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Geometry: Chapter 3 Notes
3.1: Symmetry in Polygons
Goal 1: Define a polygon.
Goal 2: Define and use reflectional symmetry and rotational symmetry
Goal 3: Define regular polygon and central angle of a regular polygon.
Polygon:
Sides of polygon:
Vertices of polygon:
Polygons Classified by number of sides
Number
of sides
Name of Polygon
Number
of sides
3
9
4
10
5
11
6
12
7
13
8
n
Equianglular polygon:
Equilateral polygon:
Regular polygon:
Central angle:
Reflectional Symmetry:
Axis of symmetry:
Name of Polygon
Triangles Classified by the Number of Congruent Sides
Equilateral triangle:
Isosceles triangle:
Scalene triangle:
Rotational Symmetry:
Central Angle of a Regular Polygon:
Example 1: Find the measure of the central angle of each regular polygon below:
a) decagon
b) heptagon
c) 15-gon
3.2: Properties of Quadrilaterals
Goal 1: Define quadrilateral, parallelogram, rhombus, rectangle, square and trapezoid.
Goal 2: Identify the properties of quadrilateral and their relationships.
Define:
1) Quadrilateral:
2) Trapezoid:
3) Parallelogram:
4) Rhombus:
5) Square:
6) Rectangle:
I) Properties of a Parallelogram:
a) Opposite sides of a parallelogram are ______________________.
b) Opposite angles of a parallelogram are ______________________.
c) Diagonals of a parallelogram ____________________________.
d) Consecutive angles of a parallelogram are ______________________.
II) Properties of a Rhombus:
a) A rhombus is a ____________________.
b) The diagonals of a rhombus are ______________________.
III) Properties of a Rectangle:
a) A rectangle is a _____________________.
b) The diagonals of a rectangle ________________________.
IV) Properties of a Square:
a) A square is a _______________________, __________________, and ___________________.
b) The diagonals of a square _____________________ each other, are _____________________
and are _________________________
Example 1: In the parallelogram GRAM, find the indicated values.
G
R
120°
10 in.
GR = ___________
MG = __________
mR = _________
M
A
15 in.
mA = _________
mM = _________
Example 2: In the rhombus ABCD, find the following:
Given:
AB = 25, AC = 20, mABC = 80°
AD = _________
DC = _________
A
mBAD = ________
E
D
B
mDCB = ________
AE = _________
C
EC = _________
mAEB = ________
3.3: Parallel Lines and Transversals
Goal 1: Define transversal, alternate interior angles, alternate exterior angles, corresponding angles, same-side
interior angles.
Goal 2: Make and use conjectures, postulates, and theorem about parallel lines.
Define:
a) Parallel Lines:
b) Transversal:
Example 1: Identify the following:
a) 1 and
8 are _________________________________
b)
7 and 3 are _________________________________
c)
3 and 6 are _________________________________
d)
3 and 5 are _________________________________
e)
6 and 3 are _________________________________
f) 2 and
1
3
5
7
2
4
6
8
7 are _________________________________
Corresponding Angles Postulate:
If two lines cut by a transversal are parallel, then ____________________________________
are _________________________.
Alternate Interior Angle Theorem
If two lines cut by a transversal are parallel, then ____________________________________
are _________________________.
Alternate Exterior Angle Theorem
If two lines cut by a transversal are parallel, then ____________________________________
are _________________________.
Same-Side Interior Angles Theorem
If two lines cut by a transversal are parallel, then ____________________________________
are _________________________.
Example 2: Find the measure of each angle.
a
c
1 ║  2 .
b
d
1
e 110°
f g
Example 3: Find the measure of each angle.
1
2
48°
2
1 ║  2 .
3.4: Proving that Lines are Parallel
Goal 1: Identify and use the converse of the corresponding angle postulate.
Goal 2: prove that lines are parallel by postulates and theorems.
I) Converse of Corresponding angles Theorem
______________________________________________________________________________________________
II) Converse of Alternate Interior Angle Theorem
______________________________________________________________________________________________
III) Converse of Alternate Exterior Angle Theorem
______________________________________________________________________________________________
IV) Converse of Same- Side Interior Angle Theorem
______________________________________________________________________________________________
Example 1: Indicate which lines are parallel and give a reason.
A) If
t
1
m
4
5
n
8 7
6
2
3
3.4: The Triangle Sum Theorem
Goal 1: Identify and use the parallel postulate and triangle sum theorem.
Parallel Postulate: Given a line and a point not on the line, there is ______________________________
Line that contains the given point and is _____________________ to the given line.
Triangle Sum Theorem: The sum of the measures of the interior angles of a triangle equals _________________.
Example 1: Find the measure of each missing angle.
