Question

What is the measure of the vertex angle in the next triangle in the pattern?

4 isosceles triangles in a row, each smaller than the one before. The first has base angles 89 degrees and vertex angle 2 degrees, the second has base angles 88 degrees and vertex angle 4 degrees, the third has base angles 87 degrees and vertex angle 6 degrees, and the fourth has base angles 86 degrees and vertex angle 8 degrees,

Not to scale

vertex angle =

Answer 1

The measure of the vertex angle of the next isosceles triangle is 10°

Explanation:

The given information are;

The first isosceles triangle; Base angles = 89°, vertex angle = 2°

The second isosceles triangle; Base angles = 88°, vertex angle = 4°

The third isosceles triangle; Base angles = 87°, vertex angle = 6°

The fourth isosceles triangle; Base angles = 86°, vertex angle = 8°

The vertices of the isosceles triangles form a arithmetic projection, with a common difference of 2°

Therefore, the vertex for next isosceles triangle = 8° + 2° = 10°

The base angles are derived from the sum of interior angles of a triangle which is 180°

Which gives;

2 × Base angles + Vertex angle = 180°

2 × Base angles + 10° = 180°

2 × Base angles = 180° - 10° = 170°

Base angles = 170°/2 = 85°

Base angles of the fifth isosceles triangle = 85°

From which we have;

The fifth isosceles triangle; Base angles = 85°, vertex angle = 10°

The measure of the vertex angle of the next (fifth) isosceles triangle = 10°