Angle of Intersecting Chords Theorem

Chapter 47

Angle of Intersecting Chords Theorem Concept Explanation:

The Angle of Intersecting Chords Theorem states that if two chords intersect inside a circle, the measure of the angle formed by these intersecting chords is half the sum of the measures of the arcs intercepted by the angle and its vertical angle. In other words, if two chords $AB$ and $CD$ intersect at a point $E$ inside a circle, then the measure of $\angle AEC$ is given by:

$m \angle AEC = \frac{1}{2}(m \overset{\frown}{AC} + m \overset{\frown}{BD})$

This theorem is useful for calculating angles formed by intersecting chords in circles.

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\node at (-2,2) [left] {A};
\node at (2,-2) [right] {B};
\node at (-2,-2) [left] {C};
\node at (2,2) [right] {D};
\node at (0,0) [below] {E};
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In this diagram, $AB$ and $CD$ are intersecting chords, and $E$ is their point of intersection.

Common Mistakes:

Confusing the Sum of Arcs: Students sometimes mistakenly use the arc opposite the angle instead of the intercepted arcs.
Forgetting to Halve the Sum: After calculating the sum of the arcs, it’s common to forget dividing by 2.
Misinterpreting Vertical Angles: Vertical angles formed by intersecting chords are congruent, and this can be overlooked.

Helpful Tips:

Draw Arcs Clearly: Always mark the intercepted arcs to avoid confusion.
Check Vertical Angles: If needed, use the fact that vertical angles are equal when chords intersect.

Hard Questions:

Q1: In a circle, two chords $AB$ and $CD$ intersect at point $E$ . The measure of $\overset{\frown}{AC}$ is 80°, and the measure of $\overset{\frown}{BD}$ is 60°. Find $m \angle AEC$ .

Step-by-Step Solution:

Apply the Angle of Intersecting Chords Theorem:

$m \angle AEC = \frac{1}{2}(m \overset{\frown}{AC} + m \overset{\frown}{BD})$

Substitute the known values:

$m \angle AEC = \frac{1}{2}(80^\circ + 60^\circ) = \frac{1}{2}(140^\circ) = 70^\circ$

Answer:

$m \angle AEC = 70^\circ$

Q2: If the intercepted arcs in a circle are $m \overset{\frown}{AC} = 100^\circ$ and $m \overset{\frown}{BD} = 40^\circ$ , and the chords intersect at point $E$ , what is the measure of $\angle AEC$ ?

Step-by-Step Solution:

Use the Angle of Intersecting Chords Theorem:

$m \angle AEC = \frac{1}{2}(m \overset{\frown}{AC} + m \overset{\frown}{BD})$

Substitute the given arc measures:

$m \angle AEC = \frac{1}{2}(100^\circ + 40^\circ) = \frac{1}{2}(140^\circ) = 70^\circ$

Answer:

$m \angle AEC = 70^\circ$

Q3: In a circle, two chords intersect, and the intercepted arcs are $m \overset{\frown}{AC} = 110^\circ$ and $m \overset{\frown}{BD} = 50^\circ$ . Find the measure of the angle formed by the intersecting chords.

Step-by-Step Solution:

Use the Angle of Intersecting Chords Theorem:

$m \angle AEC = \frac{1}{2}(m \overset{\frown}{AC} + m \overset{\frown}{BD})$

Substitute the known values:

$m \angle AEC = \frac{1}{2}(110^\circ + 50^\circ) = \frac{1}{2}(160^\circ) = 80^\circ$

Answer:

$m \angle AEC = 80^\circ$

Q4: Two chords $AB$ and $CD$ intersect at point $E$ inside a circle. The intercepted arcs $\overset{\frown}{AC}$ and $\overset{\frown}{BD}$ are 90° and 70° respectively. What is $m \angle AEC$ ?

Step-by-Step Solution:

Apply the Angle of Intersecting Chords Theorem:

$m \angle AEC = \frac{1}{2}(m \overset{\frown}{AC} + m \overset{\frown}{BD})$

Substitute the arc measures:

$m \angle AEC = \frac{1}{2}(90^\circ + 70^\circ) = \frac{1}{2}(160^\circ) = 80^\circ$

Answer:

$m \angle AEC = 80^\circ$

Q5: If two chords intersect inside a circle such that $m \overset{\frown}{AC} = 120^\circ$ and $m \overset{\frown}{BD} = 100^\circ$ , find $m \angle AEC$ .

Step-by-Step Solution:

Use the Angle of Intersecting Chords Theorem:

$m \angle AEC = \frac{1}{2}(m \overset{\frown}{AC} + m \overset{\frown}{BD})$

Substitute the values:

$m \angle AEC = \frac{1}{2}(120^\circ + 100^\circ) = \frac{1}{2}(220^\circ) = 110^\circ$

Answer:

$m \angle AEC = 110^\circ$

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