Question

1. angle htn is bisected by (overline{tr}).

a. find the value of ( x ) if ( mangle htr = (7x - 8.5)^circ ) and ( mangle rtn = (5x + 3.5)^circ ).

b. what is the measure of ( angle htn )?

Explanation:

Part a

Step1: Recall angle bisector property

Since \(\overrightarrow{TR}\) bisects \(\angle HTN\), then \(m\angle HTR = m\angle RTN\).
So we set the two angle expressions equal: \(7x - 8.5 = 5x + 3.5\)

Step2: Solve for \(x\)

Subtract \(5x\) from both sides: \(7x - 5x - 8.5 = 5x - 5x + 3.5\)
Simplify: \(2x - 8.5 = 3.5\)

Add \(8.5\) to both sides: \(2x - 8.5 + 8.5 = 3.5 + 8.5\)
Simplify: \(2x = 12\)

Divide both sides by \(2\): \(\frac{2x}{2}=\frac{12}{2}\)
Simplify: \(x = 6\)

Step1: Find \(m\angle HTR\) and \(m\angle RTN\)

First, substitute \(x = 6\) into \(m\angle HTR=(7x - 8.5)^\circ\):
\(m\angle HTR = 7(6)-8.5 = 42 - 8.5 = 33.5^\circ\)

Then, substitute \(x = 6\) into \(m\angle RTN=(5x + 3.5)^\circ\):
\(m\angle RTN = 5(6)+3.5 = 30 + 3.5 = 33.5^\circ\)

Step2: Find \(m\angle HTN\)

Since \(\overrightarrow{TR}\) bisects \(\angle HTN\), \(m\angle HTN=m\angle HTR + m\angle RTN\)
So \(m\angle HTN = 33.5^\circ+33.5^\circ = 67^\circ\)

Answer:

\(x = 6\)

Part b