Congruence of a triangles

Triangles Chapter 2 Triangles2.4 Attitude, Median And Angle Bisector 2.5 Congruence Of Triangles 2.6 Midsegment Theorem

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seg. CD is _______ of ABC

20

20

B

A

C

D

Angle Bisector

If BE = 8 cm, and CE = 8 cm. then AE is a/an _______ of ABC

B

A

C

E


Median

If BF AC, then AF is a/an _______ of ABC

B

A

CF


Altitude

Copyright © 2000 by Monica Yuskaitis

SECONDARY PARTS OF A TRIANGLE

Every Triangle has secondary parts



• ANGLE BISECTOR- Is a segment that

DIVIDES (bisects) any angle of a triangle into 2 angles of equal measures.

M N

G SB

A

AG, BN & SM are angle bisector of BAS.

20°20°

40°40°30°

30°



• ALTITUDE

-The height of a triangle.



• ALTITUDE- It is a segment

drawn from any vertex of a triangle perpendicular to the opposite side.

S

C

D

H

N

O



• ALTITUDE

EXAMPLE,

SH, NC, OD are altitudes of

SON.S

C

D

H

N

O



• MEDIANNOTE:

like markingsindicates

congruent or equal parts.

A B

C NM

O



• MEDIANTHUS, IN THE

FIGURE

OA = MA, OB = NB, MC = NC.

A B

C NM

O



A is the midpoint of MO.

B is the midpoint of NO

C is the midpoint of MN

A B

C NM

O



• MEDIAN- Is a segment drawn

from any vertex of a triangle to the MIDPOINT of the opposite side.

A B

C NM

O

NA, MB & OC are median of MON.

2.5 Congruence of trianglesTwo triangles are said to be congruent, if all the corresponding parts are equal. The symbol used for denoting congruence is and PQR STU implies that


i.e. corresponding angles and corresponding sides are equal.

S S S PostulateIf all the sides of one triangle are congruent to the corresponding sides of another triangle then the triangles are congruent (figure 2.15 ).


seg. AB = seg. PQ , seg. BC = seg. QR andseg. CA = seg. RP

\ D ABC @ D PQR by S S S.

S A S PostulateIf the two sides and the angle included in one triangle are congruent to the corresponding two sides and the angle included in another triangle then the two triangles are congruent (figure 2.16).


seg. AB = seg. PQ , seg. BC = seg. QR and m Ð ABC = m Ð PQR

\ D ABC @ D PQR by S A S postulate.

A S A PostulateIf two angles of one triangle and the side they include are congruent to the corresponding angles and side of another triangle the two triangles are congruent (figure 2.17 ).


m Ð B = m Ð R, m Ð C = m Ð P and seg. BC = seg. RP

\D ABC @ D QRP by A S A postulate.

A A S PostulateIf two angles of a triangle and a side not included by them are congruent to the corresponding angles and side of another triangle the two triangles are congruent (figure 2.18)


m Ð A = m Ð P m Ð B = m Ð Q and AC = PR

\ D ABC @ D PQR by A A S.

H S PostulateThis postulate is applicable only to right triangles. If the hypotenuse and any one side of a right triangle are congruent to the hypotenuse and the corresponding side of another right triangle then the two triangles are congruent (figure 2.19).

then hypotenuse AC = hypotenuse PR

Side AB = Side PQ

\ D ABC @ D PQR by HS postulate.


Midsegment TheoremA midsegment of a triangle is a segment connecting the midpoints of two sides of a triangle


T

RX

Z

Y

The segment connecting the midpoints of two sides of a triangle is parallel to the 3rd side and is half as long.

YZRTandRTYZ21

|| T R

X

Z Y

Midsegment Theorem


Using Midsegments of a Triangle

10

6

KJ

LA

B

C

Find JK and AB


Using Midsegments of a Triangle

Given: DE = x + 2; BC =

Find DE

D E

B C

A

192

1x


Education

Congruence of a triangles