Angle Investigations Slides Angle Investigations
The Inner Circle: Part I
The Blueprint: Vocabulary
Central Angle
Vertex is at the center of the circle.
Inscribed Angle
Vertex is ON the circle; sides are chords.
Intercepted Arc
The section of the circle captured by an angle.
Central Inscribed Arc
Theorem 1: Central Strength
The measure of a central angle is equal to the measure of its intercepted arc.
\[ m\angle ABC = m\text{ arc } AC \]
B C A 80° 80°
Theorem 2: Inscribed Insights
The measure of an inscribed angle is half the measure of its arc.
\[ m\angle ABC = \frac{1}{2} m\text{ arc } AC \]
B C A 40° 80°
Special Case: The Semicircle
An inscribed angle that intercepts a diameter is always a right angle.
90°
90°
Cyclic Quadrilaterals
Opposite angles of a quadrilateral inscribed in a circle are supplementary.
Sum = 180°
A C
∠A + ∠C = 180°
Rapid Discovery
If a central angle is 110°, what is the intercepted arc?
110°
If an inscribed angle is 35°, what is the intercepted arc?
70°
Angle Investigations Notes Angle Investigations Notes
Geometry Unit: Circles | Lesson 01
Name:
Date:
1. Essential Vocabulary
Central Angle:
Inscribed Angle:
Intercepted Arc:
2. Circle Angle Theorems
Central Angle Theorem
The measure of a central angle is ____________________ to the measure of its intercepted arc.
\( m\angle \text{Central} = m\text{Arc} \)
x° x°
Inscribed Angle Theorem
The measure of an inscribed angle is ____________________ the measure of its arc.
\( m\angle \text{Inscribed} = \frac{1}{2} (m\text{Arc}) \)
x° 2x°
3. Special Geometric Cases
Angles in a Semicircle
Any angle inscribed in a semicircle is a:
Cyclic Quadrilaterals
Opposite angles are always:
4. Rapid Check-In
1. Find x:
72° x
x = _________
2. Find y:
y 48°
y = _________
3. Find z:
z 180°
z = _________
Big Idea Summary
Angles inside circles are defined by their intercepted arc. Central angles equal the arc measure, while inscribed angles are half the arc measure. Opposite angles of cyclic quadrilaterals are supplementary (180°).
Circle Vocabulary Match Circle Vocabulary Match
Master the language of the curve.
Name:
Chord
Diameter
Secant
Tangent
Central Angle
Inscribed Angle
Minor Arc
Major Arc
Semicircle
Radius
A.
A line that intersects a circle at exactly two points.
B.
An arc with a measure greater than 180 degrees.
C.
A segment whose endpoints both lie on the circle boundary.
D.
A line that intersects the circle at exactly one point.
E.
An angle with vertex on the circle and sides that are chords.
F.
A segment from the center to any point on its boundary.
G.
An arc with a measure exactly equal to 180 degrees.
H.
A chord that passes through the center of the circle.
I.
An arc with a measure less than 180 degrees.
J.
An angle with its vertex at the center of the circle.
Diagram Identification
Identify the part of the circle indicated by each number.
11 12 13 14
Angle Power Quiz Angle Power Quiz
Circle Theorem Assessment
Name:
Select the best answer for each question. All diagrams are for geometric context only.
1. If a central angle measures 82°, what is the measure of the arc it intercepts?
A) 41°
B) 82°
C) 164°
D) 90°
Work Area
2. An inscribed angle intercepts an arc of 120°. What is the measure of the inscribed angle?
A) 240°
B) 120°
C) 60°
D) 30°
Work Area
3. In a circle, an angle inscribed in a semicircle must always be:
A) Acute
B) Obtuse
C) Right
D) Scalene
Work Area
4. In cyclic quadrilateral ABCD, if ∠A = 75°, what is the measure of its opposite angle ∠C?
A) 75°
B) 15°
C) 105°
D) 285°
Work Area
5. Two inscribed angles intercept the same arc. Which statement is true?
A) The angles are complementary.
B) The angles are supplementary.
C) The angles are equal in measure.
D) One angle is double the other.
