Question

1. (mangle fcd=x + 41), (mangle bcf=x + 78), and (mangle bcd = 95^{circ}). find (x). 2. (mangle hgf=16x + 4), (mangle egf = 110^{circ}), and (mangle hge=3x + 11). find (x). 3. (z) is on the interior of (angle efg). (mangle gfz = 38^{circ}), (mangle zfe=2x + 125), (mangle gfe=x + 163). find (x). 4. (overrightarrow{ba}) bisects (angle dbc). (mangle dbc = 135^{circ}), (mangle dba=6x). find (x).

Explanation:

1.

Step1: Use angle - addition postulate

Since $\angle BCD=\angle BCF+\angle FCD$, we have the equation $(x + 78)+(x + 41)=95$.

Step2: Combine like - terms

$2x+119 = 95$.

Step3: Isolate the variable term

Subtract 119 from both sides: $2x=95 - 119=-24$.

Step4: Solve for x

Divide both sides by 2: $x=-12$.

Step1: Use angle - addition postulate

Since $\angle HGF=\angle HGE+\angle EGF$, we have the equation $16x + 4=3x+11 + 110$.

Step2: Combine like - terms

$16x+4=3x + 121$.

Step3: Move variable terms to one side

Subtract 3x from both sides: $16x-3x+4=3x-3x + 121$, which simplifies to $13x+4=121$.

Step4: Move constant term to the other side

Subtract 4 from both sides: $13x=121 - 4 = 117$.

Step5: Solve for x

Divide both sides by 13: $x = 9$.

Step1: Use angle - addition postulate

Since $\angle GFE=\angle GFZ+\angle ZFE$, we have the equation $x + 163=38+(2x + 125)$.

Step2: Simplify the right - hand side

$x + 163=2x+163$.

Step3: Move variable terms to one side

Subtract x from both sides: $x-x + 163=2x-x+163$, which gives $0=x$.

Answer:

$x=-12$

2.