Question

1. (mangle abc = 17x + 8), (mangle abk = 42^{circ}), and (mangle kbc = 12x - 4). find (mangle abc).

Explanation:

Step1: Use angle - addition postulate

Since $\angle ABC=\angle ABK+\angle KBC$, we have the equation $17x + 8=42+(12x - 4)$.

Step2: Simplify the right - hand side of the equation

$42+(12x - 4)=12x+38$. So the equation becomes $17x + 8=12x+38$.

Step3: Solve for $x$

Subtract $12x$ from both sides: $17x-12x + 8=12x-12x+38$, which simplifies to $5x+8 = 38$. Then subtract 8 from both sides: $5x+8 - 8=38 - 8$, getting $5x=30$. Divide both sides by 5: $x=\frac{30}{5}=6$.

Step4: Find $m\angle ABC$

Substitute $x = 6$ into the expression for $m\angle ABC$. $m\angle ABC=17x + 8=17\times6+8=102 + 8=110^{\circ}$.

Answer:

$110^{\circ}$