Lesson 3-1 to 3-7Accelerated Algebra/Geometry
Mrs. Crespo 2012-2013
Lesson 3-1 Vocabulary & Key Concepts
Postulate 3-1
Corresponding Angles Postulate
If a transversal intersects two parallel lines, then corresponding angles are congruent.
l
m
1
2
Lesson 3-1 Vocabulary & Key Concepts
Theorem 3-1Alternate Interior Angles Theorem
If a transversal intersects two parallel lines, then alternate interior angles are
congruent.a
b1
2
t
3
4
5 6
Lesson 3-1 Vocabulary & Key Concepts
Theorem 3-1Alternate Interior Angles Theorem
If a transversal intersects two parallel lines, then alternate interior angles are
congruent.
Theorem 3-2Same Side Interior Angles Theorem
If a transversal intersects two parallel lines, then same side interior angles are
supplementary.
a
b1
2
t
3
4
5 6
Lesson 3-1 Vocabulary & Key Concepts
Theorem 3-3Alternate Exterior Angles Theorem
If a transversal intersects two parallel lines, then alternate exterior angles are
congruent.a
b1
2
t
3
4
5 6
Lesson 3-1 Vocabulary & Key Concepts
Theorem 3-3Alternate Exterior Angles Theorem
If a transversal intersects two parallel lines, then alternate exterior angles are
congruent.
Theorem 3-4Same Side Exterior Angles Theorem
If a transversal intersects two parallel lines, then same side exterior angles are
supplementary.
a
b1
2
t
3
4
5 6
A transversal is a line that intersects two coplanar lines at two distinct points.
Lesson 3-1 Vocabulary & Key Concepts
l
m
t
Alternate interior angles are non-adjacent interior angles that lie on opposite sides of the transversal.
Lesson 3-1 Vocabulary & Key Concepts
l
m
1
8
t
3
2
5
4
7
6
Same-side interior angles are interior angles that lie on the same side of the transversal.
Lesson 3-1 Vocabulary & Key Concepts
l
m
1
8
t
3
2
5
4
7
6
Corresponding angles are angles that lie on the same side of the transversal and in corresponding positions relative to the coplanar lines.
Lesson 3-1 Vocabulary & Key Concepts
l
m
1
8
t
3
2
5
4
7
6
A transversal is a line that intersects two coplanar lines at two distinct points.
Alternate interior angles are non-adjacent interior angles that lie on opposite sides of the transversal.
Same-side interior angles are interior angles that lie on the same side of the transversal.
Corresponding angles are angles that lie on the same side of the transversal and in corresponding positions relative to the coplanar lines.
Lesson 3-1 Vocabulary & Key Concepts
l
m
1
8
t
3
2
5
4
7
6
<1¿ <2
<1¿<4
<1¿<7
Alternate Interior Angles
Same Side Interior Angles
Corresponding Angles
3
1
Applying Properties of Parallel Lines
In the diagram of LRA, the black segments are runways.
Compare <2 and the angle vertical to <1. Classify the angles.
Lesson 3-1 Example 1
2 X
Alternate Interior Angles
Lesson 3-1 Example 2
2
3
1
Finding Measures of Angles
In the diagram l//m and p//q. Find m<1 and m<2.
<1 and the 42⁰ angle are
l
m4
5
pq
corresponding angles
42⁰8
7
6
= 42⁰
by Corresponding < Post
m<1+¿m<2 = 180⁰ by < Add’n Post
Lesson 3-1 Example 3
a Alternate Interior <s Thm
< Add’n Post
= 65⁰
c = 40⁰ Alternate Interior <s Thm
a + b + c = 180⁰
65 + b + 40 = 180⁰
b = 75⁰
Substitution
Subtraction
l
m
a⁰b⁰
65⁰
c⁰
40⁰
65⁰ 40⁰
75⁰
Lesson 3-2 Vocabulary and Key Concepts
Postulate 3-2
Converse of the Corresponding Angles Postulate
If two lines and a transversal form corresponding angles that are congruent, then the two lines are parallel.
l
m
1
2l // m
l
m
Lesson 3-2 Vocabulary & Key Concepts
Theorem 3-5 Converse of theAlternate Interior Angles Theorem
If two lines and a transversal form alternate interior angles that are congruent, then the
two lines are parallel. 1
2
tthen l // m
l
m
Lesson 3-2 Vocabulary & Key Concepts
Theorem 3-5 Converse of theAlternate Interior Angles Theorem
If two lines and a transversal form alternate interior angles that are congruent, then the
two lines are parallel. 1
2
t
4
Theorem 3-6 Converse of theSame Side Interior Angles Theorem
If two lines and a transversal form same side interior angles that are supplementary,
then the two lines are parallel.
