The Proof of the Second Case of the Inscribed Angle Theorem

In December last year, we have proved one case of the Inscribed Angle Theorem. In this post, we prove the second case. Note that the order of the cases does not matter; we just placed ordinal numbers to distinguish one from the other.

inscribed angle theorem

Recall that the first case of the theorem involved drawing an auxiliary line. We drew a line segment passing through the center. In this case, one of the sides of the triangle is passing through the center, so it is not possible to repeat this strategy. However, we will use another line to prove the theorem. Continue reading

Linking Triangle Angle Sum and Inscribed Angle Theorems

We have shown that the angle sum of a triangle is $latex 180^\circ$. We have also shown that the measure of an inscribed angle is half the measure the central angle that intercepts the same arc. In this post, we use the Inscribed Angle Theorem to show that the Triangle Angle Sum Theorem holds.

central-angle

Theorem

The angle sum of a triangle is $latex 180^\circ$.

Proof

Consider the figures above. In the first figure, the triangle is divided into three central angles. Clearly the three angles add up to a complete rotation about the center so  Continue reading

The Inscribed Angle Theorem

Introduction

An inscribed angle is formed when two secant lines intersect on a circle. It can also be formed using a secant line and a tangent line intersecting on a circle. A central angle, on the other hand, is an angle whose vertex is the center of the circle and whose sides pass through a pair of points on the circle, therefore subtending an arc. In this post, we explore the relationship between inscribed angles and central angles having the same subtended arc. The angle of the subtended arc is the same as the measure of the central angle (by definition).

inscribed-angle-theorem

In the first circle, $latex \angle O$ is a central angle subtended by arc $latex AB$. Angle $latex Q$ is an inscribed angle subtended by arc $latex PR$. In the second circle, $latex \angle T$ is an inscribed angle and $latex \angle V$ is a central angle. Both angles are subtending arc $latex SU$. Continue reading