Todays Handouts:

1
What are the properties of inscribed angles?

• Todays Handouts
• Worksheet 5-5 (HW 28)
• Class Activity Worksheet 5-5 (Inscribed Angles)
• take a protractor ruler
• Warm-up
• Find the measure of each arc in ?O.
• 1) AB
• 2) BC
• 3) AD

2
HW 27 Review

• Answers to HW 27, L5-4 Properties of Chords
  Arcs
• /Worksheets/Unit 5/HW 5-4 Answers.doc

3
What are the properties of inscribed angles?
Review of some basic circle geometry definitions
to date
Circle the locus of points a given distance
from a given point. The given point being the
center of the circle.
Chord a line segment with endpoints on a
circle.
Tangent to a circle a line (or ray or line
segment) in the plane of the circle that
intersects the circle at exactly one point.
Central Angle an angle whose vertex is the
center of a circle.
Arc a part of a circle (three types major,
minor semicircle).
Arc measure the degree measure of an arc.
4
What are the properties of inscribed angles?
Compare contrast central and inscribed angles

• ?AOB is a central angle.
• Attributes of central angles
• Formed by two radii (vertex is center of ?)
• Measure of central angle measure of intercepted
  arc... m?AOB mAB

?E is an inscribed angle. What could be some
attributes of an inscribed angle?
5
What are the properties of inscribed angles?
Basic circle geometry definitions
Inscribed Angles An inscribed angle is an angle
formed by two chords that have a common endpoint
(the common endpoint being the vertex of the
angle).

• In the figure, the vertex of ?C is on ?O, and the
  sides of ?C are chords of the circle.
• ?C is an inscribed angle.
• AB is the intercepted arc of ?C.

6
What are the properties of inscribed angles?
read and do class worksheet

• Work in pairs.
• Answer the questions relating to the figure
  containing the triangle inscribed in the circle.
• ? Make a conjecture about the relationship
  between an inscribed angle and its intercepted
  arc.

• ?ABC is an __________________ triangle.
• What do you know about the interior angles of
  this kind of triangle?____________________________
  ________________
• 3. m?A _______?.
• 4. The degree measure of a complete circle is
  _________?.
• 5. ?ABC divides ?O into three _________________
  arcs.
• 6. mBC _______?.
• Make a conjecture about the reationship between
  the inscribed angle ?A and the degree measure of
  the arc it intercepts (mBC).
• _________________________________________________
  _____

7
What are the properties of inscribed angles?
read and do class worksheet

• Work in groups of 3-4.
• Use a protractor to find the measure of the
  numbered inscribed angles in ?O.
• ? Is the the conjecture you made previously
  confirmed?

8
What are the properties of inscribed angles?
Inscribed Angle Theorems
Theorem 12-8 The measure of an inscribed angle
is half the measure of its intercepted arc.
What can we say about two inscribed angles that
intersect the same arc? Corollary 1 Two
inscribed angles that intersect the same arc are
congruent.
How about angles inscribed in a semicircle?
Corollary 2 An angle inscribed in a semicircle
is a right angle.
What about the opposite angles of a quadrilateral
inscribed in a circle? Corollary 3 The
opposite angles of a quadrilateral inscribed in a
circle are supplementary.
9
What are the properties of inscribed angles?
Using Inscribed Angle Theorems

• Find the values of a and b in the
• diagram at the right.

10
What are the properties of inscribed angles?
Chord-Tangent Theorems
What conjecture can you make regarding the angle
formed by an intersecting chord and a tangent and
the arc they intersept? Theorem 12-9 The
measure of an angle formed by a chord and a
tangent that intersect on a circle is half the
measure of its intercepted arc.
11
What are the properties of inscribed angles?
Using Inscribed Angle Chord-Tangent Theorems

• 6. Find the value of each variable.

12
What are the properties of inscribed angles?
Using Inscribed Angle Theorems

• 7. Find the value of each variable. a.
• b.

13
What are the properties of inscribed angles?

• Wrap-up
• Writing What are you going to remember about
  inscribed angles?

14
What are the properties of inscribed angles?

• Homework Assignment
• Homework Worksheet 5-5
• Class Reminders
• Next class
• Periods Diagonal, 1, and 3 meet in Room 1015
• Period 7 meets in Computer Lab (Room 3006)