then l // m
then l // m
Lesson 3-2 Vocabulary & Key Concepts
Theorem 3-7 Converse of theAlternate Exterior Angles Theorem
If two lines and a transversal form alternate exterior angles that are congruent, then the
two lines are parallel.l
m
t
3
5
then l // m
Lesson 3-2 Vocabulary & Key Concepts
Theorem 3-7 Converse of theAlternate Exterior Angles Theorem
If two lines and a transversal form alternate exterior angles that are congruent, then the
two lines are parallel.l
m
t
3
5 6
Theorem 3-8 Converse of theSame Side Exterior Angles Theorem
If two lines and a transversal form same side exterior angles that are
supplementary, then the two lines are parallel.
then l // m
then l // m
Lesson 3-2 Example 1
K
3
1
Using Postulate 3-2
Which lines if any, must be parallel if <3 and <2 are supplementary? Justify.
E C<3 and <2 are supplementary
D 4
2By Congruent Supplements Thm
By Converse of Corresponding <s Post.
<4 and <2 are supplementary
ray EC // ray DK
Lesson 3-2 Example 2
Using Algebra
Find the value of x for which l//m.
congruent.
The labeled angles are
alternate interior angles.
If l//m, the alternate interior <s are
Thus, equal.
5x – 66 = 14 + 3x
5x = 80 + 3x
2x = 80
x = 40
l
m
(14+3x)⁰
(5x-66)⁰
Lesson 3-3 Key Concepts
c
Theorem 3-9
If two lines are parallel to the same line, then they are parallel to each other.
Theorem 3-10
In a plane, if two lines are perpendicular to the same line, then they are parallel to
each other.
Theorem 3-11
In a plane, if a line is perpendicular to one of two parallel lines, then it is also
perpendicular to the other.
b
a
a//b
m
n
t
m//n
m
l
a
n
Lesson 3-3 Example 2
Using Theorem 3-11
Write a paragraph proof.
Prove: The transversal is perpendicular to line m.
the transversal is also perpendicular to line m.
Given: In a plane, k//l and k//m.
Also, m<1 = 90⁰.
Since k//m,
Since
by Theorem 3-11
m<1 = 90,
the transversal is perp. to
line k.
k
m
l
1
Lesson 3-4 Vocabulary & Key Concepts
Theorem 3-12Triangle Angle Sum Theorem
The sum of the measures of the angles of a triangle is 180⁰.
Theorem 3-13Triangle Exterior Angle Theorem
The measure of each exterior angle of a triangle equals the sum of the measures of
its remote interior angles.
m<A + m<B + m<C = 180⁰.
A
C
B
1
2
3
Interior <
Exterior <
Interior <
An exterior angle of a polygon is an angle formed by a side and an extension of an adjacent side.
Remote interior angles are two non-adjacent interior angles corresponding to each exterior angle of a triangle.
Lesson 3-4 Vocabulary & Key Concepts
1
23
Exterior Angle
Remote Interior Angles
Lesson 3-4 Example 1
Applying the Triangle Angle-Sum Theorem
In triangle ABC, <ACB is a right angle; segment CD perpendicular to segment AB. Find the values of a and c.
m<ACB =90⁰ Definition of a Right <
c + 70 = 90 Angle Addition Postulate
c = 20 Subtract 70 from each side.
Find c
Find a
a + m<ADC + c = 180
m<ADC
= 90
c + 90 + 20 = 180
a + 110 = 180
a = 70
Triangle Angle Sum Theorem
Def’n of Perpendicular Lines
Sub. 90 for m<ADC & 20 for c.
Simplify.
Subtract 110 from each side.
Lesson 3-4 Example 2
Applying the Triangle Exterior Angle Theorem
Explain what happens to the angle formed by the back of the chair and the armrest as you make a lounge chair recline more.
The exterior angle and the angle formed by the back of the chair and the armrest are adjacent angles, which together form a
straight angle. As one measure increases, the other decreases. The angle formed by the back of the chair and the armrest
increases as you make a lounge chair recline more.
x⁰
(180⁰ - x)⁰
Back
Arm
x⁰
(180⁰ - x)⁰
Lesson 3-5 Vocabulary & Key Concepts
Theorem 3-14Polygon Angle-Sum Theorem
The sum of the measures of the angles of an n-gon = (n-2)180⁰.
Theorem 3-15Polygon Exterior Angle-Sum Theorem
The sum of the measures of the exterior angles of a polygon, one at each vertex is
360⁰.
m<1 + m<2 + m<3 + m<4 + m<5 = 360⁰.
4
5
1
3
2Pentagon
Lesson 3-5 Vocabulary & Key Concepts
A polygon is a closed plane figure with at least three sides that are segments. The sides intersect only at their endpoints, and no two adjacent sides are collinear.
