Question

what are the values of c and t?

c = 37°
t = 149°

Explanation:

Step1: Use the exterior angle theorem

The exterior angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles. So, \(t - 50^{\circ}=(c + 12^{\circ})+(2c-24^{\circ})\).
Simplify the right - hand side: \((c + 12^{\circ})+(2c-24^{\circ})=3c-12^{\circ}\). So, \(t-50^{\circ}=3c - 12^{\circ}\), and \(t=3c + 38^{\circ}\).

Step2: Use the linear pair relationship

We know that \((t - 50^{\circ})+(c + 44^{\circ}) = 180^{\circ}\) (linear pair of angles). Substitute \(t = 3c+38^{\circ}\) into \((t - 50^{\circ})+(c + 44^{\circ}) = 180^{\circ}\).
\((3c + 38^{\circ}-50^{\circ})+(c + 44^{\circ})=180^{\circ}\).
\((3c-12^{\circ})+(c + 44^{\circ})=180^{\circ}\).
Combine like terms: \(4c+32^{\circ}=180^{\circ}\).
Subtract \(32^{\circ}\) from both sides: \(4c=180^{\circ}-32^{\circ}=148^{\circ}\).
Divide both sides by \(4\): \(c=\frac{148^{\circ}}{4}=37^{\circ}\).

Step3: Find the value of \(t\)

Substitute \(c = 37^{\circ}\) into \(t=3c + 38^{\circ}\).
\(t=3\times37^{\circ}+38^{\circ}\).
\(t = 111^{\circ}+38^{\circ}=149^{\circ}\).

Answer:

\(c = 37^{\circ}\), \(t = 149^{\circ}\)