Work Area
6. A central angle and an inscribed angle intercept the same 50° arc. What is the difference in their measures?
A) 25°
B) 50°
C) 0°
D) 75°
Work Area
7. Arc AB measures 100°. If ∠ACB and ∠ADB are inscribed angles, what is their sum ∠ACB + ∠ADB?
A) 50°
B) 100°
C) 200°
D) 150°
Work Area
8. An inscribed angle measures 45°. What is the measure of its corresponding central angle?
A) 45°
B) 22.5°
C) 90°
D) 180°
Work Area
9. A quadrilateral is inscribed in a circle. If its angles are x, y, z, and w in order, which is true?
A) x + y = 180°
B) x + z = 180°
C) x + y + z + w = 180°
D) x = y = z = w
Work Area
10. If an arc measure is 360°, what kind of angle intercepts it at the center?
A) Acute angle
B) Right angle
C) Straight angle
D) Full rotation
Work Area
Angle Key Teacher Resource Teacher Key: Angle Investigations
Lesson 01 Mastery Key
Part 1: Vocabulary & Identification
Matching Answers
1. Chord C
6. Inscribed E
2. Diameter H
7. Minor Arc I
3. Secant A
8. Major Arc B
4. Tangent D
9. Semicircle G
5. Central J
10. Radius F
11. Diameter | 12. Radius | 13. Tangent | 14. Secant
Part 2: Quiz Solutions (10 MCQs)
1. B (82°) Central = Arc
6. A (25°) 50 - 25 = 25
2. C (60°) Inscribed = 1/2 Arc
7. B (100°) Sum (50+50)
3. C (Right) Semicircle Rule
8. C (90°) Double inscribed
4. C (105°) 180 - 75
9. B (x+z=180°) Opposite pair
5. C (Equal) Intercept same arc
10. D (Rotation) Full 360° circle
Segment Secrets Slides Segment Secrets
The Inner Circle: Part II
The Tangent Touch
A tangent line is always perpendicular to the radius at the point of tangency.
Angle = 90°
External Equality
Tangent segments from a common external point are congruent.
\[ \text{Tangent}_A = \text{Tangent}_B \]
x x
Intersecting Chords
The product of the segments of one chord equals the product of the segments of the other.
a * b = c * d
a b c d
The External Rule
Whole × Outside = Constant
\[ w_1 \cdot e_1 = w_2 \cdot e_2 \]
e1 whole w1 e2
Segment Secrets Notes Segment Secrets Notes
Geometry Unit: Circles | Lesson 02
Name:
I. Tangent Relationships
1. Tangent-Radius
Meets at point of tangency to form a:
2. Two-Tangent Rule
External segments to circle are:
II. Segment Calculations
Chord-Chord
The product of the segments of one chord equals the other.
a * b = c * d
a b c d
Secant-Secant
Whole * Outside = Constant
W1 * O1 = W2 * O2
O1 Whole W1
Tangent-Secant
Tan2 = Whole * Out
Tan Out
Discovery Challenge
1. Solve for x:
4 x 6 2
x = _________
2. Solve for y:
12 y
y = _________
Segment Strength Practice Worksheet Segment Strength Practice
Chord, Secant, and Tangent Mastery
Name:
Problem 01
Find the value of x for the intersecting chords.
6 4 8 x
Calculations:
x = _________
Problem 02
PA and PB are tangents to circle O. Solve for x.
P A B O 2x + 4 16
Calculations:
x = _________
Problem 03
Solve for x using the Secant rule.
4 8 6 x
Calculations:
x = _________
Problem 04
A chord of 12 cm is bisected by a perpendicular radius. Find the length of each half.
Explain reasoning:
Segments:
_____ & _____
Segment Key Teacher Resource Teacher Key: Segment Secrets
Lesson 02 Mastery Key
1 Chord-Chord Solution
6 * 4 = 8 * x
6 4 8 x
Intersecting Chords
2 Two-Tangent Solution
2x + 4 = 16
2x+4 16
Tangent Equality
3 Secant-Secant Solution
Whole * Out = Whole * Out
4 8 6 x
Whole x Out
4 Bisected Chord Solution
12 cm / 2 = 6 cm each
Radius Bisection
