Question

given $overline{ab}paralleloverline{dc}$, $mangle1 = 112$, $mangle2 = 4x$, and $mangle3 = 3x + 12$, find the measure of each angle below.
(c)15. $angle2$ (16)

1. $angle3$ (16)

(d)17. $angle d$ (24)
answer each question below.

1. if $l - m - n$, and if $lm = 2x$, $mn = 3x - 1$, and $ln = 24$, then find $mn$. (8)

Explanation:

Step1: Use property of alternate - interior angles

Since $\overline{AB}\parallel\overline{DC}$, $\angle1$ and $\angle D$ are supplementary (same - side interior angles), and $\angle2=\angle3$ (alternate - interior angles). So we set up the equation $4x = 3x + 12$.
$4x-3x=3x + 12-3x$
$x = 12$

Step2: Find the measure of $\angle2$

Substitute $x = 12$ into the expression for $\angle2$. Since $m\angle2 = 4x$, then $m\angle2=4\times12 = 48$.

Step3: Find the measure of $\angle3$

Substitute $x = 12$ into the expression for $\angle3$. Since $m\angle3 = 3x + 12$, then $m\angle3=3\times12 + 12=36 + 12=48$.

Step4: Find the measure of $\angle D$

Since $\angle1$ and $\angle D$ are supplementary ( $\angle1+\angle D = 180^{\circ}$) and $m\angle1 = 112^{\circ}$, then $m\angle D=180 - 112=68$.

Step5: Solve for $x$ in the second part

Given $LM = 2x$, $MN = 3x - 1$, and $LN=LM + MN = 24$. So we set up the equation $2x+(3x - 1)=24$.
$2x+3x-1 = 24$
$5x-1=24$
$5x=24 + 1$
$5x=25$
$x = 5$

Step6: Find the measure of $MN$

Substitute $x = 5$ into the expression for $MN$. Since $MN = 3x - 1$, then $MN=3\times5-1=15 - 1 = 14$.

Answer:

1. $m\angle2 = 48$
2. $m\angle3 = 48$
3. $m\angle D = 68$
4. $MN = 14$