This document discusses various properties of arcs and angles related to circles: - Arcs are portions of a circle and can be measured in degrees or length. Two arcs are congruent if subtended by congruent chords. - Parallel lines that intersect a circle intercept congruent arcs. - An inscribed angle is equal to half its intercepted arc. A diameter subtends 180 degrees. - An angle inside a circle is half the sum of intercepted arcs. An outside angle is half the difference of arcs. - Secant and tangent theorems relate lengths of chords and segments. - Example problems demonstrate using properties to find measures of unknown arcs.

1. Arcs

2. Arcs
• Any semi-circle, minor arc or major arc is
called an arc.
• An arc is subtended by the chord that
connects it endpoints.
• Arcs can be measured
• In degrees (360˚)
• In length (cm)

3. Congruent Arcs and Chords
Theorem
• In a circle, two arcs are congruent if and
only if they are subtended by congruent
chords.

4. Parallel Lines
• Two parallel lines that are secant or
tangent to a circle intercept congruent arcs
on the circle.
• See page 300-301 for special cases.

5. Inscribed Angle Theorem
• The measure of an inscribed angle equals one
half the arc angle.
• <ADC = ½ Arc AC
• NB Central Angle = Arc Angle
• A diameter subtends 180˚.

6. Interior Point Theorem
• An angle whose vertex is inside a circle
measures one-half the sum of the
subtended arc angles.
• <NAX = Arc NX + ArcBZ
2

7. Exterior Point Theorem
• An angle whose vertex is outside the circle
measures one half the difference of the
subtended arcs.
• <TPM = ArcTBN-ArcTAM
2

8. Secant Chord Theorem
• If 2 chords intersect in a circle, the product of the
lengths of the segments of one chord is equal to the
product of the lengths of the segments of the other
chord.
• If 2 secants are drawn from an external point P, the
product of the lengths of the SECANT and its
EXTERNAL part is equal to the product of the
length of the OTHER secant and its EXTERNAL
part.
• If a secant segment and a tangent segment are
drawn from an external point P, the length of the
tangent segment is the GEOMETRIC MEAN
between the lengths of the SECANT and its
EXTERNAL part.

9. •

10. Exam Question
In the figure below, arc AD measures 45 and angle P measures 10.
A
B
C
D
O
P
What is the measure of arc BC?
A) 20 C) 30
B) 25 D) 35

11. Exam QuestionIn the figure below, the angle at center O measures 60 and angle P measures 30.
A
B
C
D
O
P
What is the measure of arc AB?
A) 90 C) 110
B) 100 D) 120

12. Exam Question
In the figure to the right, line AB is tangent at C to the circle
with centre O and is parallel to the secant line DE.
Chord CF intersects the secant line DE at point G.
Which pair of arcs is necessarily congruent?
A
B
C
D
E
F
G
O
A) CD and CE C) CE and EF
B) CD and EF D) DF and DCE

13. Activity
• Page 301, 1-3, 9, 11
• Page 311, 1, 3, 4, 5
• Page 318, 1, 2, 7, 9
• Page 332, 1-7, 8-14