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Grade 9 Math
Unit 7 – Circle Geometry
7.3 – Properties of Angles in a Circle
Arcs
A section of the circumference of a circle is an arc.
The shorter arc AB is the minor arc.
The longer arc AB is the major arc.
Angles
The angle formed by joining the endpoints of an arc to the centre of the circle
is called a central angle.
The angle formed by joining the endpoints of an arc to a point on the circle is
called an inscribed angle.
The inscribed and central angles in this circle are subtended by the arc AB.
Central and Inscribed Angle Property:
The central angle is twice the size of the inscribed angle when both angles are
subtended by the same arc.
 POQ = 2  PRQ
 PRQ = 1/2  POQ
Inscribed Angles Property:
All inscribed angles subtended by the same arc are congruent (equal).
Angles in a Semicircle Property:
All inscribed angles subtended by a semicircle are right angles.
Since  AOB = 180°, then  AFB =  AGB =  AHB = 90°
Ex. 1 Point O is the centre of a circle. Determine the values of  x and  y.
Answer:
Since  ADB and  ACB are inscribed angles
subtended by the same arc, AB, these angles are
equal. So  x = 55°.
Since both the central angle  AOB and the
inscribed angle  ADB are subtended by the same
arc, AB,  AOB = 2  ADB
So  AOB = 2 (55°) = 110°.
Ex. 2 Rectangle ABCD has its vertices on a circle with radius 8.5 cm.
The width of the rectangle is 10.0 cm. What is its length?
Give the answer to the nearest tenth.
Answer:
The length of the rectangle is AD.
Each angle is 90°, so each angle is subtended
by a semicircle.
 ADC is subtended by semicircle ABC.
So the diameter of the circle is the same as the diagonal of the rectangle.
Therefore, length AC = 2 x 8.5 cm = 17 cm.
Use Pythagorean’s Theorem to find length BC.
x 2  10 2  17 2
x 2  289  100
x 2  189
x 2  189
x  13.7
Therefore, the length of the rectangle is 13.7 cm.
Ex. 3 Triangle ABC is inscribed in a circle, centre O.
 AOB = 100° and  COB = 140°
Determine the values of x°, y° and z°.
Answer:
To solve  x:
All angles in a circle add up to 360°, so
360 – 100 – 140 = 120°
To solve  y:
Since  ABC is an inscribed angle and
 AOC is a central angle subtended by
the same arc,  ABC = ½  AOC
Therefore,  ABC = ½ (110) = 55°
To solve  z:
Since OB, AO and OC are radii, all of the triangles are isosceles.
Therefore  z =  OAC. Since all angles of a triangle add up to 180°,
120 + z + z = 180
2z = 180 – 120
2z = 60
z = 30°
Ex. 4 Point O is the centre of the circle. Determine the value of x° and y°.
Which circle properties did you use?
Answer:
Since AO, OC, and OB are all radii, they are
isosceles triangles.
Therefore,  OAC =  ACO = 30°
So  AOC = 180 – 30 – 30 = 120°.
And  x = 180 – 120 = 60°
 ACB = 900, so  ABC = 180 – 30 – 90 = 600
Assignment
Do #3-6, 11 p. 410
Do #1 – 10 p. 418 Chapter Review
This is the end of Unit 7 – Circle Geometry. The unit test will be on ______________________!
Grade 9 Math
Unit 7 – Circle Geometry
Student Copy
7.3 – Properties of Angles in a Circle
Arcs
A section of the ______________ of a circle is an _____.
The shorter arc AB is the ____________ arc.
The longer arc AB is the _____________ arc.
Angles
The angle formed by joining the ______________ of an ______ to the
__________ of the circle is called a _______________ angle.
The angle formed by joining the endpoints of an arc to a __________ on the
circle is called an _______________ angle.
The inscribed and central angles in this circle are ________________ by the arc _____.
Central and Inscribed Angle Property:
The ____________ angle is ___________ the size of the ________________
angle when both angles are subtended by the same arc.
 POQ = __________
 PRQ = ___________
Inscribed Angles Property:
All inscribed angles _______________ by the same arc are _____________
(_________).
Angles in a Semicircle Property:
All inscribed angles subtended by a ______________ are _______ angles.
Since  AOB = _____°, then  AFB =  AGB =  AHB = ____°
Ex. 1 Point O is the centre of a circle. Determine the values of  x and  y.
Ex. 2 Rectangle ABCD has its vertices on a circle with radius 8.5 cm.
The width of the rectangle is 10.0 cm. What is its length?
Give the answer to the nearest tenth.
Ex. 3 Triangle ABC is inscribed in a circle, centre O.
 AOB = 100° and  COB = 140°
Determine the values of x°, y° and z°.
Ex. 4 Point O is the centre of the circle. Determine the value of x° and y°.
Which circle properties did you use?
Assignment
Do #3-6, 11 p. 410
Do #1 – 10 p. 418 Chapter Review
This is the end of Unit 7 – Circle Geometry. The unit test will be on ______________________!