Question

find $m\angle qrs$, if $m\angle qrd = 5x - 10$, $m\angle drs = 110$ deg, and $m\angle qrs = 14x - 8$.

options:
128 degrees
12 degrees
112 degrees
160 degrees

Explanation:

Step1: Set up angle sum equation

$m\angle QRD + m\angle QRS = m\angle DRS$
Substitute given values:
$5x - 10 + 14x - 8 = 110$

Step2: Simplify and solve for $x$

Combine like terms:
$19x - 18 = 110$
Add 18 to both sides:
$19x = 128$
Solve for $x$:
$x = \frac{128}{19} \approx 6.7368$

Step3: Calculate $m\angle QRS$

Substitute $x$ into $14x - 8$:
$m\angle QRS = 14\times\frac{128}{19} - 8 = \frac{1792}{19} - \frac{152}{19} = \frac{1640}{19} = 86.315...$
*Note: Correcting based on angle composition (QRD + QRS = DRS is incorrect; correct relation: $m\angle DRS = m\angle QRD + m\angle QRS$ was misapplied. Correct relation: $m\angle QRS = m\angle DRS + m\angle QRD$
Revised Step1: Correct angle equation
$m\angle QRS = m\angle QRD + m\angle DRS$
$14x - 8 = (5x - 10) + 110$

Revised Step2: Solve for $x$
$14x - 8 = 5x + 100$
$14x - 5x = 100 + 8$
$9x = 108$
$x = 12$

Revised Step3: Find $m\angle QRS$
$m\angle QRS = 14(12) - 8 = 168 - 8 = 160$

Answer:

160 degrees