Download Class Notes Triangles

Geometry Notes T - 1: Parallel Lines and Transversals
Definition: Two coplanar lines are parallel if
.
Postulate: Through a point not on a given line, there
is exactly one line parallel to the given line.
t
Vocabulary (Know these!)
Corresponding angles: Angles in the same relative positions.
Ex: 1 and 5 (both in the “upper right”)
Also, 2 and 6, 3 and 7, and 4 and 8.
1
2
Transversal: A line, t, that intersects two other lines, l1 and l2,
at different points.
6
7
l1
4
3
5
l2
8
Alternate interior angles: Angles between the two lines and on opposite
sides of the transversal.
Ex: 3 and 5, 4 and 6
Same side interior angles: Angles between the two lines and on the same side of the transversal.
Ex: 3 and 6, 4 and 5
Two Facts
1. If a line is translated in its own direction, its image will be the same line.
P
P'
2.
If two lines, l1 and l2, are parallel and P is any point on l1 and Q is any point on
l2, then after a translation along the vector PQ , the image of l1will coincide with
l2. (If l1 and l2, are not parallel , then no translation will make the image of l1
coincide with l2.)
P
l1
PQ
Q
l2
Corresponding Angles Theorem and Converse
t
Given: l1 and l2, transversal t intersects l1 at P and l2 at Q
2
3
a. If l1 || l2, then under the translation along PQ ,
6
7
the image of t is
1
l1
P
4
5
Q
8
l2
the image of l1 is
The is means that the image of 1 is
Therefore, 1
, 2
, the image of 2 is
, etc.
, etc. because in each pair
b. If l1  l2, then under the translation along PQ , the image of t will still be t but the image of l1 will not be l2
and so the image of 1 will not be 5 and the angles will not be congruent. (Same for the other three pairs.)
Theorems: When parallel lines are cut by a transversal,
and (converse)
When two lines are cut by a transversal and corresponding angles are congruent,
t
Ex: If l1 || l2, find the measures of all seven unknown angles on the diagram.
l1
130
t
Ex: Which lines are parallel?
47
48
49
48
Alternate Interior Angles Theorem and Converse
a
b
c
d
l2
Theorem: When parallel lines are cut by a transversal, alternate interior angles are congruent.
t
Given:
2
Prove:
3
6
7
1
l1
4
5
l2
8
The converse of this theorem is also true: When two lines are cut by a transversal and alternate interior angles
are congruent, the lines are parallel.
Ex: In the diagram at right, find the measures of the three marked angles and
determine if l1 || l2.
6x – 4 15x – 5
5x + 5
l2
l1
Geometry Notes T - 2: Parallel Lines and Transversals 2
Summary of Important Facts
1.
If lines are parallel, corresponding angles are congruent.
Ex: If l1 || l2, 1  5.
t
1C. If corresponding angles are congruent, then lines are parallel.
Ex: If 2  6, then l1 || l2.
2.
2
3
If lines are parallel, alternate interior angles are congruent.
Ex: If l1 || l2, 3  5
6
2C. If alternate interior angles are congruent, then lines are parallel.
Ex: If 4  6, then l1 || l2.
3.
7
1
l1
4
5
l2
8
If lines are parallel, same side interior angles are supplementary.
Ex: If l1 || l2, 4 and 5 are supplementary.
3C. If same side interior angles are supplementary, then lines are parallel
Ex: If 3 and 6 are supplementary, then l1 || l2.
Ex: Given: Quadrilateral ABCD with AB || CD D B
Prove: ABCD is a parallelogram
A
D
B
C
Ex: Given: AB || CDE , BDF , CD bisects ADF
Prove: DAB DBA
F
D
C
A
E
B
Geometry Notes T - 3: Triangles
Theorem: The sum of the interior angles of a triangle is
.
C
Given: ABC
2
Prove: m1 + m2 + m3 =
A
1
3
B
Geometry Notes T - 4: Exterior Angle Theorem
Definition: An exterior angle of a polygon is an angle outside the polygon formed by extending one side.
b
c
a
Theorem: The measure of an exterior angle of a triangle is equal to
Corollary: The measure of an exterior angle of a triangle is
Ex: Find the value of x in each diagram:
a.
x
30
3x
b.
80
x
c.
x
3x – 25
Geometry Notes T - 5: Triangle Inequalities
The Triangle Inequality Theorem
Postulate: The shortest distance between two points is a straight line.
In a triangle, this means: 1)
C
and 3)
Theorem: In a triangle,
Corollary: In a triangle,
Ex: Which of the following could be the sides of a triangle?
1. {2, 3, 7}
2. {6, 8, 10}
3. {6, 6, 1}
4. {4, 8, 4}
Ex: Using the same choices above:
a. Which could be the sides of an isosceles triangle?
b. Which could be the sides of a right triangle?
a
c
b
and 2)
A
c
a
B
Bounds on Third Side of a Triangle
Ex: Two sides of a triangle measure 4 and 7. What are all the possible lengths for the third side, c?
c
4
If c is the longest side, then:
7
If 7 is the longest side, then
Fact: If we know the lengths of two sides of a triangle, a and b, with a  b, then the length of the third side c
must be
b
(Known)
c ?
a
(Known)
Ex: Two sides of a triangle measure 6 and 13.
a. How long can the third side be?
b. How long is the third side if the triangle is isosceles?
Ex: The sides of a triangle measure x, 2x and 5x – 12. What are all the possible values for x?
2x
x
5x – 12
Angles and Sides of a Triangle
Ex: In the triangle below, indicate the largest and smallest angles; indicate the longest and shortest sides.
FACT: The longest side of a triangle is
Ex: SML has sides ML = 6, SL = 7 and SM = 8.
L
Which is the largest angle of the triangle?
7
6
The smallest angle?
M
S
8
P
Ex: Which is the longest side of PQR?
60
The shortest side?
R
125
Q
Geometry Class Work: Medians of a Triangle
PQR with vertices P(0, 4), Q(6, 0) and R(18, 8) is graphed below.
y
R
8
6
P
4
2
0
2
4
6
Q
8
10
12
14
16
18
x
1. Let L, M and N, be the midpoints of PQ , QR and RP . Graph medians RL , PM and QN . Use a
straight-edge.
2. The point where the three medians intersect is called the centroid. Label it C and find its coordinates. (If
your medians do not all intersect in a single point, check your work.)
3. Find the following lengths. Irrational lengths should be in simplest radical form.
PC =
CM =
QC =
CN =
RC =
CL =
4. Find the following ratios in simplest form:
PC:CM =
QC:CN =
RC:CL =
Geometry Notes T - 6: Centroid of a Triangle
Theorem: The medians of a triangle are concurrent.
C
Medians AD , BE and CF all intersect
at point G, called the centroid.
E
Note: The centroid of a triangle is its center of gravity
(a.k.a the the center of mass), the point on which the
triangle can be perfectly balanced.
D
G
A
F
B
Theorem: The centroid divides each median in the ratio 2:1.
AG:GD = BG:GE = CG:GF = 2:1
Theorem: If the vertices of a triangle are (x1, y1), (x2, y2), (x3, y3), the coordinates of the centroid are
 x  x  x y  y2  y3 
G 1 2 3 , 1

