1. Introduction
Circles have several special properties, conjectures, postulates, and theorems associated with them. This lesson focuses on the relationship between chords and the angles and arcs they create.
3.1.2: Chord Central Angles Conjecture

2. Key Concepts
Chords are segments whose endpoints lie on the circumference of a circle.
Three chords are shown on the circle to the right.
3.1.2: Chord Central Angles Conjecture

3. Key Concepts, continued
Congruent chords
of a circle create
one pair of
congruent central
angles.
3.1.2: Chord Central Angles Conjecture

4. Key Concepts, continued
When the sides of
the central angles
create diameters
of the circle, vertical
angles are formed.
This creates two
pairs of congruent
central angles.
3.1.2: Chord Central Angles Conjecture

5. Key Concepts, continued
Congruent chords also intercept congruent arcs.
An intercepted arc is an arc whose endpoints intersect the sides of an inscribed angle and whose other points are in the interior of the angle.
Remember that the measure of an arc is the same as the measure of its central angle.
Also, recall that central angles are twice the measure of their inscribed angles.
Central angles of two different triangles are congruent if their chords and circles are congruent.
3.1.2: Chord Central Angles Conjecture

6. Key Concepts, continued
In the circle below, chords and are congruent chords of
3.1.2: Chord Central Angles Conjecture

7. Key Concepts, continued
When the radii are constructed such that each endpoint of the chord connects to the center of the circle, four central angles are created, as well
as two congruent isosceles triangles by the SSS Congruence Postulate.
3.1.2: Chord Central Angles Conjecture

8. Key Concepts, continued
Since the triangles are congruent and both triangles include two central angles that are the vertex angles of the isosceles triangles,
those central angles are also congruent because Corresponding Parts of Congruent Triangles are Congruent (CPCTC).
3.1.2: Chord Central Angles Conjecture

9. Key Concepts, continued
The measure of the arcs intercepted by the chords is
congruent to the measure of the central angle because arc measures are determined by their central angle.
3.1.2: Chord Central Angles Conjecture

10. Common Errors/Misconceptions
mistakenly believing that central angles of different circles will be congruent if their chords are congruent
3.1.2: Chord Central Angles Conjecture

11. Guided Practice Example 1
3.1.2: Chord Central Angles Conjecture

12. Guided Practice: Example 1, continued Find the measure of
The measure of ∠BAC is equal to the measure of
because central angles are congruent to their
intercepted arc; therefore, the measure of is also 57°.
3.1.2: Chord Central Angles Conjecture

13. Guided Practice: Example 1, continued Find the measure of
Subtract the measure of from 360°.
3.1.2: Chord Central Angles Conjecture

14. Guided Practice: Example 1, continued State your conclusion.✔
3.1.2: Chord Central Angles Conjecture

15. Guided Practice: Example 1, continued
3.1.2: Chord Central Angles Conjecture

16. Guided Practice Example 2 What conclusions can you make?
3.1.2: Chord Central Angles Conjecture

17. Guided Practice: Example 2, continued
What type of angles are ∠G and ∠E, and what does that tell you about the chords?
∠G and ∠E are congruent central angles of congruent circles, so and are congruent chords.
3.1.2: Chord Central Angles Conjecture

18. ✔ Guided Practice: Example 2, continued
Since the central angles are congruent, what else do you know?
The measures of the arcs intercepted by congruent chords are congruent, so minor arc HI is congruent to minor arc JK.
Deductively, then, major arc IH is congruent to major arc KJ. ✔
3.1.2: Chord Central Angles Conjecture

19. Guided Practice: Example 2, continued
3.1.2: Chord Central Angles Conjecture