Question

in △jkl, if (overline{jk}congoverline{jl}), (mangle j=(23x - 4)^{circ}), (mangle k=(4x - 1)^{circ}), and (mangle l=(9x - 31)^{circ}), find (x) and the measure of each angle.
(x=)
(mangle j=)
(mangle k=)
(mangle l=)
qs = 36

Explanation:

Step1: Use angle - sum property of a triangle

The sum of the interior angles of a triangle is 180°. So, \(m\angle J + m\angle K+m\angle L=180^{\circ}\). Substitute the given angle - measures: \((23x - 4)+(4x - 1)+(9x - 31)=180\).

Step2: Combine like - terms

\((23x+4x + 9x)+(-4-1 - 31)=180\), which simplifies to \(36x-36 = 180\).

Step3: Add 36 to both sides

\(36x-36 + 36=180 + 36\), resulting in \(36x=216\).

Step4: Solve for x

Divide both sides by 36: \(x=\frac{216}{36}=6\).

Step5: Find \(m\angle J\)

Substitute \(x = 6\) into the expression for \(m\angle J\): \(m\angle J=23x-4=23\times6 - 4=138 - 4 = 134^{\circ}\).

Step6: Find \(m\angle K\)

Substitute \(x = 6\) into the expression for \(m\angle K\): \(m\angle K=4x-1=4\times6 - 1=24 - 1 = 23^{\circ}\).

Step7: Find \(m\angle L\)

Substitute \(x = 6\) into the expression for \(m\angle L\): \(m\angle L=9x-31=9\times6 - 31=54 - 31 = 23^{\circ}\).

Answer:

\(x = 6\), \(m\angle J=134^{\circ}\), \(m\angle K = 23^{\circ}\), \(m\angle L=23^{\circ}\)