📄 10th Grade Geometry: Angles of Intersecting Chords Worksheet
📌 1. True / False
1. The measure of an angle formed by two chords intersecting inside a circle is half the sum of the measures of the intercepted arcs.
2. If two chords intersect at the center of a circle, the angle formed is always 90 degrees.
3. The angles formed by intersecting chords are vertical angles.
4. The formula for angles formed by intersecting chords applies only when the chords are perpendicular.
5. The measure of an angle formed by two chords intersecting inside a circle is half the difference of the intercepted arcs.
✏️ 2. Fill in the Blanks
1. The point where two chords intersect inside a circle is called the .
2. The measure of an angle formed by two chords intersecting inside a circle is the sum of the measures of the intercepted arcs.
3. If two chords intersect inside a circle, the vertical angles formed are .
4. The arcs intercepted by an angle formed by intersecting chords are the arcs that lie the angle and its vertical angle.
5. In a circle, a is a line segment whose endpoints lie on the circle.
🔗 3. Matching
« A line segment whose endpoints lie on the circle.
« The portion of a circle that lies between two lines, rays, or segments that intersect the circle.
« A pair of opposite angles formed by two intersecting lines.
« States that the measure of an angle formed by two chords intersecting inside a circle is half the sum of the measures of the intercepted arcs.
« The set of all points in a plane that are equidistant from a given point, called the center.
✍️ 4. Short Answer Questions
1. State the theorem for the measure of an angle formed by two chords intersecting inside a circle.
💡 Suggested Answer: The measure of an angle formed by two chords intersecting inside a circle is half the sum of the measures of the intercepted arcs.
2. What is the relationship between the two pairs of vertical angles formed by intersecting chords?
💡 Suggested Answer: The two angles in each pair of vertical angles are congruent.
🎯 5. Multiple Choice
1. Chords AB and CD intersect at point E inside a circle. If \(m\overset{\frown}{AC} = 60^\circ\) and \(m\overset{\frown}{BD} = 80^\circ\), what is \(m\angle AEC\)?
2. Chords FG and HJ intersect at point K inside a circle. If \(m\angle FKH = 100^\circ\) and \(m\overset{\frown}{FH} = 120^\circ\), what is \(m\overset{\frown}{GJ}\)?
3. If two chords intersect inside a circle, how many angles are formed at their intersection point?
📝 6. Open-Ended Questions
1. Chords PQ and RS intersect at point T inside a circle. If \(m\overset{\frown}{PR} = 70^\circ\) and \(m\overset{\frown}{QS} = 110^\circ\), find \(m\angle PTR\) and \(m\angle PTS\).
💡 Solution Steps:
Step 1: Identify the intercepted arcs for \(\angle PTR\). These are \(\overset{\frown}{PR}\) and \(\overset{\frown}{QS}\).
Step 2: Apply the intersecting chords angle theorem: \(m\angle PTR = \frac{1}{2}(m\overset{\frown}{PR} + m\overset{\frown}{QS})\).
Step 3: Substitute the given values: \(m\angle PTR = \frac{1}{2}(70^\circ + 110^\circ) = \frac{1}{2}(180^\circ) = 90^\circ\).
Step 4: Recognize that \(\angle PTS\) and \(\angle PTR\) are supplementary angles because they form a linear pair.
Step 5: Calculate \(m\angle PTS = 180^\circ - m\angle PTR = 180^\circ - 90^\circ = 90^\circ\).
Final Answer: \(m\angle PTR = 90^\circ\) and \(m\angle PTS = 90^\circ\).
2. Chords MN and OP intersect at point Q inside a circle. If \(m\angle MQO = 85^\circ\) and \(m\overset{\frown}{MO} = 95^\circ\), find \(m\overset{\frown}{NP}\).
💡 Solution Steps:
Step 1: Identify the angle and its intercepted arcs. \(\angle MQO\) intercepts \(\overset{\frown}{MO}\) and \(\overset{\frown}{NP}\).
Step 2: Apply the intersecting chords angle theorem: \(m\angle MQO = \frac{1}{2}(m\overset{\frown}{MO} + m\overset{\frown}{NP})\).
Step 3: Substitute the given values: \(85^\circ = \frac{1}{2}(95^\circ + m\overset{\frown}{NP})\).
Step 4: Multiply both sides by 2: \(170^\circ = 95^\circ + m\overset{\frown}{NP}\).
Step 5: Solve for \(m\overset{\frown}{NP}\): \(m\overset{\frown}{NP} = 170^\circ - 95^\circ = 75^\circ\).
Final Answer: \(m\overset{\frown}{NP} = 75^\circ\).
3. Chords AB and CD intersect at E. If \(m\overset{\frown}{AC} = (3x + 5)^\circ\), \(m\overset{\frown}{BD} = (5x - 15)^\circ\), and \(m\angle AEC = 70^\circ\), find the value of \(x\) and the measures of the arcs.
💡 Solution Steps:
Step 1: Identify the angle and its intercepted arcs. \(\angle AEC\) intercepts \(\overset{\frown}{AC}\) and \(\overset{\frown}{BD}\).
