Question

6.2 - angle bisectors - spiral review
find each measure.

1. $m\angle abf$
2. $m\angle yba$
3. $mk$
4. $m\angle ewl$
5. $iu$
6. $m\angle myw$

(the corresponding geometric figures with angle markings and algebraic expressions: $43^\circ$, $47^\circ$, $2x + 1$, $3x - 3$, $(7x + 5)^\circ$, $(3x + 21)^\circ$, $19^\circ$, $2x + 5$, $7x$, $(4x - 1)^\circ$, $(2x + 5)^\circ$ are present in the figures)

Explanation:

Step1: Solve for $\boldsymbol{m\angle ABF}$

$AC \parallel EF$, $BF$ bisects $\angle ABC$.
$\angle ABC = 43^\circ$, so $\angle ABF = \frac{1}{2}\angle ABC$
$\angle ABF = \frac{1}{2} \times 43^\circ = 21.5^\circ$

Step2: Solve for $\boldsymbol{m\angle YBA}$

$YB \parallel AF$, $\angle ADF = 47^\circ$, so $\angle YAB = \angle ADF = 47^\circ$.
$\angle YBA = 90^\circ - \angle YAB$
$\angle YBA = 90^\circ - 47^\circ = 43^\circ$

Step3: Solve for $\boldsymbol{x}$ (Question 3)

$KM$ and $KT$ are angle bisectors, so $2x+1=3x-8$
$3x-2x=1+8$
$x=9$

Step4: Solve for $\boldsymbol{m\angle EWL}$

$WL$ is an angle bisector, so $7x+5=3x+21$
$7x-3x=21-5$
$4x=16 \implies x=4$
$m\angle EWL = 7x+5 = 7(4)+5=33^\circ$

Step5: Solve for $\boldsymbol{IU}$

$RU$ is an angle bisector, so $2x+5=7x$
$7x-2x=5$
$5x=5 \implies x=1$
$IU=7x=7(1)=7$

Step6: Solve for $\boldsymbol{m\angle MYW}$

$YM$ is an angle bisector, so $4x-1=2x+5$
$4x-2x=5+1$
$2x=6 \implies x=3$
$m\angle MYW = 2x+5 = 2(3)+5=11^\circ$

Answer:

1. $21.5^\circ$
2. $43^\circ$
3. $9$
4. $33^\circ$
5. $7$
6. $11^\circ$