Question

in the triangle below, \\( \angle i \\) is a right angle. suppose that \\( m \angle j=(2 x+26)^{\circ} \\) and \\( m \angle k=(3 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 i= \\)
\\( m \angle j= \\)
\\( m \angle k= \\)

Explanation:

Step1: Use the triangle angle - sum property

In a triangle, the sum of the interior angles is \(180^{\circ}\). Since \(\angle I = 90^{\circ}\), we have \(m\angle J+m\angle K + m\angle I=180^{\circ}\). Substituting the given expressions for \(m\angle J=(2x + 26)^{\circ}\) and \(m\angle K=(3x + 34)^{\circ}\) and \(m\angle I = 90^{\circ}\) into the equation:
\((2x + 26)+(3x + 34)+90=180\)
Simplify the left - hand side:
\(2x+3x+26 + 34+90=180\)
\(5x+(26 + 34+90)=180\)
\(5x + 150=180\)

Step2: Solve the equation for \(x\)

Subtract \(150\) from both sides of the equation \(5x+150 = 180\):
\(5x+150-150=180 - 150\)
\(5x=30\)
Divide both sides by \(5\):
\(x=\frac{30}{5}=6\)

Step3: Find the measure of \(\angle J\)

Substitute \(x = 6\) into \(m\angle J=(2x + 26)^{\circ}\)
\(m\angle J=2\times6+26=12 + 26=38^{\circ}\)

Step4: Find the measure of \(\angle K\)

Substitute \(x = 6\) into \(m\angle K=(3x + 34)^{\circ}\)
\(m\angle K=3\times6+34=18+34 = 52^{\circ}\)

Answer:

(a) Equation: \(5x+150 = 180\)
(b) \(m\angle I = 90^{\circ}\), \(m\angle J=38^{\circ}\), \(m\angle K = 52^{\circ}\)