Real-World Examples of Angles Formed by Intersecting Chords

📚 Quick Study Guide

• 📐 When two chords intersect inside a circle, the measure of the angle formed is half the sum of the intercepted arcs.
• 📝 Formula: If chords AB and CD intersect at point E inside the circle, then $ \angle AEC = \frac{1}{2} (arc(AC) + arc(DB)) $.
• 💡 The intercepted arcs are the arcs that lie between the sides of the angle and its vertical angle.
• ✨ Remember that the vertical angle (e.g., $ \angle DEB $) will have the same measure.
• 🔑 If you know the angle and one arc, you can solve for the other arc.

🧪 Practice Quiz

1. A circular garden has two intersecting pathways. If the intercepted arcs measure 60° and 80°, what is the angle formed by the intersection of the pathways?

   1. 70°
   2. 140°
   3. 35°
   4. 60°

2. A carpenter is designing a circular tabletop with inlaid wooden chords forming a design. Two chords intersect, creating an angle of 50°. If one intercepted arc measures 40°, what does the other intercepted arc measure?

   1. 60°
   2. 140°
   3. 40°
   4. 80°

3. An artist is creating a stained-glass window in a circular frame. Two glass supports intersect within the circle. The arcs they intercept measure 95° and 45°. What is the measure of the angle formed by the intersecting supports?

   1. 70°
   2. 140°
   3. 35°
   4. 90°

4. Two chords in a clock design intersect. If one intercepted arc is 100° and the intersection angle is 75°, what is the measure of the other intercepted arc?

   1. 200°
   2. 50°
   3. 150°
   4. 100°

5. A circular park has two walking paths that intersect. The angle formed at their intersection is 65°. If one of the intercepted arcs measures 70°, what is the measure of the other intercepted arc?

   1. 60°
   2. 130°
   3. 40°
   4. 120°

6. In a circular amphitheater design, two support beams intersect. The intercepted arcs measure 75° and 85°. What is the angle formed by the intersection?

   1. 160°
   2. 80°
   3. 40°
   4. 90°

7. A circular window has two intersecting support bars. The angle of intersection is 42°. If one intercepted arc measures 50°, what is the measure of the other intercepted arc?

   1. 34°
   2. 84°
   3. 25°
   4. 90°