3
3


(Average the x-values; average the y-values.)
Ex: Find the coordinates of the centroid of ABC having vertices A(4, –2), B(7, 3) , and C(1, 8).
Ex: The coordinates of the vertices of RST are R(a, b), S(a + 5, b – 1) and T(2a, 3b + 8). The coordinates of
the centroid of the triangle are (–1, 9). Find the values of a and b.
Ex: In PQR, M is the midpoint of QR and C is the centroid. If PC = 4x + 4 and CM = 3x – 3, find the length
of median PM .
Geometry Notes T - 8: Prooflets
A
Ex: Given: DAE  BCF, BAE  DCF
Prove: DAB  BCD
B
F
E
D
Statement
C
Reason
A
Ex. Given: DEFB , DF  EB
Prove: DE  FB
B
F
E
D
Statement
C
Reason
A
B
Ex. Given: AE  DEFB , CF  DEFB
Prove: DEA  BFC
F
E
D
Statement
Reason
C
A
B
Ex. Given: AB || CD , DEFB
Prove: ABF  CDE
F
E
D
Statement
C
Reason
F
Ex. Given: AEB , CD , CEF , FEB  ECD
Prove: AB || CD
E
A
B
C
D
Statement
Reason
F
Ex. Given: AEB , CEF
Prove: AEF  CEB
E
A
C
Statement
Reason
B
D
F
Ex. Given: AEBG , AEF  GBF
Prove: FEB  FBE
E
A
C
Statement
Reason
B
D
G