Question

n is the incenter of \\( \triangle rst \\). if \\( m \angle rst = ( 3 x + 17 ) ^ { \circ } \\), \\( m \angle str = ( 8 x - 32 ) ^ { \circ } \\), and \\( m \angle trs = ( 2 x ) ^ { \circ } \\), find \\( m \angle rsn \\).
answer attempt 1 out of 2
\\( x = \\)
\\( m \angle rst = \\)
\\( m \angle rsn = \\)

Explanation:

Step1: Use the triangle angle - sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). So, \(m\angle RST+m\angle STR + m\angle TRS=180^{\circ}\).
Substitute the given expressions: \((3x + 17)+(8x-32)+(2x)=180\).
Combine like terms: \(3x+8x + 2x+17 - 32=180\), which simplifies to \(13x-15 = 180\).
Add \(15\) to both sides: \(13x=180 + 15\), so \(13x=195\).
Divide both sides by \(13\): \(x=\frac{195}{13}=15\).

Step2: Find \(m\angle RST\)

Substitute \(x = 15\) into the expression for \(m\angle RST\): \(m\angle RST=(3x + 17)^{\circ}\).
\(m\angle RST=(3\times15 + 17)^{\circ}=(45+17)^{\circ}=62^{\circ}\).

Step3: Use the property of the in - center

Since \(N\) is the in - center of \(\triangle RST\), \(SN\) is the angle bisector of \(\angle RST\).
So, \(m\angle RSN=\frac{1}{2}m\angle RST\).
Substitute \(m\angle RST = 62^{\circ}\): \(m\angle RSN=\frac{1}{2}\times62^{\circ}=31^{\circ}\).

Answer:

\(x = 15\)
\(m\angle RST=62^{\circ}\)
\(m\angle RSN = 31^{\circ}\)