Inscribed Angles & Intercepted Arcs: Geometry

Ahoy math friends and welcome to MathSux! In this post, we are going to go over the inscribed angle theorem which dives into inscribed angles and intercepted arcs. We’ll break down the main basic rule for inscribed angle theorem and the three inscribed angle theorems associated with this rule. If you are looking for more circle theorems, check out these posts on the Intersecting Secants Theorem and Central Angles Theorem. Also, don’t forget to check out the video and practice questions to truly master the topic below. Happy calculating! 🙂

Inscribed Angles:

When two chords come together to touch the outline of a circle, they create something called an inscribed angle. Unlike central angles, an inscribed angle is equal to half the measure of the arc length. In the example below, we can see that the inscribed angle ACB is equal to 25º while its corresponding intercepted arc AB is twice the inscribed angle at 50º. Knowing this allows us to find the value of unknown arcs or inscribed angles.

Inscribed Angle Theorem:

There are three parts to the inscribed angle theorem to know based on the rule stated above, check them out below!

The inscribed angle theorem states that, in a circle when two inscribed angles intercept the same arc, the angles are congruent. Notice below how angle A and angle B both have the same arc CD (hi-lighted in pink). In this theorem, we have two inscribed angles, angle A and angle B, that intercept the same arc, arc CD.

Theorem #2: In a circle when an angle is inscribed by a semicircle, it forms a 90º angle (or a right angle). If you look at angle BAC below, you’ll notice that the arc it corresponds with is a semicircle and that angle BAC forms a right angle.

Theorem #3: When a quadrilateral is inscribed within a circle, the opposite angles formed are supplementary, meaning that they add to 180º. The proof below shows angles A and C (hi-lighted in green) as supplementary, but this proof would also work for opposite angles B and D (hi-lighted in pink).

Let’s look at how to apply these rules with an Example:

a) Step 1: To find the value of angle CDB we need to look at our given information. We know that angle CAB=85º, notice that this follows theorem number 3, “When a quadrilateral is inscribed in a circle, opposite angles are supplementary.” Therefore, we must subtract 110º from 180º to find the value of angle CDB.

b) Step 2: For finding angle ABD, we’re going to use the same theorem we used in part a, opposite supplementary angles of an inscribed quadrilateral are supplementary.

c) Step 3: Next, to find the value of arc ABD, we need to use the basic inscribed angle theorem that tells us an inscribed angle is equal to half the measure of its arc. Then use some basic algebra to solve for arc ABD.

d) Step 4: To find arc ACD, we need to use the basic inscribed angle theorem that tells us an inscribed angle is equal to the value of its arc, then use algebra to solve similar to part c.

If this looks confusing, check out the video above! And when you are ready master this topic with the practice questions below!

Practice Questions:

Use the inscribed angle theorem outlined above to answer the following questions.

Solutions:

Still got questions? No problem! Check out the video above or comment below for any questions and follow for the latest MathSux posts. Happy calculating! 🙂

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Looking for more posts on Circle Theorems? Check out the posts below!