Question

in the diagram, ( mangle fli ) is ( 106^{circ} ), ( mangle flg=(2x - 1)^{circ} ), ( mangle glh=(x + 17)^{circ} ), and ( mangle hli=(4x - 15)^{circ} ). what is the measure of the smallest angle in the diagram? ( 15^{circ} ) ( 29^{circ} ) ( 32^{circ} ) ( 45^{circ} )

Explanation:

Step1: Set up the equation

Since \(m\angle FLI=m\angle FLG + m\angle GLH+m\angle HLI\), we substitute the given expressions:
\(106=(2x - 1)+(x + 17)+(4x-15)\)

Step2: Simplify the right - hand side

Combine like terms: \((2x+x + 4x)+(-1 + 17-15)=7x + 1\)
So the equation becomes \(7x+1 = 106\)

Step3: Solve for \(x\)

Subtract 1 from both sides: \(7x=106 - 1=105\)
Divide both sides by 7: \(x=\frac{105}{7}=15\)

Step4: Find the measures of the angles

• For \(\angle FLG\): \(m\angle FLG=(2x - 1)^{\circ}\), substitute \(x = 15\), \(m\angle FLG=(2\times15-1)^{\circ}=29^{\circ}\)
• For \(\angle GLH\): \(m\angle GLH=(x + 17)^{\circ}\), substitute \(x = 15\), \(m\angle GLH=(15 + 17)^{\circ}=32^{\circ}\)
• For \(\angle HLI\): \(m\angle HLI=(4x-15)^{\circ}\), substitute \(x = 15\), \(m\angle HLI=(4\times15-15)^{\circ}=45^{\circ}\)

Answer:

\(29^{\circ}\)