Question

1. angle htn is bisected by 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: Use angle bisector property

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

Step2: Solve for \( x \)

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

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 \( \angle HTN = \angle HTR + \angle RTN \), we add the two angles:
\( m\angle HTN = 33.5^\circ + 33.5^\circ = 67^\circ \)

Answer:

\( x = 6 \)

Part b