This document discusses different types of triangle parts including: - Angle bisectors, which divide any angle of a triangle into two equal angles - Altitudes, which are perpendicular segments from a vertex to the opposite side - Medians, which connect vertices to midpoints of the opposite sides It also covers triangle congruence, noting triangles are congruent if corresponding sides and angles are equal. Five postulates for triangle congruence are presented relating equal sides, angles, and combinations of these. The midsegment theorem is introduced, stating a midsegment is parallel to the third side and half as long. Examples demonstrate using midsegments and finding unknown sides.

1. Triangles
Chapter 2 Triangles
2.4 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

3. If BE = 8 cm, and CE = 8 cm. then AE is a/an _______ of ABC
B
A
C
E
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Median

4. If BF  AC, then AF is a/an _______ of ABC
B
A
C
F
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Altitude

5. Copyright © 2000 by Monica Yuskaitis
SECONDARY PARTS OF A TRIANGLE
Every Triangle has secondary parts

6. Copyright © 2000 by Monica Yuskaitis
SECONDARY PARTS OF A TRIANGLE
• 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°

7. Copyright © 2000 by Monica Yuskaitis
SECONDARY PARTS OF A TRIANGLE
• ALTITUDE
-The height
of a
triangle.

8. Copyright © 2000 by Monica Yuskaitis
SECONDARY PARTS 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

9. Copyright © 2000 by Monica Yuskaitis
SECONDARY PARTS OF A TRIANGLE
• ALTITUDE
EXAMPLE,
SH, NC, OD are
altitudes of
SON.
S
C
D
H
N
O

10. Copyright © 2000 by Monica Yuskaitis
SECONDARY PARTS OF A TRIANGLE
• MEDIAN
NOTE:
like markings
indicates
congruent or
equal parts.
A B
C NM
O

11. Copyright © 2000 by Monica Yuskaitis
SECONDARY PARTS OF A TRIANGLE
• MEDIAN
THUS, IN THE
FIGURE
OA = MA, OB =
NB, MC = NC.
A B
C NM
O

12. Copyright © 2000 by Monica Yuskaitis
SECONDARY PARTS OF A TRIANGLE
A is the midpoint of
MO.
B is the midpoint of
NO
C is the midpoint of
MN
A B
C NM
O

13. Copyright © 2000 by Monica Yuskaitis
SECONDARY PARTS OF A TRIANGLE
• 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.

14. 2.5 Congruence of triangles
Two 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
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i.e. corresponding angles and corresponding sides are equal.

15. S S S Postulate
If all the sides of one triangle are congruent to the corresponding
sides of another triangle then the triangles are congruent (figure
2.15 ).
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seg. AB = seg. PQ , seg. BC = seg. QR and
seg. CA = seg. RP
 ABC @  PQR by S S S.

16. S A S Postulate
If 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).
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seg. AB = seg. PQ , seg. BC = seg. QR and m  ABC = m  PQR
 ABC @  PQR by S A S postulate.

17. A S A Postulate
If 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 ).
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m  B = m  R, m  C = m  P and seg. BC = seg. RP
 ABC @  QRP by A S A postulate.

18. A A S Postulate
If 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)
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m  A = m  P m  B = m  Q and AC = PR
 ABC @  PQR by AA S.

19. H S Postulate
This 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
 ABC @  PQR by HS postulate.
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20. Midsegment Theorem
A midsegment of a triangle is a segment
connecting the midpoints of two sides of
a triangle
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T
R
X
Z
Y

21. The segment connecting the midpoints of two sides
of a triangle is parallel to the 3rd side and is half as
long.
YZRTandRTYZ
2
1
|| 
T R
X
Z Y
Midsegment Theorem
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22. Using Midsegments of a Triangle
10
6
KJ
LA
B
C
Find JK and AB
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23. Using Midsegments of a Triangle
Given: DE = x + 2; BC =
Find DE
D E
B C
A
19
2
1
x
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