10.3 and 10.4 Arcs and Chords and Inscribed Angles

Copyright © McGraw Hill

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

10.3 Arcs and Chords

10.4 Inscribed Angles

Copyright © McGraw Hill

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

WARM UP

McGraw Hill |

Inscribed Angles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

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

McGraw Hill |

Inscribed Angles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

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.

McGraw Hill |

Inscribed Angles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

1.

Congruent Arcs

Congruent Chords

2.

If Radius is ⊥ to a chord

The chord is bisected

3.

Congruent Chords are
equidistant from center

McGraw Hill |

Inscribed Angles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

1.

Congruent Arcs

Congruent Chords

McGraw Hill |

Inscribed Angles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

McGraw Hill |

Inscribed Angles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

YOU TRY!

Multiple Choice

What is the measure of segment RS?

• 20

• 4

• 11

• 180

McGraw Hill |

Inscribed Angles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

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

McGraw Hill |

Inscribed Angles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

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

McGraw Hill |

Inscribed Angles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

Congruent chords
are equidistant to
the center

McGraw Hill |

Inscribed Angles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

McGraw Hill |

Inscribed Angles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

Multiple Choice

Check 4.

In Circle P, CE=FH=48. Find PG.

• 3

• 9

• 15

• 30

McGraw Hill |

Inscribed Angles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

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.

McGraw Hill |

Inscribed Angles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

Central Angle = Arc Measure

A

r

c

M

ea

s

u

r

e

McGraw Hill |

Inscribed Angles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

Central Angle = Arc Measure

A

r

c

M

ea

s

u

r

e

Inscribed Angle =½ Arc Measure

Inscribed Angle =½ Arc Measure

McGraw Hill |

Inscribed Angles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

McGraw Hill |

Inscribed Angles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

Multiple Choice

Find arc BC

• 96

• 48

• 24

• 180

Multiple Choice

Find angle ADB.

• 76

• 45

• 38

• 90

Multiple Choice

Find arc DC.

• 76

• 38

• 45

• 90

Multiple Choice

Find angle YZX.

• 52

• 104

• 55

• 73

Multiple Choice

Find arc FH.

• 35

• 70

• 140

• 180

McGraw Hill |

Inscribed Angles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

A

B

What can we conclude about
∠𝐴 𝑎𝑛𝑑 ∠𝐵?

McGraw Hill |

Inscribed Angles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

A

B

What can we conclude about
∠𝐴 𝑎𝑛𝑑 ∠𝐵?

Inscribed angles that
share an intercepted arc
are congruent

McGraw Hill |

Inscribed Angles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

A

B

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

C

McGraw Hill |

Inscribed Angles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

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

McGraw Hill |

Inscribed Angles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

A

B

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

C

D

McGraw Hill |

Inscribed Angles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

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.

McGraw Hill |

Inscribed Angles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

Example 2
Find Measures of Congruent and Inscribed Angles

Find 𝒎∠𝑨.

McGraw Hill |

Inscribed Angles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

Example 4
Find Angle Measures in Inscribed Triangles

Find𝒎∠𝑲.

McGraw Hill |

Inscribed Angles

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

Example 4
Find Angle Measures in Inscribed Quadrilaterals

Multiple Choice

True or False: The red arcs are congruent.

• True

• False

Multiple Choice

Find the measure of the given arc.

• $$276^o$$

• $$155^o$$

• $$138^o$$

• $$84^o$$

Multiple Choice

What is x?

• 5

• 23

• 2

• 28

Multiple Choice

Solve for x.

• 86

• 12

• 11

• 94

Multiple Choice

Find a and b.

• a = 74°

  b = 93°

• a = 93°

  b = 74°

• a = 87°

  b = 106°

• a = 106°

  b = 93°

Multiple Choice

then what is the value of x?

• 13

• 3

• 24

• 6