1. Chord Central Angles Conjecture
Adapted from Walch Education

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. Congruent Chords 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.
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
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

7. 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

8. 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

9. Practice
3.1.2: Chord Central Angles Conjecture

10. Step 1 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

11. Step 2 Find the measure of Subtract the measure of from 360°.
3.1.2: Chord Central Angles Conjecture

12. Your turn… What conclusions can you make?
3.1.2: Chord Central Angles Conjecture

13. Thanks for Watching!!!
~ms. dambreville