' \parallel EE'$ is true.
ii. Reflections preserve segment length: $DD' = EE'$ is true.
iii. The segment from $E$ to $(0,0)$ and $E'$ to $(0,0)$ are not parallel (they form a V shape).
iv. Reflections preserve distance from a point: length of $E$ to $(0,0)$ equals $E'$ to $(0,0)$ is true.
v. The segments $E$ to $(0,0)$ and $E'$ to $(0,0)$ are not perpendicular.

Answer:

$m\angle7 = 87^\circ$

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Problem 10