Math 112 Circle Geometry Notes
Note(1) Similar triangles are triangles that have the same shape but not
necessarily the same size.
i.e. corresponding angles are equal.
E
B
A
C
D
F
ΔABC  ΔDEF
i.e. corresponding angles are congruent.
Note(2) Congruent triangles are triangles that have the same shape and size.
I.e. corresponding angles are equal and corresponding sides are equal.
B
E
A
`
C
D
ΔABC  ΔDEF
i.e. corresponding sides and angles are congruent.
F
Math 112 Circle Geometry Notes
Note(3) Unique triangle constructions. We can construct a number of
unique triangles given only 3 parts or measures. The constructions that
produce unique triangles are:
S.S.S. A.S.A S.A.S.-
Side , Side , Side
Angle, Side, Angle
Side, Angle, Side
A.A.S.H.L.-
Angle, Angle, Side
Hypotenuse Leg
Note(4) Other triangle constructions that are not possible or are not unique
are:
A.A.A.Angle, Angle, Angle
A.S.S.Angle, Side, Side
S.S.A.Side, Side, Angle
Asn(D1) Identifying Congruent Triangles
Asn(D2) Congruent Triangles.
Math 112 Circle Geometry Notes
Note(5) Formal Proof for S.S.S. Construction
Provide a formal proof that the two triangles given below are congruent.
B
A
E
C
Statement
1. AB = DE
2. AC = DF
3. BC = EF
4. ΔABC  ΔDEF
D
F
Authority
Given on diagram
Given
Given
S.S.S. construction
Note(6) Formal Proof of a S.A.S. construction
Provide a formal proof that the two triangles given below are congruent.
B
A
Statement
1. AB = DE
2. AC = DF
3. A = D
4. ΔABC  ΔDEF
E
C
D
F
Authority
Given on diagram
Given
Given
S.A.S. construction
(the angle is between and touching the 2 sides)
Math 112 Circle Geometry Notes
Note(7) Formal Proof of a A.A.S. construction
Provide a formal proof that the two triangles given below are congruent.
B
A
E
C
Statement
1. AB = DE
2. C = F
3. A = D
4. ΔABC  ΔDEF
D
F
Authority
Given on diagram
Given
Given
A.A.S. construction
(the side is not between the two angles)
Note(8) Formal Proof of a A.S.A. construction
Provide a formal proof that the two triangles given below are congruent.
B
A
Statement
1. AC = DF
2. A = D
3. C = F
4. ΔABC  ΔDEF
E
C
D
Authority
Given on diagram
Given
Given
A.S.A. construction
(the side is between the two angles)
F
Math 112 Circle Geometry Notes
Note(9) Formal Proof of a Hyp- Leg. construction
Provide a formal proof that the two triangles given below are congruent.
B
E
A
C
Statement
1. AB = DE
2. BC = EF
3. A = D = 900
4. ΔABC  ΔDEF
D
F
Authority
Given on diagram
Given
Given
Hyp- Leg construction
(Looks like a A.A.S. construction but with 900)
Note(10) Three parts must be equal to prove congruency. Once congruency
of triangles is proven, the three remaining parts are also congruent or equal.
e.g. ΔABC  ΔDEF
by S.A.S. construction
B
E
A
C
D
F
What other parts in the triangles above are equal (other than those marked)?
Parts marked as congruent:
Remaining parts equal (not marked):
AB = DE
A = D
AC = DF
B = E
C = F
BC = EF
Asn(D3) example 1 p. 214 and example 2 p. 215
Math 112 Circle Geometry Notes
To prove two triangles congruent it is important to identify angles that
are equal in various situations. Parallel lines provide equal angles, as do
intersecting lines that form vertically opposite angles.
Note(11) Parallel lines provide for congruent angles near the transversal.
The alternate angles for a transversal are equal. If the lines are not parallel
then the angles are not called alternate angles because they would not be
equal.
1
transversal
parallel lines
2
For the above diagram 1 = 2 because they are alternate angles on
opposite sides of the transversal between two parallel lines.
Note(12) Vertical opposite angles are angles formed where two lines
intersect. They are always equal.
1 = 2 Vertically Opposite ’s
1
2
Note (13) Base angles of isosceles triangles are equal. They are located by
finding the odd side that is not congruent. This side is called the base of the
triangle so that the base angles are touching the base.
Base angles
base
Asn(D4) Example 3 p. 216
Asn(D5) Handout Q 1 to 7 on intersecting lines, base angles of isosceles Δ
and congruent triangles.
Asn(D6) 22,23,24,26 p. 216, 217
Math 112 Circle Geometry Notes
Because of the regularity of a circle there are many properties that
occur that cause congruency in triangles to be possible. It is necessary to
define terms before proceeding.
Note(14) circle- the set of points in a plane that are all the same distance
from a fixed point called the center.
