Question

in the triangle below, \\( \angle v \\) is a right angle. suppose that \\( m \angle w=(4 x+15)^{\circ} \\) and \\( m \angle x=(2 x+15)^{\circ} \\).
(a) write an equation to find \\( x \\). make sure you use an \=\ sign in your answer.
equation: \\( \square \\)
(b) find the degree measure of each angle.
\\( m \angle v=\square^{\circ} \\)
\\( m \angle w=\square^{\circ} \\)
\\( m \angle x=\square^{\circ} \\)

Explanation:

Step1: Use the triangle angle - sum property

The sum of the interior angles of a triangle is \(180^{\circ}\). In right - triangle \(V W X\), \(\angle V = 90^{\circ}\), \(\angle W=(4x + 15)^{\circ}\), and \(\angle X=(2x + 15)^{\circ}\).
So, \(m\angle V+m\angle W+m\angle X = 180^{\circ}\).
Substitute the angle measures: \(90+(4x + 15)+(2x + 15)=180\).
Simplify the left - hand side: \(90+4x+15 + 2x+15=180\), which becomes \(6x+120 = 180\).

Step2: Solve the equation for \(x\)

Subtract \(120\) from both sides of the equation \(6x+120 = 180\):
\(6x=180 - 120\), so \(6x=60\).
Divide both sides by \(6\): \(x=\frac{60}{6}=10\).

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

Substitute \(x = 10\) into the formula for \(\angle W\): \(m\angle W=(4x + 15)^{\circ}\).
\(m\angle W=(4\times10 + 15)^{\circ}=(40+15)^{\circ}=55^{\circ}\).

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

Substitute \(x = 10\) into the formula for \(\angle X\): \(m\angle X=(2x + 15)^{\circ}\).
\(m\angle X=(2\times10+15)^{\circ}=(20 + 15)^{\circ}=35^{\circ}\).

Answer:

(a) Equation: \(6x + 120=180\)
(b) \(m\angle V = 90^{\circ}\), \(m\angle W = 55^{\circ}\), \(m\angle X = 35^{\circ}\)