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

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.

 triangle

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

The Angle Secant Theorem

A secant is a line that intersects a circle at two points. In the figure below, $latex \angle E$ is formed by two secants. The angle intercepts two arcs $latex \overline{AB}$ and $latex \overline{CD}$. In this post, we will prove that the measure of the angle formed by two secants intersecting outside a circle is half the difference of the arcs intercepted by it.

angle secant theorem

To prove this theorem we will connect $latex \overline{BC}$ and use the Inscribed Angle Theorem and Exterior Angle Theorem.

Angle Secant Theorem

The angle measure formed by two secants intersecting outside a circle is half the difference of the arcs intercepted by it Continue reading