Inscribed Angles & Intercepted Arcs: Geometry

Ahoy math friends and welcome to MathSux! In this post, we are going to go over inscribed angles and intercepted arcs. We’ll break down the main basic rule for inscribed angles and the three 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. An inscribed angle is equal to half the value of the arc length.

Inscribed Angles & Intercepted Arcs

Inscribed Angle Theorems:

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

Theorem #1: (Intercepted Arcs) In a circle when inscribed angles intercept the same arc, the angles are congruent.

Inscribed Angles & Intercepted Arcs

Theorem #2: In a circle when an angle is inscribed by a semicircle, it forms a  90º angle.

Theorem #3: When a quadrilateral is inscribed in a circle, opposite angles are supplementary (add to 180º). (The proof below shows angles A and C as supplementary, but this proof would also work for opposite angles B and D).

Inscribed Angles & Intercepted Arcs

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 value 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:

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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