Geometry 1 Why Study Chapter 3? Knowledge of triangles is a key application for: Support beams...

1

GeometryWhy Study Chapter 3?

Knowledge of triangles is a key application for:• Support beams• Theater• Kaleidoscopes• Painting• Car stereos• Rug design• Tile floors• Gates

2

Geometry

• Baseball field• Sign making• Architecture• Sailboats• Fire and lifeguard towers• Sports • Airplanes• Bicycles• Surveying

3

Geometry

• Canyon• Snowboarding• Advertising (logos)• Kites• Chess• Stenciling

4

Geometry

Section 3.1Congruent Figures

• Definitions Congruent: having the same size and shape. Congruent triangles: all pairs of corresponding parts are

congruent. Corresponding parts

– If triangles ABC and DEF are congruent, then what parts must match up? Refer to page 111 in text.

A

C B

D

F E

ABC DEF# #

5

GeometryProblem

• Is ? Explain your answer.• Refer to page 112 in the text.

ABC FED

A

C B

D

F E

6

GeometryDefinitions

• Plane: a two-dimensional figure usually represented by a shape that looks like a wall or floor even though the plane extends without end

• Polygon: a closed plane figure with the following properties:

1) It is formed by three or more line segments

called sides.

2) Each side intersects exactly two sides, one at

each endpoint, so that no two sides with a

common endpoint are collinear.

7

GeometryA Triangle is a Polygon with Three Sides

•Congruent Polygons all pairs of

corresponding parts are congruent

8

Geometry

Section 3.1Congruent Figures

• Reflection when a figure has a mirror line Refer to page 112 in text.

9

Geometry

Section 3.1Congruent Figures

• Other Types of Correspondences Slide: where a copy of the figure has been shifted by some set

amount Refer to page 113 in text.

A B

C

DE

P Q

R

ST

ABCDE PQRST

10

Geometry

Section 3.1Congruent Figures

• Other Types of Correspondences Rotational: when the figure has been rotated around a common

point Refer to page 113 in text.

R

T

AY

B

RAT BAY# #

11

Geometry

Section 3.1Congruent Figures

• These correspondences can be combined! Try a reflection and a rotation on the figure below.

12

Geometry

Section 3.1Congruent Figures

• Reflexive property any segment or angle is congruent to itself.

Since the triangles overlap, this angle is reflexive to triangles!

13

Geometry

Section 3.23 Ways to Prove Triangles Congruent

• Introduction Included sides and angles

To be included means to be flanked by or trapped between – Thus, the points of a line segment are included between

its endpoints.– In a triangle, sides can be included by angles and angles

can be included by sides.

M

RZ

List the inclusions in triangle MRZ.

Refer to page 115 in text.

14

Geometry

Section 3.23 Ways to Prove Triangles Congruent

• Side Side Side (SSS) Postulate SSS: If three sides of one triangle are congruent to three sides of

another triangle then the triangles are congruent

A

C B

D

F E

15

Geometry

Section 3.23 Ways to Prove Triangles Congruent

XW XZ

WY ZY

W

Z

Y VX

57

68

Given:

Prove: XWY XZY# #

16

Geometry

Section 3.23 Ways to Prove Triangles Congruent

Given: FA FB

AD BE

G is the midpoint of DE

Prove: DFG EFG

DG

E

F

12

A B

Statements Reasons

17

Geometry

Section 3.23 Ways to Prove Triangles Congruent

Given: AB bisects CD CD bisects AB AC BD Prove: ACP BDP

A C

B

1

2 P

D

Statements Reasons

18

Geometry

Section 3.23 Ways to Prove Triangles Congruent

• Angle Side Angle (ASA) Postulate ASA: If two angles and the included side of one triangle are

congruent to two angles and the included side of another triangle, then the triangles are congruent.

A

C B

D

F E

19

Geometry

Section 3.23 Ways to Prove Triangles Congruent

Given: WYV ZYV

XY bisects WXZ

Prove: XWY XZY

W

Z

Y VX

57

68

Statements Reasons

20

Geometry

Section 3.23 Ways to Prove Triangles Congruent

Given: PQ AB

PQ Bisects APB Prove: APQ BPQ

P

12

34A

QB

Statements Reasons

21

Geometry

Section 3.23 Ways to Prove Triangles Congruent

Given: XY bisects WXZYX bisects WYZ

Prove: XWY XZY

W

Z

Y VX

57

68

Statements Reasons

22

Geometry

Section 3.23 Ways to Prove Triangles Congruent

Given: FAFB

ADBE

m1 m2 mD mE Prove: DFG EFG

DG

E

F

12

A B

Statements Reasons

23

Geometry

Section 3.23 Ways to Prove Triangles Congruent

Given: RS XY

RS PQ

1 4 Prove: PRS QRS

P R Q5 6

X S Y

1

23

4

Statements Reasons

24

Geometry

Section 3.23 Ways to Prove Triangles Congruent

• Side Angle Side (SAS) Postulate SAS: If two sides and the included angle of one triangle are

congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

A

C B

D

F E

25

Geometry

Section 3.23 Ways to Prove Triangles Congruent

Given: AB bisects CD CD bisects AB Prove: ACP BDP

A C

B

1

2 P

D

Statements Reasons

26

Geometry

Section 3.23 Ways to Prove Triangles Congruent

Given:

