10.3 and 10.4 Arcs and Chords and Inscribed Angles

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WARM UP

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Inscribed Angles

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Florida’s B.E.S.T. Standards for
Mathematics

MA.912.GR.5.3

Construct the inscribed and circumscribed
circles of a triangle.

MA.912.GR.6.2

Solve mathematical and real-world problems
involving the measures of arcs and related
angles.

MA.912.GR.6.3

Solve mathematical problems involving
triangles and quadrilaterals inscribed in a
circle

MA.K12.MTR.4.1
Engage in discussions that reflect on the
mathematical thinking of self and others.

MA.K12.MTR.5.1
Use patterns and structure to help understand
and connect mathematical concepts.

MA.K12.MTR.6.1
Assess the reasonableness of solutions.

Florida’s Mathematical Thinking and
Reasoning Standards

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Inscribed Angles

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Lesson Objectives

Content Objective

Students solve problems using inscribed angles.

Language Objectives

• Students talk about inscribed angles using because and

therefore.

• To support sense-making, students participate in MLR1: Stronger

and Clearer Each Time.

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Inscribed Angles

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1.

Congruent Arcs

Congruent Chords

2.

If Radius is ⊥ to a chord

The chord is bisected

3.

Congruent Chords are
equidistant from center

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1.

Congruent Arcs

Congruent Chords

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Inscribed Angles

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YOU TRY!

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Inscribed Angles

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If the radius is
perpendicular to a
chord, the chord
is bisected

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QS is perpendicular to PR.
RT is 6. Find PR.

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Congruent chords
are equidistant to
the center

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Inscribed Angles

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Inscribed Angles

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Inscribed Angles

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Learn
Inscribed Angles

An inscribed angle is an angle that has its vertex on a circle and sides that contain chords of the circle.

In ⊙𝐶, ∠𝑄𝑅𝑆 is an inscribed angle, and
minor arc ෢
𝑄𝑆is intercepted by ∠𝑄𝑅𝑆.

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Inscribed Angles

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Central Angle = Arc Measure

A

r

c

M

ea

s

u

r

e

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Inscribed Angles

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Central Angle = Arc Measure

A

r

c

M

ea

s

u

r

e

Inscribed Angle =½ Arc Measure

Inscribed Angle =½ Arc Measure

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Inscribed Angles

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Inscribed Angles

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A

B

What can we conclude about
∠𝐴 𝑎𝑛𝑑 ∠𝐵?

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A

B

What can we conclude about
∠𝐴 𝑎𝑛𝑑 ∠𝐵?

Inscribed angles that
share an intercepted arc
are congruent

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A

B

If AB is the diameter of the circle,
what can we conclude about ∠𝐶?

C

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A

B

If AB is the diameter of the circle,
what can we conclude about ∠𝐶?

C

An inscribed angle whose
intercepted arc is a right angle

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A

B

Quadrilateral ABCD is
inscribed in a circle, what
can we conclude about
∠𝐴 𝑎𝑛𝑑 ∠𝐶?

C

D

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A

B

Quadrilateral ABCD is
inscribed in a circle, what
can we conclude about
∠𝐴 𝑎𝑛𝑑 ∠𝐶?∠𝐵 𝑎𝑛𝑑 ∠𝐷?

C

D

Opposite angles in an INSCRIBED
quadrilateral are supplementary.

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Example 2
Find Measures of Congruent and Inscribed Angles

Find 𝒎∠𝑨.

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Example 4
Find Angle Measures in Inscribed Triangles

Find𝒎∠𝑲.

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Example 4
Find Angle Measures in Inscribed Quadrilaterals