Geometry 1 Why Study Chapter 3? Knowledge of triangles is a key application for: Support beams Theater Kaleidoscopes Painting Car stereos Rug design Tile

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

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Geometry

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

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Geometry

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

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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# #

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GeometryProblem

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

ABC FED

A

C B

D

F E

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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.

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GeometryA Triangle is a Polygon with Three Sides

•Congruent Polygons all pairs of

corresponding parts are congruent

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Geometry


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

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Geometry


• 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

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Geometry


• 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# #

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Geometry


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

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Geometry


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

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

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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.

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Geometry


• 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

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Geometry


XW XZ

WY ZY

W

Z

Y VX

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68

Given:

Prove: XWY XZY# #

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Geometry


Given: FA FB

AD BE

G is the midpoint of DE

Prove: DFG EFG

DG

E

F

12

A B

Statements Reasons

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Geometry


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

A C

B

1

2 P

D

Statements Reasons

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Geometry


• 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

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Geometry


Given: WYV ZYV

XY bisects WXZ

Prove: XWY XZY

W

Z

Y VX

57

68

Statements Reasons

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Geometry


Given: PQ AB

PQ Bisects APB Prove: APQ BPQ

P

12

34A

QB

Statements Reasons

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Geometry


Given: XY bisects WXZYX bisects WYZ

Prove: XWY XZY

W

Z

Y VX

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68

Statements Reasons

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Geometry


Given: FAFB

ADBE

m1 m2 mD mE Prove: DFG EFG

DG

E

F

12

A B

Statements Reasons

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Geometry


Given: RS XY

RS PQ

1 4 Prove: PRS QRS

P R Q5 6

X S Y

1

23

4

Statements Reasons

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Geometry


• 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

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Geometry


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

A C

B

1

2 P

D

Statements Reasons

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Geometry


Given:

Prove: APQ BPQ

Statements Reasons

Q is the midpont of

PQ AB

AB

P

12

3 4A

QB

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Geometry


Given: Prove: XWY XZY

WY ZY

VYW VYZ

W

Z

Y VX

57

68

Statements Reasons

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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# #

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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.

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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.

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Geometry


• 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.

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Geometry


• 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

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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.

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Geometry

Section 3.4Practice Proof

• Given:

is an altitude

Prove:

is a median

CFD EFD

FD

FD

D

E

C

F

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Geometry

• Given:

• Prove:

O

GJ HJ

G H

.O

G H

J

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Geometry

• Given: is a median

ST = x + 40

SW = 2x + 30

WV = 5x – 6• Find: SW, WV, and ST

TW

S

T

VW

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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

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Geometry

Section 3.5Overlapping Triangles

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Geometry

Section 3.6Types of Triangles

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

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Geometry


• 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

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Geometry


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

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Geometry


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

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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

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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.

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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.

Documents

Geometry 1 Why Study Chapter 3? Knowledge of triangles is a key application for: Support beams Theater Kaleidoscopes Painting Car stereos Rug design Tile