Question

1. in \\( \triangle qrs, \overline{qs} \\) is extended through point \\( s \\) to point \\( t \\), \\( m\angle rst=(6x + 7)^{circ}, m\angle qrs=(x - 3)^{circ}, \\) and \\( m\angle sqr=(3x + 20)^{circ} \\). find \\( m\angle sqr \\).

Explanation:

Problem 1 (Top Triangle):

Step1: Find adjacent interior angle

The exterior angle (141°) and adjacent interior angle are supplementary:
$180^\circ - 141^\circ = 39^\circ$

Step2: Calculate $x$ using right triangle angles

Sum of angles in right triangle is 180°:
$x + 90^\circ + 39^\circ = 180^\circ$
$x = 180^\circ - 129^\circ = 51^\circ$

Step1: Apply exterior angle theorem

Exterior angle = sum of remote interior angles:
$6x + 7 = (x - 3) + (3x + 20)$

Step2: Solve for $x$

Simplify equation:
$6x + 7 = 4x + 17$
$2x = 10$
$x = 5$

Step3: Calculate $m\angle SQR$

Substitute $x=5$ into $(3x + 20)$:
$3(5) + 20 = 15 + 20 = 35^\circ$

Answer:

51

Problem 13 (ΔQRS):