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 Definition

Since \(\overline{TR}\) bisects \(\angle HTN\), \(\angle HTR=\angle RTN\). So, \(7x - 8.5=5x + 3.5\).

Step2: Solve for \(x\)

Subtract \(5x\) from both sides: \(7x-5x - 8.5=5x - 5x+ 3.5\) \(\Rightarrow 2x - 8.5 = 3.5\).
Add \(8.5\) to both sides: \(2x-8.5 + 8.5=3.5 + 8.5\) \(\Rightarrow 2x=12\).
Divide by \(2\): \(x = \frac{12}{2}=6\).

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

Substitute \(x = 6\) into \(m\angle HTR=(7x - 8.5)^\circ\): \(7(6)-8.5=42 - 8.5 = 33.5^\circ\).
Substitute \(x = 6\) into \(m\angle RTN=(5x + 3.5)^\circ\): \(5(6)+3.5=30 + 3.5 = 33.5^\circ\).

Step2: Calculate \(m\angle HTN\)

Since \(\angle HTN=\angle HTR+\angle RTN\), \(m\angle HTN = 33.5^\circ+33.5^\circ = 67^\circ\).

Answer:

\(x = 6\)

Part (b)