Given a triangle ABC with AB = AC, the angles opposite the equal sides are equal. Let a = angle BAC and let b = angle ABC = angle ACB.

Using the angle sum theorem, if a is known, then b is determined, and if b is given, then a is determined.

Write down the relations:

a =

b =

Some important examples

We know that the sum of the vertex angles of a triangle in the plane is always 180 degrees. A theorem about angle sums for polygons in general will be developed carefully later, but for now this will be a quick informal introduction.

Let ABCD be a quadrilateral. If the diagonal AC (extended to a line) is such that B is on one side of AC and D is on the other, then ABCD is divided into the union of two triangles ABC and CDA.

In this case, the sum of the angles of ABCD is 360 degrees, which is the sum of the angles of the two triangles, since 180 + 180 = 360 degrees.

For a convex quadrilateral such as the one on the left, this works for either choice of diagonal. For the non-convex quadrilateral on the right, we chose one diagonal that divides the quadrilateral into two triangles.

If we divide a pentagon into triangles as in the figure on the left below, the pentagon is made up of 3 triangles, so the angle sum is 180 + 180 + 180 = 3*180 = 540 degrees.

However, the non-convex pentagon on the right is a trickier case. If we have the figure on the page, we can always find a way to draw segments to divide the pentagon into 3 triangles, but how can we prove this in all cases? How can we prove triangle subdivision for polygons with a large number of vertices? This is not only a theoretical problem, but it is a practical problem in computer science. If a polygon is defined in the plane using coordinates, how can one instruct a computer to divide it into triangles. We will return to this question later. For now, we make the reasonable assumption that any pentagons we encounter can be divided into 3 triangles. This is certainly true for convex ones, as we see in the figure on the left.

Question: A regular pentagon is defined to be a pentagon that has all angles equal and all sides equal. What must the angle be at each vertex?

Answer:

Given a regular polygon, we have seen that each vertex angle is 108 = 3*180/5 degrees.

In this figure, draw the diagonal AC.

• Explain why triangle ABC is an isosceles triangle.

• Write down the measure of the angles of the triangle ABC. In the figure, label each angle of triangle ABC with the number of degrees in the angle.

• Then draw diagonal AD and likewise label the measures of the angles in triangle ADE. Why is triangle ADE congruent to triangle ABC?

• Explain why triangle CAD is an isosceles triangle.

• Finally, using what you now about all the angles with vertex at A, write the measure of angle CAD and then label the measures of the other angles of triangle ACD.

• Let s = ||AB| and let d =|AC|. Tell what are the angles in a triangle similar to (1) a triangle with sides s-s-d and (2) a triangle with sides d-d-s.

The familiar 5-pointed star or pentagram is also a regular figure with equal sides and equal angles.

The pentagram can be drawn by drawing all the diagonals of the regular pentagon.

We have seen on the previous page the angles of some isosceles triangles. In particular, the angles 36 degrees, 72 degrees and 108 degrees appeared. In this figure above, mark all the angles of 36 degrees with a single mark, mark the angles of 72 degrees with a double mark, and all angles of 108 degrees with a triple mark.

Let us label the intersection of AC and BD as F.

Now temporarily ignoring the rest of the figure, concentrate on this triangle with sub-triangle. Label all the angles in the figure with their measures.

Based on the angles, explain why each of the sub-triangles is an isosceles triangle.

• Given that |AC| = d and |CD| = s, what is |CD|?

• Using similar triangles, find an equation relating s and d.

• Now let the ratio r = d/s. Rewrite the equation as an equation in r. (There should be no other variables left.) Then solve for r.