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 F
ΔABC @ ΔDEF
` i.e. corresponding sides and angles are congruent.
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. - Side , Side , Side A.A.S.- Angle, Angle, Side
A.S.A - Angle, Side, Angle H.L.- Hypotenuse Leg
S.A.S.- Side, Angle, Side
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 E
A C D F
Statement Authority
1. AB = DE Given on diagram
2. AC = DF Given
3. BC = EF Given
4. \ΔABC @ ΔDEF 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 E
A C D F
Statement Authority
1. AB = DE Given on diagram
2. AC = DF Given
3. ÐA = ÐD Given
4. \ΔABC @ ΔDEF 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 E
A C D F
Statement Authority
1. AB = DE Given on diagram
2. ÐC = ÐF Given
3. ÐA = ÐD Given
4. \ΔABC @ ΔDEF 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 E
A C D F
Statement Authority
1. AC = DF Given on diagram
2. ÐA = ÐD Given
3. ÐC = ÐF Given
4. \ΔABC @ ΔDEF A.S.A. construction
(the side is between the two angles)
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 D F
Statement Authority
1. AB = DE Given on diagram
2. BC = EF Given
3. ÐA = ÐD = 900 Given
4. \ΔABC @ ΔDEF 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 ÐB = ÐE
ÐA = ÐD ÐC = ÐF
AC = DF 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 Ð1 = Ð2 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.
Ð1 = Ð2 Vertically Opposite Ð’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 M
N 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 Chord length
2.1 cm 18 cm
3.7 cm 16.4 cm
6.0 cm increasing 13.4 cm decreasing
8.0 cm 8.2 cm
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 Authority
OM is a perpendicular line Given in the diagram
from the center of the circle
AM = BM 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 Authority
AM = BM Given in diagram
OM is a perpendicular A perpend. bisector of a chord
bisector of AB that passes passes through the center
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 1. Radii of a circle
2. AM = BM 2. Given (M as midpoint)
3. OM = OM 3. Common side
4. ΔOMA @ ΔOMB 4. S.S.S. congruency
5. ÐOMA = ÐOMB 5. Remaining parts are congruent
6. ÐOMA + ÐOMB = 1800 6. Supplementary angle sum
7. ÐOMA = 900 = ÐOMB 7. Division by 2
8. OM ^AB 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