Question

1. in $\triangle def$, $angle d cong angle f$. the measure of $angle d$ is nine more than nine times a number. the measure of $angle f$ is twelve less than twelve times the number. find the measures of $angle d$, $angle e$, and $angle f$.

1. in the diagram, $\triangle pqr$ is the image of $\triangle pqr$ under a dilation with center $o$ and scale factor $\frac{2}{3}$. find the length of $overline{pq}$.

Explanation:

For Problem 4:

Step1: Define variable & equations

Let the number be $n$.
$m\angle D = 9n + 9$, $m\angle F = 12n - 12$
Since $\angle D \cong \angle F$, set equal:
$$9n + 9 = 12n - 12$$

Step2: Solve for $n$

Rearrange terms:
$$9 + 12 = 12n - 9n$$
$$21 = 3n$$
$$n = 7$$

Step3: Calculate $\angle D$ and $\angle F$

Substitute $n=7$:
$m\angle D = 9(7) + 9 = 72$
$m\angle F = 12(7) - 12 = 72$

Step4: Calculate $\angle E$

Use triangle angle sum ($180^\circ$):
$$m\angle E = 180 - 72 - 72 = 36$$

Step1: Set up dilation ratio

Dilation scale factor $\frac{2}{3}$ means:
$$\frac{P'Q'}{PQ} = \frac{2}{3}$$
Substitute $PQ = x+1$, $P'Q' = \frac{8}{3}$:
$$\frac{\frac{8}{3}}{x+1} = \frac{2}{3}$$

Step2: Solve for $x$

Cross-multiply:
$$2(x+1) = 8$$
$$x+1 = 4$$
$$x = 3$$

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

Substitute $x=3$:
$$PQ = 3 + 1 = 4$$

Answer:

$m\angle D = 72^\circ$, $m\angle E = 36^\circ$, $m\angle F = 72^\circ$

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For Problem 5: