View
220
Download
1
Category
Preview:
Citation preview
Warm Up
Lesson 3.1 AIM: Properties of Parallel Lines
DO NOW: Name an angle congruent to angle 2.
Angles 2, 3, 6, and 7 are
congruent.
A transversal is… A line that intersects two or more lines at different
points.
Which angles are alternate interior angles?
Angles 3 and 6.
Angle 4 and 5.
Alternate Interior Angles
are congruent.
Which angles are same side interior angles?
Angles 3 and 5.
Angle 4 and 6.
Same Side Interior Angles
are supplementary.
Which angles are corresponding angles?
Angles 1 and 5.
Angle 2 and 6.
Angles 3 and 7.
Angles 4 and 8.
Corresponding Angles
are congruent.
Which angles are alternate exterior angles?
Angles 1 and 8.
Angle 2 and 7.
Alternate Exterior Angles
are congruent.
What is the measure of angle 5?
Angle 5 = 135º
Corresponding Angles
are congruent.
What is the measure of angle PQR?
Angle PQR = 120º
Alternate Interior Angles
are congruent.
What is the measure of angle 9?
Angle 9 = 90º
Alternate Exterior Angles
are congruent.
What is the measure of angle 6?
Angle 6 = 50º
Same Side Interior Angles
are supplementary.
Finding Measures of Angles What are the measures of the missing angles?
3 105m 4 105 180m
4 75m
Find the value of x. Relationship:
Corresponding Angles
125 = x + 15
-15 -15
110 = x
Finding an Angle Measure What is the value of y?
( 40) 80 180y
120 180y 60y
Summary Question
Practice!!
Pgs. 118-120 #1-15 odd #27 and #29
Turn in #27 and 29!
Warm Up
18
Lesson 2 - 2
Proving Lines
Parallel
19
Proving Lines Parallel - Postulates & Theorems
If two lines are cut by a transversal and corresponding
angles are congruent, then the lines are parallel.
DC
BA
20
Proving Lines Parallel - Postulates &Theorems
If two lines are cut by a transversal and alternate
interior angles are congruent, then the lines are parallel.
DC
BA
21
Proving Lines Parallel - Postulates &Theorems
If two lines are cut by a transversal and consecutive
interior angles are supplementary, then the lines are
parallel.
DC
BA
22
Proving Lines Parallel - Postulates &Theorems
If two lines are cut by a transversal and consecutive
exterior angles are supplementary, then the lines are
parallel.
DC
BA
23
Examples: Proving Lines Parallel
Find the value of x which will make lines a and lines b parallel.
4. 20° Answers: 1.
20°
2.
50°
3.
90°
2.
80
2x
b
a1.
60
3x
b
a
3.
70
(x-20)
b
a 4.
60
3x
b
a
24
Ways to Prove Two Lines Parallel
Show that corresponding angles are equal.
Show that alternative interior angles are equal.
Show that consecutive interior angles are supplementary.
Show that consecutive exterior angles are supplementary.
In a plane, show that the lines are perpendicular to the same line.
Practice
Pgs. 125-127 # 2-16 even
#18-24 all
#25
Section 3-3 Parallel Line and the Triangle Angle-Sum Theorem
Purpose:
Students will classify triangles, find the measures of their
angles, and use exterior angles of triangles to solve problems.
Finding Angle Measures in Triangles
Theorem 3-7: Triangle Angle-Sum Theorem:
The sum of the measures of the angles of a triangle is 180.
mAmBmC 180
A
C
B
50° 35°
Finding Angle Measures in Triangles
A
C
B
50° 35°
Find the measure of
the three angles.
Justify your work.
Finding Angle Measures in Triangles
F
G
H J
65°
39° 21°
x° y° z°
Find the values of x, y, and z.
Justify your work.
Finding Angle Measures in Triangles
A
B
D C
62° 49°
x° y°
z°
Find the values of x, y, and z.
Classifications of Triangles
Equiangular
All 's are
Acute
All 's acute
Right
One right Obtuse
One obtuse
Equilateral
All sides
Isosceles
At least 2 sides
Scalene
No sides
Classifications of Triangles
37°
37°
Classify the following triangles by their angles and sides.
Draw an equilateral right triangle.
NOT POSSIBLE!!!
1
2
3
Using Exterior Angles of Triangles
Exterior
Angle
Exterior Angle of a Polygon: An angle formed by a side and an
extension of an adjacent side.
Remote Interior Angles: The two nonadjacent interior angles
corresponding to each exterior angle of a triangle.
Remote
Interior
Angles
Using Exterior Angles of Triangles
Theorem 3-8: Triangle Exterior Angle Theorem:
The measure of each exterior angle of a triangle equals the sum
of the measures of its two remote interior angles.
m1m2m3
1
2
3
Using Exterior Angles of Triangles
x°
100°
50°
Find the measure of
the x. Justify your
work.
Using Exterior Angles of Triangles
53° w° 86°
x° y°
z°
Find the values of w, x, y, and z. 97°
Little More Practice
(80 – 2x)°
(70 – 3x)°
Find the value of x.
Practice:
Pg. 134-1371-51 odd
#51 can be turned in for 5 extra credit points!!
Warm Up
3-4 The Polygon Angle-Sum Theorem
Theorem 3-9 Polygon Angle-Sum Theorem
The sum of the measures of the interior angles of an n-gon is (n – 2) * 180.
Finding a Polygon Angle Sum What is the sum of the interior angle measures of a
heptagon?
Sum = (n – 2) * 180
= (7 – 2) * 180
= 5 * 180
= 900°
The sum of the interior angle measures of a heptagon is 900°.
An equilateral polygon is a polygon with all sides congruent.
