Question

1. in triangle abc, the interior angles at a and b are 65 degrees and 45 degrees, respectively. what is the measure of the exterior angle at c?
2. a triangle has a base 12 and height 3. what is the area of the triangle?
3. the vertices a, b, and c of triangle abc correspond to the vertices d, e, and f of similar triangle def. if the length of side ab is 8, the length of side ac is 14, and the length of side de is 12, what is the length of side df?

Explanation:

Step1: Recall the exterior angle theorem

The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles.

Step2: Calculate the exterior angle at \(C\)

Given interior angles at \(A = 65^{\circ}\) and \(B=45^{\circ}\). Using the formula \(E=\angle A+\angle B\) (where \(E\) is the exterior angle at \(C\)).
\(E = 65^{\circ}+45^{\circ}\)

Step3: Recall the area formula for a triangle

The area formula for a triangle is \(A=\frac{1}{2}\times base\times height\). Given \(base = 12\) and \(height = 3\).
\(A=\frac{1}{2}\times12\times3\)

Step4: Recall the property of similar triangles

For similar triangles \(\triangle ABC\) and \(\triangle DEF\), \(\frac{AB}{DE}=\frac{AC}{DF}\). Given \(AB = 8\), \(AC = 14\), \(DE = 12\). Let \(DF=x\), then \(\frac{8}{12}=\frac{14}{x}\)

Answer:

1. The measure of the exterior angle at \(C\) is \(110^{\circ}\).
2. The area of the triangle is \(18\).
3. Cross - multiply in \(\frac{8}{12}=\frac{14}{x}\), we get \(8x=12\times14\), \(8x = 168\), \(x=\frac{168}{8}=21\). So the length of side \(DF\) is \(21\).