Question

in the triangle below, \\( \angle k \\) is a right angle. suppose that \\( m \angle l = ( 5 x ) ^ { \circ } \\) and \\( m \angle m = ( 2 x + 34 ) ^ { \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.
\\( m \angle k = \\)\\( \circ \\)
\\( m \angle l = \\)\\( \circ \\)
\\( m \angle m = \\)\\( \circ \\)

Explanation:

Step1: Use the triangle angle - sum property

In a triangle, the sum of the interior angles is \(180^{\circ}\). Since \(\angle K = 90^{\circ}\), \(\angle L=(5x)^{\circ}\), and \(\angle M=(2x + 34)^{\circ}\), we have the equation \(90+5x+(2x + 34)=180\).
Simplify the left - hand side: \(90+34+5x+2x=180\), which is \(124 + 7x=180\).

Step2: Solve the equation for \(x\)

Subtract \(124\) from both sides of the equation \(124 + 7x=180\):
\(7x=180 - 124\)
\(7x=56\)
Divide both sides by \(7\): \(x = 8\)

Step3: Find the measure of each angle

• For \(\angle L\):

Since \(m\angle L=(5x)^{\circ}\) and \(x = 8\), then \(m\angle L=5\times8=40^{\circ}\)

• For \(\angle M\):

Since \(m\angle M=(2x + 34)^{\circ}\) and \(x = 8\), then \(m\angle M=2\times8+34=16 + 34=50^{\circ}\)

• \(\angle K\) is given as \(90^{\circ}\)

Answer:

(a) Equation: \(5x+(2x + 34)+90=180\) (or \(7x+124 = 180\))
(b) \(m\angle K = 90^{\circ}\), \(m\angle L=40^{\circ}\), \(m\angle M = 50^{\circ}\)