An equiangular polygon is a polygon with all angles congruent.
A regular polygon is a polygon that is both equilateral and equiangular.
Equilateral Polygon Equiangular
Polygon
Regular
Polygon
Corollary to the Polygon Angle-Sum Theorem
The measure of each interior angle of a regular n-gon is ( 2) 180
.n
n
Using the Polygon Angle-Sum Theorem The common housefly, Musca domestica, has eyes that
consist of approximately 4000 facets. Each facet is a regular hexagon. What is the measure of each interior angle in one hexagonal facet?
( 2) 180n
n
(6 2) 180
6
4 180
6
120
Using the Polygon-Angle Theorem
What is the measure of angle Y in pentagon TODAY?
Using the Polygon-Angle Theorem (5 2) 180m T m O m D m A m Y
110 90 120 150 3 180m Y
470 540m Y
70m Y
Polygon Exterior Angle-Sum Theorem
The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360.
1 2 3 4 5 360m m m m m
Finding an Exterior Angle Measure
What is the measure of angle 1 in the regular octagon?
3601
8m
1 45m
Practice!!!!! Homework – Textbook p. 147-149
1-25 odd
28-31 all
32-55 odd
Warm-Up
Parallel and Perpendicular Lines
Parallel Lines Two lines with the same slope are said to be parallel lines. If
you graph them they will never intersect.
We can decide algebraically if two lines are parallel by finding
the slope of each line and seeing if the slopes are equal to each
other.
We can find the equation of a line parallel to a given line and
going through a given point by: a.)
first finding the slope m of the given line; b.) finding
the equation of the line through the given point with slope
m.
Testing if Lines are Parallel Are the lines parallel? 12 3 9 and -8 2 14x y x y
Find the slope of 12 3 9
3 12 9
4 3
x y
y x
y x
The slope m = -4
Find the slope of 8 2 14
2 8 14
4 7
x y
y x
y x
The slope m = -4
Since the slopes are equal the lines are parallel.
Graphs of Parallel Lines
The red line is the graph of
y = – 4x – 3
and the blue line is the graph of
y = – 4x – 7
Practice Testing if Lines are Parallel Are the lines 6 3 5 and 2 4 4x y y x parallel? (click mouse for answer)
6 3 5
3 6 5
523
2
x y
y x
y x
m
2 4 4
2 2
2
y x
y x
m
Since the slopes are different
the lines are not parallel.
Are the lines 2 4 and 2 4 12x y x y parallel? (click mouse for answer)
2 4
2 4
1 22
12
x y
y x
y x
m
2 4 12
4 2 12
1 32
12
x y
y x
y x
m
Since the slopes are equal
the lines are parallel.
Constructing Parallel Lines Find the equation of a line going through the point (3, -5) and
parallel to 2 83
y x
Using the point-slope equation where the slope m = -2/3
and
the point is (3, -5) we get 25 33
25 23
2 33
y x
y x
y x
Practice Constructing Parallel Lines Find the equation of the line going through the point (4,1) and
parallel to (click mouse for answer) 3 7y x
1 3 4
1 3 12
3 13
y x
y x
y x
Find the equation of the line going through the point (-2,7) and
parallel to (click mouse for answer) 2 8x y
7 2 2
7 2 2
7 2 4
2 3
y x
y x
y x
y x
Perpendicular Lines Perpendicular lines are lines that intersect in a right angle.
We can decide algebraically if two lines are perpendicular by finding the
slope of each line and seeing if the slopes are negative reciprocals of each
other. This is equivalent to multiplying the two slopes together and seeing if
their product is –1.
We can find the equation of a line perpendicular to a given line and going
through a given point by:
a.) first finding the slope m of the given line;
b.) finding the equation of the line through the given point with
slope = –1 /m.
Testing if Lines Are Perpendicular
1Are the lines 2 5 and 4 perpendicular?
2x y y x
Find the slope of 2 5 2
2 5
x y m
y x
1 1Find the slope of 4
2 2y x m
Since the slopes are negative reciprocals of each other the lines
are perpendicular. 12 1
2
Graphs of Perpendicular Lines
The red line is the graph of
y = – 2x + 5
and the blue line is the
graph of
y = – 1/2 x +4
Practice Testing if Lines Are Perpendicular Are the lines 6 3 5 and 2 4 4 perpendicular?x y y x
6 3 5
3 6 5
523
2
x y
y x
y x
m
2 4 4
2 2
2
y x
y x
m
Since the slopes are not
negative reciprocals of
each other (their product
is not -1) the lines are
not perpendicular
Are the lines 2 4 and 4 2 6 perpendicular?x y x y
2 4
2 4
1 22
12
x y
y x
y x
m
4 2 6
2 4 6
2 3
2
x y
y x
y x
m
Since the slopes are
negative reciprocals of
each other (their
product is -1) the lines
are perpendicular.
Constructing Perpendicular Lines
Find the equation of a line going through the point (3, -5) and
perpendicular to 2 83
y x
The slope of the perpendicular line will be m = 3/2 Using
the point-slope equation where the slope m = 3/2 and
the point is (3, -5) we get 35 3
2
3 952 2
3 192 2
y x
y x
y x
Practice Constructing Perpendicular Lines
Find the equation of the line going through the point (4,1) and
perpendicular to (click mouse for answer) 3 7y x
11 43
1 413 3
1 13 3
y x
y x
y x
Find the equation of the line going through the point (-2,7) and
perpendicular to (click mouse for answer) 2 8x y
17 22
17 22
17 12
1 82
y x
y x
y x
y x
Practice!!
Pg. 1-23 odd
25-30 all
#35, 38, & 41
Recommended