lesson 3-1 to 3-7 accelerated algebra/geometry mrs. crespo 2012-2013

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