Question

in the figure below, ( mangle 1 = 6x^circ ) and ( mangle 2 = (x - 23)^circ ). find the angle measures.

Explanation:

Step1: Identify supplementary angles

$\angle 1$ and $\angle 2$ form a straight line, so they are supplementary:
$$m\angle 1 + m\angle 2 = 180^\circ$$

Step2: Substitute angle expressions

Replace with given algebraic expressions:
$$6x + (x - 23) = 180$$

Step3: Simplify and solve for $x$

Combine like terms and isolate $x$:
$$7x - 23 = 180$$
$$7x = 180 + 23 = 203$$
$$x = \frac{203}{7} = 29$$

Step4: Calculate $m\angle 1$

Substitute $x=29$ into $6x$:
$$m\angle 1 = 6 \times 29 = 174$$

Step5: Calculate $m\angle 2$

Substitute $x=29$ into $x-23$:
$$m\angle 2 = 29 - 23 = 6$$

Answer:

$m\angle 1 = 174^\circ$
$m\angle 2 = 6^\circ$