1. Accelerated Algebra/Geometry Mrs. Crespo 2012-2013
Lesson 3-1 to 3-7
Accelerated Algebra/Geometry
Mrs. Crespo

2. Lesson 3-1 Vocabulary & Key Concepts
Postulate 3-1
Corresponding Angles Postulate
If a transversal intersects two parallel lines, then corresponding angles are congruent. 1 l m 2

3. Lesson 3-1 Vocabulary & Key Concepts
Theorem 3-1
Alternate Interior Angles Theorem
If a transversal intersects two parallel lines, then alternate interior angles are congruent. t 4 a b 3 2 1 5 6

4. Lesson 3-1 Vocabulary & Key Concepts
Theorem 3-1
Alternate Interior Angles Theorem
If a transversal intersects two parallel lines, then alternate interior angles are congruent. t 4 a b 3 2
Theorem 3-2
Same Side Interior Angles Theorem
If a transversal intersects two parallel lines, then same side interior angles are supplementary. 1 5 6

5. Lesson 3-1 Vocabulary & Key Concepts
Theorem 3-3
Alternate Exterior Angles Theorem
If a transversal intersects two parallel lines, then alternate exterior angles are congruent. t 4 a b 3 2 1 5 6

6. Lesson 3-1 Vocabulary & Key Concepts
Theorem 3-3
Alternate Exterior Angles Theorem
If a transversal intersects two parallel lines, then alternate exterior angles are congruent. t 4 a b 3 2
Theorem 3-4
Same Side Exterior Angles Theorem
If a transversal intersects two parallel lines, then same side exterior angles are supplementary. 1 5 6

7. Lesson 3-1 Vocabulary & Key Concepts
A transversal is a line that intersects two coplanar lines at two distinct points. l m

8. Lesson 3-1 Vocabulary & Key Concepts
Alternate interior angles are non-adjacent interior angles that lie on opposite sides of the transversal. 5 6 l m 1 3 4 2 7 8

9. Lesson 3-1 Vocabulary & Key Concepts
Same-side interior angles are interior angles that lie on the same side of the transversal. 5 6 l m 1 3 4 2 7 8

10. Lesson 3-1 Vocabulary & Key Concepts
Corresponding angles are angles that lie on the same side of the transversal and in corresponding positions relative to the coplanar lines. 5 6 l m 1 3 4 2 7 8

11. Lesson 3-1 Vocabulary & Key Concepts
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. 5 6 l m 1 3 4 2 7 8
Alternate Interior Angles
<1
&
<2
Same Side Interior Angles
<1
&
<4
Corresponding Angles
<1
&
<7

12. Alternate Interior Angles
Lesson 3-1 Example 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. 3 1 2 X
Alternate Interior Angles

13. by Corresponding < Post
Lesson 3-1 Example 2
Finding Measures of Angles
In the diagram l//m and p//q. Find m<1 and m<2. p q 8 6
42⁰ l 7
<1 and the 42⁰ angle are 4 2 1
= 42⁰ m
corresponding angles 5 3
by Corresponding < Post
m<1 +
m<2 = 180⁰
by < Add’n Post

14. Lesson 3-1 Example 3 l a = 65⁰ c = 40⁰ m a + b + c = 180⁰
75⁰ a
= 65⁰
Alternate Interior <s Thm c
= 40⁰
Alternate Interior <s Thm
a + b + c
= 180⁰
< Add’n Post
65 + b + 40 = 180⁰
Substitution b
= 75⁰
Subtraction

15. 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. 1 2 l m
l // m

16. Lesson 3-2 Vocabulary & Key Concepts
Theorem 3-5 Converse of the
Alternate Interior Angles Theorem
If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel. t
then l // m l m 1 2

17. Lesson 3-2 Vocabulary & Key Concepts
Theorem 3-5 Converse of the
Alternate Interior Angles Theorem
If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel. t
then l // m l m 1 4 2
Theorem 3-6 Converse of the
Same 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

18. Lesson 3-2 Vocabulary & Key Concepts
Theorem 3-7 Converse of the
Alternate Exterior Angles Theorem
If two lines and a transversal form alternate exterior angles that are congruent, then the two lines are parallel. t
then l // m 5 l m 3

19. Lesson 3-2 Vocabulary & Key Concepts
Theorem 3-7 Converse of the
Alternate Exterior Angles Theorem
If two lines and a transversal form alternate exterior angles that are congruent, then the two lines are parallel. t
then l // m 5 6 l m
Theorem 3-8 Converse of the
Same Side Exterior Angles Theorem
If two lines and a transversal form same side exterior angles that are supplementary, then the two lines are parallel. 3
then l // m

20. <3 and <2 are supplementary
Lesson 3-2 Example 1
Using Postulate 3-2
Which lines if any, must be parallel if <3 and <2 are supplementary? Justify. 3 E C
<3 and <2 are supplementary D 1 4 K
<4 and <2
are supplementary 2
By Congruent Supplements Thm
By Converse of Corresponding <s Post.
ray EC // ray DK

21. Lesson 3-2 Example 2 l m Using Algebra
Find the value of x for which l//m.
The labeled angles are l
alternate interior angles.
(14+3x)⁰
If l//m, the alternate interior <s are
(5x-66)⁰ m
congruent.
Thus, equal.
5x – 66 = x
5x = x
2x = 80
x = 40

22. Lesson 3-3 Key Concepts a a//b b c t m m//n n n l m a Theorem 3-9
If two lines are parallel to the same line, then they are parallel to each other. b a
a//b c t
Theorem 3-10
In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. m n
m//n a n
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. m l

23. Lesson 3-3 Example 2 k l m Using Theorem 3-11 Write a paragraph proof.
Given: In a plane, k//l and k//m.
Also, m<1 = 90⁰. k m l
Prove: The transversal is perpendicular to line m.
Since
m<1 = 90,
the transversal is perp. to
line k.
Since k//m,
by Theorem 3-11
the transversal is also perpendicular to line m.

