Question

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(student) fl geometry > building relationships in polygons with circles > interior and exterior angles of triangles > 13.5.3 activity
(student) fl geometry > building relationships in polygons with circles > interior and exterior angles of triangles > 13.5.4 guided practice
then, answer \check your understanding\ assignment below. you have three attempts.
$\triangle lwg$ is shown, where $\angle ylg$, $\angle twl$, and $\angle kgw$ are exterior angles. $m\angle lwg = 50^\circ$, $m\angle lgw = (5x + 12)^\circ$ and $m\angle ylg = (12x + 6)^\circ$
image of triangle lwg with exterior angles at each vertex: angle ylg at l is $(12x + 6)^\circ$, angle twl at w is $50^\circ$, angle kgw at g is $(5x + 12)^\circ$
what is the measure of $\angle kgw$?

Explanation:

Step1: Use exterior angle sum theorem

The sum of a triangle's exterior angles is $360^\circ$. So:
$$(12x+6) + (5x+12) + m\angle TWL = 360^\circ$$

Step2: Find $m\angle TWL$

$\angle TWL$ is supplementary to $\angle LWG$:
$$m\angle TWL = 180^\circ - 50^\circ = 130^\circ$$

Step3: Substitute and solve for $x$

Substitute $m\angle TWL=130^\circ$ into the equation:
$$(12x+6)+(5x+12)+130=360$$
Combine like terms:
$$17x + 148 = 360$$
$$17x = 360 - 148 = 212$$
$$x = \frac{212}{17} = 12.47$$

Step4: Calculate $m\angle KGW$

$\angle KGW$ is supplementary to $\angle LGW$:
$$m\angle KGW = 180^\circ - (5x+12)^\circ$$
Substitute $x=\frac{212}{17}$:
$$5x+12 = 5\times\frac{212}{17}+12 = \frac{1060}{17}+\frac{204}{17}=\frac{1264}{17}\approx74.35^\circ$$
$$m\angle KGW = 180 - \frac{1264}{17} = \frac{3060-1264}{17}=\frac{1796}{17}\approx105.65^\circ$$
*Note: Using integer check, correcting to $m\angle TWL$ is the exterior angle, so direct exterior angle sum: $(12x+6)+(5x+12)+(180-50)=360$ simplifies to $17x=360-148=212$, $x=12.47$. Alternatively, using remote interior angles for $\angle YLG$: $12x+6=50+(180-(5x+12))$, solve:
$$12x+6=50+168-5x$$
$$17x=212$$
$$x=\frac{212}{17}$$

Step5: Final calculation for $\angle KGW$

$$m\angle KGW=180-(5\times\frac{212}{17}+12)=180-\frac{1060+204}{17}=180-\frac{1264}{17}=\frac{3060-1264}{17}=\frac{1796}{17}\approx105.65^\circ$$

Answer:

The measure of $\angle KGW$ is $\frac{1796}{17}^\circ$ (or approximately $105.7^\circ$)