Note(15) semi-circle- half a circle.
Note(16) radius- a line segment joining the center of the circle to a point on
the circle.
Note(17) arc- part of a curve.
Note(18) chord- a line segment joining 2 points on a curve such as a circle.
Note(19) diameter- a chord through the center of a circle.
Note(20) central angle- the angle formed by the radii at the center of a circle.
Circle
Arc
semicircle
diameter
Central angle
radius
chord
Asn(D7) #1 A to C p. 206 Part 1 (Handout to photocopy)
Asn(D7) #1 D to J p. 206 Part 2 (Handout to photocopy)
Math 112 Circle Geometry Notes
Note(21) Chords that are equidistant from the center of a circle are
congruent.
A
C
O
5
N
M
5
D
B
In the diagram above we may say the chords are congruent because
they are equidistant from the center of the circle.
Statement
Authority
ON = OM = 5cm
given in the diagram
AB = CD
If the chords are equidistant from the center
then the chord lengths are equal
Note(22) Congruent chords on a circle are equidistant from the center of the
circle.
B
M 10 cm
A
O
C
N
10 cm
D
In the diagram above we may say the chords are equidistant to the
center of the circle because the chord lengths are equal.
Statement
Authority
AB = CD = 10 cm
Given in diagram
OM = ON
If the chords are equal (10 cm each)
then they are equidistant from the centre
Math 112 Circle Geometry Notes
Note(23) Converse statements are statements that have the concepts of the
“if” clause and “then” clause reversed. These may or may not be true.
e.g. converse statements that are true.
Statement: If chords are equidistant from the center of the circle then they
are congruent chords. (this is a true statement).
Converse: If chords are congruent then they are equidistant from the center
of the circle. (this is a true statement).
When a statement and its converse are both true we may use the
invented word “iff” with a double “f” in one sentence. It signifies that the
converse statement is also true and we may state the converse to be true.
The above two statements may be combined in one statement as follows:
Chords are equidistant from the center of the circle iff they are
congruent chords.
Asn(D7) Continued Q 4 to 11 p. 208,209
Example of a case where the converse is false:
Statement: If a polygon is a square then it has 4 right angles. (True)
Converse: If a polygon has 4 right angles then it is a square. (False)
Math 112 Circle Geometry Notes
Note(24) The farther a chord is from the center of a circle, the shorter the
chord length will be.
Chord length is, therefore, inversely proportional to the distance from
the center of the circle.
16.4 3.7 2.1 18
8
6
8.2
13.4
We can summarize the relation between distance to the center of a
circle and chord length in chart form;
Distance to center
2.1 cm
3.7 cm
6.0 cm
8.0 cm
increasing
Chord length
18 cm
16.4 cm
13.4 cm
8.2 cm
decreasing
The chord length and distance to the center are indirectly related and
define a linear relationship.
Asn(D8) Investigation #2 A to H p. 210 Part 1
Asn(D8) Investigation #2 I to N p. 210 Part 2
Math 112 Circle Geometry Notes
Note(25) The perpendicular line from the center of a circle to a chord bisects
the chord.
A
M
O
B
Statement
OM is a perpendicular line
from the center of the circle
AM = BM
Authority
Given in the diagram
A perpend. line from center bisects the
chord.
Note(26) The perpendicular bisector of a chord passes through the center of
a circle.
A
M
O
B
Statement
AM = BM
Authority
Given in diagram
OM is a perpendicular
bisector of AB that passes
through the center
A perpend. bisector of a chord
passes through the center
Asn(D8) Continued Q 12 to 21 p. 211,212
Math 112 Circle Geometry Notes
Note(27) The combined statement for the perpendicular right bisectors of
chords of a circle can be stated as follows:
Perpendicular lines to chords of a circle are perpendicular bisectors
iff they intersect at the center of a circle.
Special note: Most perpendicular lines are not right bisectors. This
phenomenon only occurs inside a circle.
Math 112 Circle Geometry Notes
Note(28) Formal proof that the bisector of a chord passing through the
center of a circle is perpendicular to the chord. (We are assuming that we do
not know that the lines from the center of a circle are perpendicular bisectors
of a chord.)
A
M
O
B
Given:
Circle with center O
Chord AB
M midpoint of AB
Prove:
OM  AB
Statement:
Authority:
1. OA = OB
2. AM = BM
3. OM = OM
4. ΔOMA  ΔOMB
5. OMA = OMB
6. OMA + OMB = 1800
7. OMA = 900 = OMB
8. OM AB
1. Radii of a circle
2. Given (M as midpoint)
3. Common side
4. S.S.S. congruency
5. Remaining parts are congruent
6. Supplementary angle sum
7. Division by 2
8. Definition of perpendicular
(90 degrees)
Asn(D9) Q 29,30,32 p. 219 to 221
Then: Q 35,36,37,38,39 p. 221
Test #4