24. Lesson 3-4 Vocabulary & Key Concepts
Theorem 3-12
Triangle Angle Sum Theorem
The sum of the measures of the angles of a triangle is 180⁰.
m<A + m<B + m<C = 180⁰.
Interior <
Theorem 3-13
Triangle Exterior Angle Theorem
The measure of each exterior angle of a triangle equals the sum of the measures of its remote interior angles.
Exterior < 2 1 3
Interior <

25. Lesson 3-4 Vocabulary & Key Concepts
ANGLES
An acute triangle has three acute angles.
A right triangle has one right angle.
An obtuse triangle has one obtuse angle.
An equiangular triangle has three congruent angles.
SIDES
An equilateral triangle has three congruent sides.
An isosceles triangle has at least two congruent sides.
A scalene triangle has no congruent sides.

26. Lesson 3-4 Vocabulary & Key Concepts
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.
Exterior Angle 1 3 2
Remote Interior Angles

27. 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.
Find c
m<ACB
=90⁰
Definition of a Right <
c + 70
= 90
Angle Addition Postulate
c = 20
Subtract 70 from each side.
Find a
a + m<ADC + c =
180
Triangle Angle Sum Theorem
m<ADC
= 90
Def’n of Perpendicular Lines
c +
= 180
Sub. 90 for m<ADC & 20 for c.
a + 110
= 180
Simplify.
a = 70
Subtract 110 from each side.

28. Lesson 3-4 Example 2 Back x⁰ (180⁰ - x)⁰ Arm
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.
Back x⁰ x⁰
(180⁰ - x)⁰
Arm
(180⁰ - x)⁰
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.

29. Lesson 3-5 Vocabulary & Key Concepts
Theorem 3-14
Polygon Angle-Sum Theorem
The sum of the measures of the angles of an n-gon = (n-2)180⁰.
Theorem 3-15
Polygon Exterior Angle-Sum Theorem
The sum of the measures of the exterior angles of a polygon, one at each vertex is 360⁰. 4 5 1 3 2
Pentagon
m<1 + m<2 + m<3 + m<4 + m<5 = 360⁰.

30. 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. A C D E B E B C A D A B D C E
Not a polygon;
two sides intersect between endpoints & two adjacent sides are collinear.
A polygon.
Not a polygon;
not a closed figure

31. 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.
YES!!!

32. Lesson 3-5 Vocabulary & Key Concepts
Polygons are either convex or concave.
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.
An equilateral polygon has all sides congruent.
An equiangular polygon has all angles congruent.
A regular polygon is both equilateral and equiangular. R D Y T A P M S W Q K G

33. Lesson 3-5 Vocabulary & Key Concepts
Convex
line segments connecting any two points on the shape lie entirely inside the shape
Concave
at least, one line segment connecting any two points on the shape pass outside the shape

34. Lesson 3-5 Example 1 Classify the polygon by its sides.
Identify it as convex or concave.
Number of sides: 12
Name:
Dodecagon
Convex?
No.
Concave?
Yes.

35. Lesson 3-5 Example 2
Find the sum of the measures of the angles of a decagon.
Number of sides: 10
So, n = 10
Sum =
(n-2)180
Polygon Angle Sum Theorem
= (10-2)180
Substitute 10 for n
= 8(180)
Subtract
= 1440
Simplify

36. m<X + m<Y + m<Z + m<W = (4-2)180=360.
Lesson 3-5 Example 3
Use the Polygon Angle Sum Theorem.
Find m<x in quadrilateral XYZW.
Number of sides: 4
So, n = 4 Y Z X W
100⁰
m<X + m<Y + m<Z + m<W = (4-2)180=360.
Polygon Angle Sum Theorem
m<X + m<Y = 360.
m<X + m<Y = 170.
m<X + m<X = 170.
2m<X = 170.
m<X = 85.

37. 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. 2 1

38. where m is the slope and b is the y-intercept.
Lesson 3-6 Vocabulary
Ex. y=2x+3
The slope-intercept form of a linear equation is
𝒚=𝒎𝒙+𝒃
where m is the slope and b is the y-intercept.
Ex. 2x-y=1
The standard form of a linear equation is
𝑨𝒙+𝑩𝒚=𝑪
Ex. y+2=2(x-1)
The point-slope form for a non-vertical line is
𝒚− 𝒚 𝟏 =𝒎(𝒙− 𝒙 𝟏 )
where m is the slope and 𝒙 𝟏 and 𝒚 𝟏 are point coordinates.

39. Lesson 3-6 Quickie y x
Intercepts are points of intersection where the graph crosses either or both the axes.
y-intercept
x-intercept

40. 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).

41. Lesson 3-6 Example 1 Graphing Lines Using Intercepts.y x
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).

42. Lesson 3-6 Example 2 Transforming to Slope-Intercept Form.
Transform -6x +3y =12 to slope intercept form. Then, graph.
-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.

43. Lesson 3-6 Example 2 Transforming to Slope-Intercept Form.
Transform -6x +3y =12 to slope intercept form. Then, graph. y x
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.

44. Lesson 3-6 Examples 3 & 4 on the board.
Lesson 3-7 on the board.

45. PowerPoint by Mrs. Crespo for Accelerated Algebra/Geometry 2012-2013
Acknowledgement
Prentice Hall Mathematics Geometry by Bass, Charles, Hall, Johnson and Kennedy 2007
PowerPoint by Mrs. Crespo for Accelerated Algebra/Geometry