The Proof of the Third Case of the Inscribed Angle Theorem

This is the third and the last case of the Proofs of the Inscribed Angle Theorem. In the first case, we used an auxiliary line between the sides of the angles to come up with the proof. In the second case, we drew a line segment from the center to the intersection of the circle and one of the sides to form an isosceles triangle. In this case, we cannot do both.

To prove the third case, we will draw an auxiliary line outside the circle.  We draw segment $latex \overline{BD}$ through the center of the circle. The detail of the proof is as follows. Continue reading

Triangles with Non Congruent Sides

We have shown that in an isosceles triangle, the angles opposite the congruent sides are congruent. This was the Isosceles Triangle Theorem which we proved two weeks ago.

In this post, we are gong to learn a slightly related theorem: a theorem that states that if in a triangle, two sides are not congruent, then the angles opposite these sides are not congruent and the angle opposite to the larger side is the larger angle.

In the triangle above, $latex \overline{AB}$ is greater than $latex \overline{BC}$. We will show that they are not equal and $latex \angle C > \angle B$. Continue reading