Step 2: Apply the intersecting chords angle theorem: \(m\angle AEC = \frac{1}{2}(m\overset{\frown}{AC} + m\overset{\frown}{BD})\).
Step 3: Substitute the given expressions and value: \(70^\circ = \frac{1}{2}((3x + 5)^\circ + (5x - 15)^\circ)\).
Step 4: Multiply both sides by 2: \(140 = (3x + 5) + (5x - 15)\).
Step 5: Combine like terms: \(140 = 8x - 10\).
Step 6: Add 10 to both sides: \(150 = 8x\).
Step 7: Solve for \(x\): \(x = \frac{150}{8} = \frac{75}{4} = 18.75\).
Step 8: Calculate the measure of arc \(\overset{\frown}{AC}\): \(m\overset{\frown}{AC} = (3(18.75) + 5)^\circ = (56.25 + 5)^\circ = 61.25^\circ\).
Step 9: Calculate the measure of arc \(\overset{\frown}{BD}\): \(m\overset{\frown}{BD} = (5(18.75) - 15)^\circ = (93.75 - 15)^\circ = 78.75^\circ\).
Step 10: Verify the answer: \(\frac{1}{2}(61.25^\circ + 78.75^\circ) = \frac{1}{2}(140^\circ) = 70^\circ\).
Final Answer: \(x = 18.75\), \(m\overset{\frown}{AC} = 61.25^\circ\), and \(m\overset{\frown}{BD} = 78.75^\circ\).
Name Surname: .................................. Date: .... / .... / 202...
Angles of Intersecting Chords Worksheet
SCORE
A. True (T) / False (F)
( .... )
The measure of an angle formed by two chords intersecting inside a circle is half the sum of the measures of the intercepted arcs.
( .... )
If two chords intersect at the center of a circle, the angle formed is always 90 degrees.
( .... )
The angles formed by intersecting chords are vertical angles.
( .... )
The formula for angles formed by intersecting chords applies only when the chords are perpendicular.
( .... )
The measure of an angle formed by two chords intersecting inside a circle is half the difference of the intercepted arcs.
B. Fill in the Blanks
1)
The point where two chords intersect inside a circle is called the .....................
2)
The measure of an angle formed by two chords intersecting inside a circle is .................... the sum of the measures of the intercepted arcs.
3)
If two chords intersect inside a circle, the vertical angles formed are .....................
4)
The arcs intercepted by an angle formed by intersecting chords are the arcs that lie .................... the angle and its vertical angle.
5)
In a circle, a .................... is a line segment whose endpoints lie on the circle.
C. Matching Concepts
( .... )
A line segment whose endpoints lie on the circle.
- Intercepted Arc
( .... )
The portion of a circle that lies between two lines, rays, or segments that intersect the circle.
- Angle of Intersecting Chords Theorem
( .... )
A pair of opposite angles formed by two intersecting lines.
- Circle
( .... )
States that the measure of an angle formed by two chords intersecting inside a circle is half the sum of the measures of the intercepted arcs.
- Vertical Angles
( .... )
The set of all points in a plane that are equidistant from a given point, called the center.
- Chord
D. Short Answer Questions
1)
State the theorem for the measure of an angle formed by two chords intersecting inside a circle.
2)
What is the relationship between the two pairs of vertical angles formed by intersecting chords?
E. Multiple Choice Questions
1)
Chords AB and CD intersect at point E inside a circle. If \(m\overset{\frown}{AC} = 60^\circ\) and \(m\overset{\frown}{BD} = 80^\circ\), what is \(m\angle AEC\)?
A) \(30^\circ\)B) \(70^\circ\)C) \(140^\circ\)D) \(10^\circ\)
2)
Chords FG and HJ intersect at point K inside a circle. If \(m\angle FKH = 100^\circ\) and \(m\overset{\frown}{FH} = 120^\circ\), what is \(m\overset{\frown}{GJ}\)?
A) \(80^\circ\)B) \(100^\circ\)C) \(120^\circ\)D) \(140^\circ\)
3)
If two chords intersect inside a circle, how many angles are formed at their intersection point?
A) 1B) 2C) 3D) 4
F. Open-Ended Questions
1)
Chords PQ and RS intersect at point T inside a circle. If \(m\overset{\frown}{PR} = 70^\circ\) and \(m\overset{\frown}{QS} = 110^\circ\), find \(m\angle PTR\) and \(m\angle PTS\).
2)
Chords MN and OP intersect at point Q inside a circle. If \(m\angle MQO = 85^\circ\) and \(m\overset{\frown}{MO} = 95^\circ\), find \(m\overset{\frown}{NP}\).
3)
Chords AB and CD intersect at E. If \(m\overset{\frown}{AC} = (3x + 5)^\circ\), \(m\overset{\frown}{BD} = (5x - 15)^\circ\), and \(m\angle AEC = 70^\circ\), find the value of \(x\) and the measures of the arcs.