High School Geometry Test Questions on Intersecting Chords Angles

📚 Quick Study Guide

• 📐 Intersecting Chords Theorem: If two chords intersect inside a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. If chords AB and CD intersect at point E, then $AE \cdot EB = CE \cdot ED$.
• 💡 Angles Formed by Intersecting Chords: The measure of an angle formed by two chords that intersect inside a circle is half the sum of the measures of the intercepted arcs. If angle $\angle AEC$ is formed by chords AB and CD, then $m\angle AEC = \frac{1}{2}(m\stackrel{\frown}{AC} + m\stackrel{\frown}{BD})$.
• 🧭 Intersecting Secants Theorem: If two secant lines intersect outside a circle, then the measure of the angle formed is one-half the positive difference of the measures of the intercepted arcs. If secants AB and AD intersect at A, then $m\angle A = \frac{1}{2}(m\stackrel{\frown}{BD} - m\stackrel{\frown}{BC})$.
• ✨ Secant-Tangent Angle Theorem: If a secant and a tangent intersect at the point of tangency on a circle, then the measure of the angle formed is one-half the measure of its intercepted arc. If tangent AB and secant AC intersect at A, then $m\angle BAC = \frac{1}{2}m\stackrel{\frown}{AC}$.
• 📌 Important Note: Always double-check which arcs are intercepted by the angles in question! Drawing a clear diagram can be super helpful!

Practice Quiz

1. Question 1: Two chords, AB and CD, intersect at point E inside a circle. If AE = 6, EB = 4, and CE = 3, what is the length of ED?
1. A) 2
2. B) 6
3. C) 8
4. D) 12
2. Question 2: Chords PQ and RS intersect at point T inside a circle. If $m\stackrel{\frown}{PR} = 50^\circ$ and $m\stackrel{\frown}{QS} = 70^\circ$, what is the measure of $\angle PTQ$?
1. A) 30°
2. B) 60°
3. C) 120°
4. D) 240°
3. Question 3: Two secants, PA and PC, intersect at point P outside a circle. If $m\stackrel{\frown}{AC} = 100^\circ$ and $m\stackrel{\frown}{BD} = 30^\circ$ (where B and D are the points where the secants intersect the circle), what is the measure of $\angle P$?
1. A) 35°
2. B) 50°
3. C) 65°
4. D) 85°
4. Question 4: A tangent AB and a secant AC intersect at point A on a circle. If $m\stackrel{\frown}{AC} = 150^\circ$, what is the measure of $\angle BAC$?
1. A) 30°
2. B) 75°
3. C) 150°
4. D) 300°
5. Question 5: Chords WX and YZ intersect at point V inside a circle. If WV = 5, VX = 8, and YV = 4, find the length of VZ.
1. A) 2.5
2. B) 6
3. C) 10
4. D) 16
6. Question 6: Chords AB and CD intersect at E. $m\stackrel{\frown}{AC} = 45^\circ$ and $m\stackrel{\frown}{DB} = 55^\circ$. What is the measure of $\angle AEC$?
1. A) 25°
2. B) 50°
3. C) 75°
4. D) 100°
7. Question 7: Secants PR and PT intersect at P. Arc RT measures 80 degrees, and arc QS measures 20 degrees (Q and S are the intersection points). What is the measure of angle P?
1. A) 20°
2. B) 30°
3. C) 40°
4. D) 50°