Question

1. triangle btm is shown, where m∠tmb = 78°, m∠tbm=(n + 48)°, and m∠btm=(5n - 12)°. a. write and solve an equation to find the value of n. b. find the measure of ∠btm. 2. triangle fnw is shown, where m∠nfw = 5y°, m∠vwn = 151°, and m∠wnf=(11y - 25)°. a. write and solve an equation to find the value of y. b. find the measures of ∠fnw and ∠nwf.

Explanation:

1.

a.

Step1: Use angle - sum property of a triangle

The sum of the interior angles of a triangle is 180°. So, for \(\triangle BTM\), we have \((n + 48)+(5n-12)+78 = 180\).

Step2: Simplify the left - hand side

Combine like terms: \(n+5n + 48-12 + 78=180\), which gives \(6n+114 = 180\).

Step3: Isolate the variable term

Subtract 114 from both sides: \(6n=180 - 114\), so \(6n = 66\).

Step4: Solve for \(n\)

Divide both sides by 6: \(n=\frac{66}{6}=11\).

We know that \(\angle BTM=(5n - 12)^{\circ}\). Substitute \(n = 11\) into the expression for \(\angle BTM\).

Step1: Substitute \(n\) value

\(\angle BTM=5\times11-12\).

Step2: Calculate the value

\(\angle BTM = 55-12=43^{\circ}\).

Use the exterior - angle property of a triangle. The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. For \(\triangle FNW\), \(\angle VWN=\angle NFW+\angle WNF\). So, \(151 = 5y+(11y - 25)\).

Step1: Simplify the right - hand side

Combine like terms: \(151=5y + 11y-25\), which gives \(151=16y-25\).

Step2: Isolate the variable term

Add 25 to both sides: \(151 + 25=16y\), so \(176 = 16y\).

Step3: Solve for \(y\)

Divide both sides by 16: \(y=\frac{176}{16}=11\).

Answer:

\(n = 11\)

b.