9.3: Angles of Chords

Difficulty Level: At Grade Created by: CK-12

Learning Objectives

• Find the measures of angles formed by chords in a circle.

Chords

Chords are line segments whose endpoints are both on a circle. The figure below shows an arc \begin{align*}\widehat{AB}\end{align*} and its related chord \begin{align*}\overline{AB}\end{align*}.

• A chord is a line segment whose ______________________ are both on a circle.

Angles Inside a Circle

The measure of the angle formed by two chords that intersect inside a circle is equal to half the sum of the measure of their intercepted arcs. In other words, the measure of the angle is the average (mean) of the measures of the intercepted arcs.

• Two intersecting _______________________ make 4 angles in a circle. Each angle is the average of the corresponding intercepted arcs.
• Another word for the average is the ______________________. The average of two numbers is half their sum.

In the figure above, \begin{align*}m \angle a = \frac{1}{2} (m \widehat{AB} + m \widehat{DC})\end{align*} and \begin{align*}m \angle b = \frac{1}{2} (m \widehat{AD} + m \widehat{BC})\end{align*}

Proof

Draw a segment to connect points \begin{align*}B\end{align*} and \begin{align*}C\end{align*}:

Statements Reasons
1. \begin{align*}m \angle DBC = \frac{1}{2} m \widehat{DC}\end{align*} 1.Inscribed angle
2. \begin{align*}m \angle ACB = \frac{1}{2} m \widehat{AB}\end{align*} 2.Inscribed angle
3. \begin{align*}m \angle a = m \angle ACB + m \angle DBC\end{align*} 3. The measure of an exterior angle in a triangle is equal to the sum of the measures of the remote interior angles.
4. \begin{align*}m \angle a = \frac{1}{2} m \widehat{DC} + \frac{1}{2} m \widehat{AB}\end{align*} 4. Substitution
5. \begin{align*}m \angle a = \frac{1}{2} (m \widehat{DC} + m \widehat{AB})\end{align*} 5. Factor and simplify

The intersection of two chords makes two pairs of vertical angles. Since vertical angles are congruent, you can compute the measure of either angle of each pair using the same calculation: take the average of the two intercepting arcs.

• When two chords intersect in a circle, two pairs of _______________________ angles are created.
• Each angle measure is equal to the ____________________________ of the two intercepting arc measures.
• The reason in steps 1 and 2 in the proof above says that an __________________ angle is half of the measure of its intercepted arc.

Example 1

Find \begin{align*}m \angle DEC\end{align*}.

We know that an angle formed by two intersecting chords is the average of the intercepted arc measures.

For \begin{align*}\angle DEC\end{align*}, the intercepted arcs are \begin{align*}\widehat{DC}\end{align*} and \begin{align*}\widehat{AB}\end{align*}, but instead we are given the measures of the arcs \begin{align*}\widehat{AD}\end{align*} and \begin{align*}\widehat{BC}\end{align*}. We can use this information to find what we need:

\begin{align*}m \angle AED & = \frac{1}{2} (m \widehat{AD} + m \widehat{BC}) \ \text{so} \ m \angle AED = \frac{1}{2}(40^\circ + 62^\circ)\\ m \angle AED & = \frac{1}{2}(102^\circ) = 51^\circ\end{align*}

Since \begin{align*}\angle AED\end{align*} and \begin{align*}\angle DEC\end{align*} are a linear pair (and thus supplementary):

\begin{align*}m \angle DEC & = 180^\circ - m \angle AED\\ m \angle DEC & = 180^\circ - 51^\circ = 129^\circ\end{align*}

1. True or false: Chords that intersect inside a circle create 4 angles that all have different measures.

2. Why is the statement in question #1 true or why is it false? Defend your choice.

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