Question

in the triangle below, ∠k is a right angle. suppose that m∠j=(3x + 41)° and m∠l=(4x)°.
(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.
m∠j=
°
m∠k=
°
m∠l=
°

Explanation:

Step1: Use the triangle angle - sum property

In a right - triangle \( \triangle LJK\) with \( \angle K = 90^{\circ}\), the sum of the interior angles of a triangle is \(180^{\circ}\). So, \(m\angle L+m\angle J + m\angle K=180^{\circ}\). Substituting \(m\angle L=(4x)^{\circ}\), \(m\angle J=(3x + 41)^{\circ}\), and \(m\angle K = 90^{\circ}\), we get the equation \(4x+(3x + 41)+90=180\).
Simplify the left - hand side: \(4x+3x+41 + 90=180\), which is \(7x+131 = 180\).
Subtract 131 from both sides: \(7x=180 - 131\), so \(7x=49\).
Divide both sides by 7: \(x = 7\).

Step2: Find \(m\angle L\)

Substitute \(x = 7\) into \(m\angle L=(4x)^{\circ}\). Then \(m\angle L=4\times7=28^{\circ}\).

Step3: Find \(m\angle J\)

Substitute \(x = 7\) into \(m\angle J=(3x + 41)^{\circ}\). So \(m\angle J=3\times7+41=21 + 41=62^{\circ}\).

Step4: Find \(m\angle K\)

Since \( \angle K\) is a right - angle, \(m\angle K = 90^{\circ}\).

Answer:

(a) Equation: \(4x+(3x + 41)+90=180\)
(b) \(m\angle L = 28^{\circ}\), \(m\angle K=90^{\circ}\), \(m\angle J = 62^{\circ}\)