📝 10th Grade Geometry: Angles of Intersecting Chords Study Notes
Angles of Intersecting Chords 📐
When two chords intersect inside a circle, they form four angles. The measure of each angle is related to the intercepted arcs. This relationship is a fundamental theorem in circle geometry.
Theorem: Angles Inside a Circle 💡
If two chords intersect inside a circle, then the measure of each angle formed is one-half the sum of the measures of the two intercepted arcs.
Case 1: Intersecting Chords Inside the Circle
Consider two chords, \( \overline{AC} \) and \( \overline{BD} \), intersecting at point \( P \) inside circle \( O \). The angle formed, for example, \( \angle APB \), intercepts arc \( \widehat{AB} \). The adjacent angle, \( \angle BPC \), intercepts arc \( \widehat{BC} \). The theorem states:
The measure of the angle formed by two intersecting chords is half the sum of the measures of the intercepted arcs.
Mathematically, this can be expressed as:
\[ m\angle APB = \frac{1}{2} (m\widehat{AB} + m\widehat{CD}) \]And also:
\[ m\angle BPC = \frac{1}{2} (m\widehat{BC} + m\widehat{AD}) \]Note that \( \angle APB \) and \( \angle BPC \) are supplementary angles, meaning \( m\angle APB + m\angle BPC = 180^\circ \). Also, \( \angle APB \) and \( \angle CPD \) are vertical angles, so \( m\angle APB = m\angle CPD \). Similarly, \( \angle BPC \) and \( \angle APD \) are vertical angles, so \( m\angle BPC = m\angle APD \).
Example Calculation
Suppose chord \( \overline{AC} \) and \( \overline{BD} \) intersect at \( P \) inside a circle. If \( m\widehat{AB} = 60^\circ \) and \( m\widehat{CD} = 80^\circ \), what is \( m\angle APB \)?
Using the theorem:
\[ m\angle APB = \frac{1}{2} (m\widehat{AB} + m\widehat{CD}) \] \[ m\angle APB = \frac{1}{2} (60^\circ + 80^\circ) \] \[ m\angle APB = \frac{1}{2} (140^\circ) \] \[ m\angle APB = 70^\circ \]Table of Relationships
| Angle | Intercepted Arcs | Formula |
|---|---|---|
| \( \angle APB \) | \( \widehat{AB} \) and \( \widehat{CD} \) | \( \frac{1}{2} (m\widehat{AB} + m\widehat{CD}) \) |
| \( \angle BPC \) | \( \widehat{BC} \) and \( \widehat{AD} \) | \( \frac{1}{2} (m\widehat{BC} + m\widehat{AD}) \) |
Key Takeaway 📌
The measure of an angle formed by two chords intersecting inside a circle is always half the sum of the measures of the arcs they intercept.
Pro Tip 💡
When dealing with intersecting chords, identify the two arcs that are "opposite" each other with respect to the angle you are trying to find. These are the arcs you will use in the formula.