Question

in triangle def, ( mangle d=(3x + 27)^{circ}), ( mangle e=(4x + 22)^{circ}), and ( mangle f = 68^{circ}). determine the degree measure of the exterior angle to (angle d).

Explanation:

Step1: Use the triangle - angle sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). So, \(m\angle D+m\angle E + m\angle F=180^{\circ}\).
Substitute \(m\angle D=(3x + 27)^{\circ}\), \(m\angle E=(4x + 22)^{\circ}\), and \(m\angle F = 68^{\circ}\) into the equation:
\((3x + 27)+(4x + 22)+68=180\)
Combine like terms:
\(3x+4x+27 + 22+68=180\)
\(7x+(27 + 22+68)=180\)
\(7x + 117=180\)
Subtract 117 from both sides:
\(7x=180 - 117\)
\(7x=63\)
Divide both sides by 7:
\(x = 9\)

Step2: Find \(m\angle D\)

Substitute \(x = 9\) into \(m\angle D=(3x + 27)^{\circ}\)
\(m\angle D=(3\times9+27)^{\circ}=(27 + 27)^{\circ}=54^{\circ}\)

Step3: Use the exterior - angle property

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles. Or, the measure of an exterior angle and its adjacent interior angle are supplementary (sum to \(180^{\circ}\)).
Let the exterior angle to \(\angle D\) be \(y\). Then \(y+m\angle D=180^{\circ}\)
Since \(m\angle D = 54^{\circ}\), then \(y=180^{\circ}-m\angle D\)
\(y=180 - 54\)
\(y = 126^{\circ}\)

Answer:

\(126^{\circ}\)