Question

what conjecture can you make about the measure of the base angles in the nth triangle in the pattern?
a. $180 - 2n$
b. $\frac{180(n - 1)}{2}$
c. $\frac{180 - 2n}{2}$
d. $\frac{180 - n}{2}$

Explanation:

Step1: Recall the triangle angle - sum property

The sum of the interior angles of a triangle is \(180^{\circ}\). Let the vertex angle be \(2n^{\circ}\) (when \(n = 1\), vertex angle \(=2^{\circ}\); when \(n = 2\), vertex angle \(=4^{\circ}\); when \(n=3\), vertex angle \(=6^{\circ}\); when \(n = 4\), vertex angle \(=8^{\circ}\)). Let the base angles be \(x\) (since the triangles are isosceles, the base angles are equal). Then, by the angle - sum property of a triangle, \(2n+2x=180\).

Step2: Solve the equation for \(x\)

Subtract \(2n\) from both sides of the equation \(2n + 2x=180\):
\(2x=180 - 2n\).
Divide both sides of the equation by \(2\):
\(x=\frac{180 - 2n}{2}\).

Answer:

C. \(\frac{180 - 2n}{2}\)