AC
DE
B
A
B
D
C
EE
B
CA
D
A polygon. Not a polygon;not a closed figure
Not a polygon;two sides intersect between endpoints & two adjacent
sides are collinear.
A polygon is a closed plane figure with at least three sides that are segments. The sides intersect only at their endpoints, and no two adjacent sides are collinear.
Lesson 3-5 Vocabulary & Key Concepts
Polygons are either convex or concave.
Lesson 3-5 Vocabulary & Key Concepts
A convex polygon does not have diagonal points outside of the polygon.
A concave polygon has at least one diagonal with points outside of the polygon.
RD
YT
APM
S
WQK
G
An equilateral polygon has all sides congruent.
An equiangular polygon has all angles congruent.
A regular polygon is both equilateral and equiangular.
Convex
line segments connecting any two
points on the shape lie entirely inside the
shape
Lesson 3-5 Vocabulary & Key Concepts
Concave
at least, one line segment connecting
any two points on the shape pass outside the
shape
Lesson 3-5 Example 1
Classify the polygon by its sides.
Identify it as convex or concave.
Number of sides:
Name:
Convex?
Concave?
12
Dodecagon
No.
Yes.
Lesson 3-5 Example 2
Find the sum of the measures of the angles of a decagon.
Number of sides:
Sum =
So, n = 1010
(n-2)180
Simplify
Polygon Angle Sum Theorem
= (10-2)180 Substitute 10 for n
= 8(180) Subtract
= 1440
Lesson 3-5 Example 3
Use the Polygon Angle Sum Theorem.
Find m<x in quadrilateral XYZW.
Number of sides: So, n = 44
Polygon Angle Sum Theorem
YZ
XW
100⁰
m<X + m<Y + m<Z + m<W = (4-2)180=360.
m<X + m<Y + 90 + 100 = 360.
m<X + m<Y = 170.
m<X + m<X = 170.
2m<X = 170.
m<X = 85.
Lesson 3-5 Example 4
Applying Theorem 3-15.
A regular hexagon is inscribed in a rectangle . Explain how you know that all angles labeled <1 have equal measures.
The hexagon is regular, so all its angles are congruent. An
exterior angle is the supplement of a polygon’s
angle because they are adjacent angles that form a
straight angle. Because supplements of congruent
angles are congruent, all the angles marked <1 have
equal measures.
22
1 1
1 1
2 2
Lesson 3-6 Vocabulary
The slope-intercept form of a linear equation is
where m is the slope and b is the y-intercept.
The standard form of a linear equation is
The point-slope form for a non-vertical line is
where m is the slope and and are point coordinates.
Ex. y=2x+3
Ex. 2x-y=1
Ex. y+2=2(x-1)
y
x
Lesson 3-6 Quickie
Intercepts are points
of intersection
where the graph
crosses either or
both the axes.
y-interceptx-intercept
Lesson 3-6 Example 1
Graphing Lines Using Intercepts.
Use the x-intercept and y-intercept to graph 5x-6y = 30.
Find the x-intercept.
Sub 0 for y. Solve for x.
5x – 6y = 30
5x – 6 (0) = 30
5x – 0 = 30
5x = 30
x = 6 , the x-intercept.
As a point, it is (6,0).
Find the y-intercept.
Sub 0 for x. Solve for y.
5x – 6y = 30
5(0) – 6y = 30
0 – 6y = 30
-6y = 30
y = -5 , the y-intercept.
As a point, it is (0,-5).
Lesson 3-6 Example 1
Graphing Lines Using Intercepts.
Use the x-intercept and y-intercept to graph 5x-6y = 30.
The x-intercept.
As a point, it is (6,0).
The y-intercept.
As a point, it is (0,-5).
y
x
Lesson 3-6 Example 2
-6x + 3y = 12
3y = 6x + 12
(3y)/3 = (6x)/3 + (12)/3
y = 2x + 4
The y-intercept is 4 and the slope is 2.
Plot.
Graph the y-intercept; as a point it is (0,4).
With slope, 2 is 2/1. So, up 2, right 1.
Connect the two points.
Transforming to Slope-Intercept Form.
Transform -6x +3y =12 to slope intercept form. Then, graph.
Lesson 3-6 Example 2
Plot.
Graph the y-intercept; as a point it is (0,4).
With slope, 2 is 2/1. So, up 2, right 1.
Connect the two points.
Transforming to Slope-Intercept Form.
Transform -6x +3y =12 to slope intercept form. Then, graph.
y
x
Lesson 3-6 Examples 3 & 4 on the board.
Lesson 3-7 on the board.
Acknowledgement
Prentice Hall Mathematics Geometry
by
Bass, Charles, Hall, Johnson and Kennedy
2007
PowerPoint by
Mrs. Crespo
for
Accelerated Algebra/Geometry
2012-2013
Top Related