B
A
A)
C
30°
B)
A 70°
30°
C
B
M
C)
D)
R
5x + 40
L
Q 60°
60°
S
3x
2x
N
Exterior Angle Theorem:
The measure of an exterior angle of a triangle is equal to the _____________________________________________.
Example 2: Find the measure of the indicated angle.
B
A)
80°
63°
A
B
D
B
B)
3x
2x
A
x
B
D
3.6: Angles in Polygons
Goal 1: Develop and use formulas for the sums of the measures of interior and exterior angles of a polygon.
Define:
Convex Polygon:
Concave polygon:
Examples of Convex Polygons
Polygon
Examples of Concave Polygons
Number of
Sides
Number of
Triangles
Sum of interior angles
triangle
quadrilateral
pentagon
hexagon
n-gon
Sum of the Interior Angles of a Polygon:
The sum of the measures of the interior angles of a polygon with n sides is __________________________________
Regular
Polygon
Number of
Sides
Sum of interior
angles
Measure of one interior
angle
triangle
quadrilateral
pentagon
hexagon
n-gon
Measure of an interior angle of a regular polygon with n sides is ________________________________________.
Example 1: Use a straight edge to construct a convex polygon. Then use a protractor to measure each exterior angle.
Sum of the Exterior Angles of a Polygon:
The sum of the measures of the exterior angles of a polygon is __________________________________________.
Example 2: find the indicated measures. Show work.
F
A)
E
E
B)
D
110°
A
D
C
B
A
C)
C
B
8x
B
D)
C
6x – 4
3x + 3
7x
2x + 20
A
5x + 1
4x
D
3.7 Midsegments of Triangles and Trapezoids
Goal 1: Define midsegments of triangles and trapezoids.
Goal 2: Develop and use formulas based on the properties of triangles and trapezoids.
Midsegment of a Triangle:
Midsegment of a Trapezoid:
Activity 1:
1. Draw triangle ABC. Find the midpoints, M
and N, of sides AB and
AC . Then draw
MN , the midsegment.
2. Measure MN and BC . What is the
relationship between their lengths?
3. Use a protractor to find the measure of
each angle. What do your measurements
suggest about
BC and MN ?
Triangle Midsegment Conjecture: A midsegment of a triangle is _______________________ to a side of the triangle
and has a measure equal to ________________ of that side.
A
Example 1: XY is a midsegment of triangle ABC. If BC = 15, What is the length of XY ?
X
Activity 2:
1. Draw trapezoid ABCD. Find the midpoints,
M and N, of the nonparallel sides. Draw
MN , the midsegment.
2. Measure the lengths of the bases AB and
DC and the midsegment MN . What is
the relationship between their lengths?
(Hint: Find AB + DC).
3. Use a protractor to find the measure of
each angle. What do your measurements
suggest about the relationship between
MN and the bases of the trapezoid?
Y
B
C
15
Trapezoid Midsegment Conjecture: A midsegment of a trapezoid is ________________________ to the bases of
the trapezoid and has a measure equal to ______________________________.
Example 2: XY is the midsegment of trapezoid ABCD, if BC = 15 and AD = 26, find XY.
B
15
C
X
Y
A
D
26
Example 3: Find the indicated value. Show work.
A
A)
B)
L
50
M
Q
X
N
Y
40
P
B
60
C
AB = ____________
QN = _______________
A
C)
X
B
2x + 1
Y
6x – 10
x = ________
AB = _________
HK = __________
C
O
3.8: Analyzing Polygons with Coordinates
Goal 1: Develop and use theorems about equal slopes and slopes of perpendicular lines.
Goal 2: Solve problems involving perpendicular and parallel lines.
Define:
Slope:
Example 1: Find the slope of AB with endpoints A(5, -3) and B(2, 3).
Example 2: Find the slope of MN with endpoints M(-1, 6) and N(-4, 0).
Parallel Theorem: Two nonvertical lines are parallel if and only if ___________________________________
Vertical lines are ______________________________.
Perpendicular Theorem: Two nonvertical lines are perpendicular if and only if the ______________________
of their slopes is _________________.
Any vertical line is ____________________________ to a horizontal line.
Example 3: Plot the following points for quadrilateral QUAD. Q(1, 4), U(7, 8), A(9, 5), D(3, 1). Find the slope of each
side. What type of quadrilateral is QUAD?
Slope
QU = _________
Slope DA = __________
Slope
QD = _________
Slope
UA = __________
Midpoint Formula:
Example 4: Find the midpoint of AB with endpoints A(-4, 2) and B(6, 4).
Example 5: Plot the triangle ABC with A(2, 6), B(0, 0), and C(4, 0).
a)
find the midpoint of AB
b)
find the midpoint of
c)
find the slope of the midsegment.
d)
find the slope of
e)
Is the midsegment parallel to
AC
BC
BC ?