Question

in the triangle, suppose that ( mangle k=(x + 4)^{circ},mangle l=(5x + 8)^{circ} ), and ( mangle m=(6x)^{circ} ).
(a) write an equation to find ( x ). make sure you use an \=\ sign in your answer.
equation:
(b) find the degree measure of each angle.
( mangle k=) (quad^{circ})
( mangle l=) (quad^{circ})
( mangle m=) (quad^{circ})

Explanation:

Step1: Use the triangle - angle sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). So, \(m\angle K + m\angle L+m\angle M=180^{\circ}\).
Substitute \(m\angle K=(x + 4)^{\circ}\), \(m\angle L=(5x + 8)^{\circ}\), and \(m\angle M=(6x)^{\circ}\) into the equation:
\((x + 4)+(5x + 8)+6x=180\)

Step2: Simplify the left - hand side of the equation

Combine like terms: \((x+5x + 6x)+(4 + 8)=180\), which gives \(12x+12 = 180\)

Step3: Solve for \(x\)

Subtract 12 from both sides: \(12x=180 - 12\), so \(12x=168\).
Divide both sides by 12: \(x=\frac{168}{12}=14\)

Step4: Find \(m\angle K\)

Substitute \(x = 14\) into \(m\angle K=(x + 4)^{\circ}\). Then \(m\angle K=(14 + 4)^{\circ}=18^{\circ}\)

Step5: Find \(m\angle L\)

Substitute \(x = 14\) into \(m\angle L=(5x + 8)^{\circ}\). Then \(m\angle L=(5\times14+8)^{\circ}=(70 + 8)^{\circ}=78^{\circ}\)

Step6: Find \(m\angle M\)

Substitute \(x = 14\) into \(m\angle M=(6x)^{\circ}\). Then \(m\angle M=(6\times14)^{\circ}=84^{\circ}\)

Answer:

(a) Equation: \(12x + 12=180\)
(b) \(m\angle K = 18^{\circ}\), \(m\angle L = 78^{\circ}\), \(m\angle M = 84^{\circ}\)