Download Chord Central Angles Conjecture - 3

Chord Central
Angles
Conjecture
Adapted from Walch Education
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
2
Congruent Chords
Congruent chords
of a circle create
one pair of
congruent central
angles.
3.1.2: Chord Central Angles Conjecture
3
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
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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
5
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
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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
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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
8
Practice
•
3.1.2: Chord Central Angles Conjecture
9
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
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Step 2
Find the measure of
•
Subtract the measure of
3.1.2: Chord Central Angles Conjecture
from 360°.
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Your turn…
What conclusions can you make?
3.1.2: Chord Central Angles Conjecture
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Thanks for
Watching!!!
~dr. dambreville