Question

1. in ( \triangle jkl ), if ( m angle j ) is seven less than ( m angle l ) and ( m angle k ) is 21 less than twice ( m angle l ), find the measure of each angle.

Explanation:

Step1: Set up equations based on angle relationships

Let \(m\angle L=x\). Then \(m\angle J = x - 7\) (since \(m\angle J\) is seven less than \(m\angle L\)) and \(m\angle K=2x - 21\) (since \(m\angle K\) is 21 less than twice \(m\angle L\)).
In a triangle, the sum of the interior angles is \(180^{\circ}\), so \(m\angle J+m\angle K + m\angle L=180^{\circ}\).
Substitute the expressions: \((x - 7)+(2x - 21)+x=180\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \((x+2x + x)+(-7-21)=180\), which gives \(4x-28 = 180\).

Step3: Solve for \(x\)

Add 28 to both sides: \(4x-28 + 28=180 + 28\), so \(4x=208\).
Divide both sides by 4: \(x=\frac{208}{4}=52\).

Step4: Find the measures of \(\angle J\) and \(\angle K\)

For \(\angle J\): \(m\angle J=x - 7\). Substitute \(x = 52\), so \(m\angle J=52-7 = 45^{\circ}\).
For \(\angle K\): \(m\angle K=2x - 21\). Substitute \(x = 52\), so \(m\angle K=2\times52-21=104 - 21=83^{\circ}\).

Answer:

\(m\angle L = 52^{\circ}\), \(m\angle J=45^{\circ}\), \(m\angle K = 83^{\circ}\)