TRIANGLE

There are four kinds of line in a triangle

A Perpendicular Bisector of A

triangle

A Bisector of A Triangle

A Height of A Triangle

A Median of A Triangle

A Perpendicular Bisector of A triangle

Perpendicular Bisector of AB

Perpendicular bisector of a triangle is

a perpendicular line that intersects

the midpoint of a side.

1. Firstly you have to make a triangle (scalene triangle). Assume that’s triangle as ∆ABC

2. Then suppose that A and B are the centre points. Then you can draw arcs of circle

above and below of the side of

3. Give a label of intersect point of those intersect points with D and E. Then connect

both points, so intersects to be an equal parts and also is

perpendicular to .

AB

DE AB DE

AB

How to Draw It????

A Bisector of A Triangle

An angle Bisector of B

Bisector of an interior angle of atriangle is a line drawn from avertex of triangle and divides itinto two equal angles.

D

E

F

1. Firstly you have to make a triangle (scalene triangle). Assume that’s triangle as ∆ABC

2. Then suppose that B is the centre point. Then draw arcs of circle that intersects

on point D and it also intersects on point E

3. With D and E as the centre points, draw a circular arc with equal radius so that

those circular arcs intersect on point F

4. Connect the point s B and F, so it will be as a bisector of

AB

BF

BC

ABC

How to Draw It????

A Height or An Altitude of A Triangle

A Height of A of ∆ABC

1. Firstly you have to make a triangle (scalene triangle). Assume that’s triangle as ∆ABC

2. Then suppose that A is the centre point. Then draw arc of circle so that can

intersects at points D and E

3. Then suppose that D and E are the centre points, draw arcs of a circle with equal

radius so that will intersects on a point F

4. Connect points A and F so that intersects at point R. Line is an

altitude or height line of a triangle ABC

BC

AMBCAM

How to Draw It????

A Median of A Triangle

A Median of BC

E

D

Q

1. Firstly you have to make a triangle (scalene triangle). Assume that’s triangle as ∆ABC

2. Then suppose that B and C are the centre points. Then you can draw arcs of circle

above and below of the side of

3. Give a label of intersect point of those intersect points with D and E. Then connect

both points, so intersects on point Q

4. Connect points A and q. then line is a median of ∆ABC

BC

DE AB

PQ

How to Draw It????

SIMILARITY AND CONGRUENCE OF

TRIANGLE

Definition: Similar triangles are triangles that have the same shapes but not necessarily the same sizes.

A

C

B

D

F

E

ABC DEFWhen we say that triangles are similar there are several requirements that come from it.

A D

B E

C F

ABDE

BCEF

ACDF= =

1. PPP Similarity Theorem 3 pairs of proportional sides

Six of those statements are true as a result of the similarity of the two triangles. However, if we need to prove that a pair of triangles are similar, how many of those statements do we need? Because we are working with triangles and the measure of the angles and sides are dependent on each other. We do not need all six. There are three special combinations as requirementsthat we can use to prove similarity of triangles.

2. PAP Similarity Theorem 2 pairs of proportional sides and congruent angles between them

3. AA Similarity Theorem 2 pairs of congruent angles

1. PPP Similarity Theorem 3 pairs of proportional sidesA

B C

E

F D

2514

5.

DFm

ABm

25169

12.

.FEm

BCm251

410

13.

.DEm

ACm

5

412

9.6

ABC DFE

2. PAP Similarity Theorem 2 pairs of proportional sides and congruent angles between them

G

H I

L

J K

66057

5.

.LKm

GHm

660510

7.

.KJm

HIm

7

10.5

70

70

m H = m K

GHI LKJ

The PAP Similarity Theorem does not work unless the congruent angles fall between the proportional sides. For example, if we have the situation that is shown in the diagram below, we cannot state that the triangles are similar. We do not have the information that we need.

G

H I

L

J K7

10.5

50

50

Angles I and J do not fall in between sides GH and HI and sides LK and KJ respectively.

3. AA Similarity Theorem 2 pairs of congruent angles

M

N O

Q

P R

70

70

50

50

m N = m R

m O = m P MNO QRP

It is possible for two triangles to be similar when they have 2 pairs of angles given but only one of those given pairs are congruent.

87

34

34

S

T

U

XY

Zm T = m X

m S = 180 - (34 + 87 )

m S = 180 - 121m S = 59

m S = m Z

TSU XZY

59

5959

34

34

Congruent Triangles

Two triangles are congruent if the sizes and shapes are same.

The definition of Congruent

Triangle

A C

B

DE

F

How much do you need to know. . .

. . . about two triangles that they

are congruent?

If all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent.

The symbol of congruent “ “.

Corresponding Parts

ABC DEF

1. AB DE

2. BC EF

3. AC DF

4. A D

5. B E

6. C F

Do you need all six ?

NO !

SSSSASASA

Some conditions or requirementthat are needed for a congruent triangle. Theyare:

The Requirements

Side-Side-Side (SSS)

1.

2.

3.

ABC DEF

DEAB

EFBC

DFAC

Incuded Angle is The angle betweentwo sides

Included Angle

G I H

Included Angle

SY

E

Name the included anglebetween two sides following below:

and is

and is

and is

E

S

Y

YE ES

YSES

YS YE

Side-Angle-Side (SAS)

ABC DEF

included angle

DEAB.1

DFAC.3

DA.2

Included side is the side betweentwo angles

Included Side

GI HI GH

Name the included anglebetween two angles following below:

Y and E is

E and S is

S and Y is

Included Side

SY

E

YE

ES

SY

Angle-Side-Angle (ASA)

1. A D

2. AB DE

3. B E

ABC DEF

included side

Name That requirement

SASASA

(when possible)

SSS

Let’s Practice

Indicate the additional information needed to enable us to show that those two triangles are congruent.

For ASA:

For SAS:

B D

AC FE

HW

For ASA:

For SAS:

Indicate the additional information needed to enable us to show that those two triangles are congruent.

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