 Prove: APQ BPQ

Statements Reasons

Q is the midpont of

PQ AB

AB

P

12

3 4A

QB

27

Geometry

Section 3.23 Ways to Prove Triangles Congruent

Given:   Prove: XWY XZY

WY ZY

VYW VYZ

W

Z

Y VX

57

68

Statements Reasons

28

Geometry

Section 3.3Circles and CPCTC

If you have proven via SSS can you state that 5 is congruent to 7? WHY?

• CPCTC Principle: Corresponding Parts of Congruent Triangles are Congruent

You MUST prove the triangles congruent FIRST!!!

W

Z

Y VX

57

68

XWY XZY# #

29

Geometry

Section 3.3Circles and CPCTC

• Circle: The set of all points in a plane that are a given distance from a given point in the plane.

O. P

R.. • Segment OP is a radius

• Segments OP and OR are radii

• Remember your formulas? Area of a circle Circumference of a circle

2

2

A r

C r

• Theorem: All radii of a circle are congruent.

30

Geometry

Section 3.4Beyond CPCTC

• Median: A line segment drawn from any vertex of the triangle to the midpoint of the opposite side.

How many medians does a triangle have? (3)

• A median divides into two congruent segments, or bisects the side to which it is drawn.

31

Geometry

Section 3.4Beyond CPCTC

• Altitude: A line segment drawn from any vertex of the triangle to the opposite side, extended if necessary, and perpendicular to that side.

How many altitudes does a triangle have? (3)

• An altitude of a triangle forms right (90º) angles with one of the sides.

32

Geometry

Section 3.4Beyond CPCTC

• Auxiliary Lines: A line introduced into a diagram for the purpose of clarifying a proof.

• Postulate: Two points determine a line (or ray or segment).

A

C

B D

R

S

T

U

33

GeometrySolving Proofs: Remember the Steps

• 1. Draw the diagram.• 2. Carefully read the problem and mark the diagram.• 3. Place a question mark (?) in the area that you

need to prove.• 4. Create a flow diagram.• 5. Use one given at a time and draw as much

information as possible from that given.• (Disregard information that is not needed.)• 6. Sequentially list the statements and reasons.• 7. The last statement should be the prove statement.• 8. Check the proof. It should follow a logical order.

34

Geometry

Section 3.4Practice Proof

• Given:

is an altitude

Prove:

is a median

CFD EFD

FD

FD

D

E

C

F

35

Geometry

• Given:

• Prove:

O

GJ HJ

G H

.O

G H

J

36

Geometry

• Given: is a median

ST = x + 40

SW = 2x + 30

WV = 5x – 6• Find: SW, WV, and ST

TW

S

T

VW

37

GeometryTest Tomorrow

• Study lessons 3.1 to 3.4 and class notes• Define the following:

1. Reflexive Property

2. SSS, SAS, and ASA Postulates

3. CPCTC, radius, radii, and diameter

4. Formulas for the area and circumference

of a circle

5. Median, altitude, auxiliary lines• Study PowerPoint slides 1-36

38

Geometry

Section 3.5Overlapping Triangles

39

Geometry

Section 3.6Types of Triangles

• Scalene Triangle: a triangle in which no two sides are congruent.

40

Geometry

Section 3.6Types of Triangles

• Isosceles Triangle: a triangle in which at least two sides are congruent.

Congruent sides are called legsNon-congruent side is called the baseAngles included between leg and base are called base angles

Leg

Base Angle

Vertex

Base

41

Geometry

Section 3.6Types of Triangles

• Equilateral Triangle: a triangle in which all sides are congruent.

42

Geometry

Section 3.6Types of Triangles

• Equiangular Triangle: a triangle in which all angles are congruent.

43

Geometry

Section 3.7Angle-Side Theorems

• Theorem: If two sides of a triangle are congruent, then the angles opposite the sides are congruent.

• Theorem: If two angles of a triangle are congruent, the sides opposite the angles are congruent.

Ways to Prove that a Triangle is Isosceles:

1. If at least two sides of a triangle are congruent

2. If at least two angles of a triangle are congruent

44

Geometry

Section 3.7Angle-Side Theorems

• Theorem: If two sides of a triangle are not congruent, then the angles opposite them are not congruent, and the larger angle is opposite the longer side.

• Theorem: If two angles of a triangle are not congruent, then the sides opposite them are not congruent, and the longer side is opposite the larger angle.

45

Geometry

Section 3.8The HL Postulate

• HL Postulate: If there exists a correspondence between the vertices of two right triangles such that the hypotenuse and a leg of one triangle are congruent to the corresponding parts of the other triangle, the two right triangles are congruent.