The document provides information about congruent triangles: - Two triangles are congruent if their corresponding sides are congruent and they have the same shape and size. - Examples are provided to demonstrate using properties of congruent triangles to find missing angle measures and prove triangles are congruent by showing corresponding parts are equal. - One example proves two triangles are congruent by showing bisectors of angles bisect the opposite sides, making corresponding parts congruent.

1. 4-3 Congruent Triangles
Drill 2.07 11/13/13
1. Name all sides and angles of ∆FGH.
FG, GH, FH,
F,
2. What is true about
G,
H
K and
L? Why?
;Third
s Thm.
3. What does it mean for two segments to
be congruent?
They have the same length.
Holt McDougal Geometry

2. 4-3 Congruent Triangles
Objectives
Use properties of congruent triangles.
Prove triangles congruent by using the
definition of congruence.
Holt McDougal Geometry

3. 4-3 Congruent Triangles
Vocabulary
corresponding angles
corresponding sides
congruent polygons
Holt McDougal Geometry

4. 4-3 Congruent Triangles
Geometric figures are congruent if they are
the same size and shape. Corresponding
angles and corresponding sides are in the
same position in polygons with an equal
number of sides.
Two polygons are congruent polygons if
and only if their corresponding sides are
congruent. Thus triangles that are the same
size and shape are congruent.
Holt McDougal Geometry

5. 4-3 Congruent Triangles
Holt McDougal Geometry

6. 4-3 Congruent Triangles
Helpful Hint
Two vertices that are the endpoints of a
side are called consecutive vertices.
For example, P and Q are consecutive
vertices.
Holt McDougal Geometry

7. 4-3 Congruent Triangles
To name a polygon, write the vertices
in consecutive order. For example, you
can name polygon PQRS as QRSP or
SRQP, but not as PRQS.
In a congruence statement, the order
of the vertices indicates the
corresponding parts.
Holt McDougal Geometry

8. 4-3 Congruent Triangles
Helpful Hint
When you write a statement such as
ABC
DEF, you are also stating
which parts are congruent.
Holt McDougal Geometry

9. 4-3 Congruent Triangles
Example 1: Naming Congruent Corresponding Parts
Given: ∆PQR
∆STW
Identify all pairs of corresponding congruent parts.
Angles:
P
Sides: PQ
S,
Q
ST, QR
Holt McDougal Geometry
T,
R
TW, PR
W
SW

10. 4-3 Congruent Triangles
Check It Out! Example 1
If polygon LMNP polygon EFGH, identify all
pairs of corresponding congruent parts.
Angles:
L
Sides: LM
E,
M
EF, MN
Holt McDougal Geometry
F,
N
FG, NP
G,
GH, LP
P
H
EH

11. 4-3 Congruent Triangles
Example 2A: Using Corresponding Parts of Congruent
Triangles
Given: ∆ABC
∆DBC.
Find the value of x.
BCA and
BCA
BCD are rt.
BCD
m BCA = m BCD
(2x – 16)° = 90°
2x = 106
x = 53
Holt McDougal Geometry
s.
Def. of
Rt.
Def. of
lines.
Thm.
s
Substitute values for m BCA and
m BCD.
Add 16 to both sides.
Divide both sides by 2.

12. 4-3 Congruent Triangles
Example 2B: Using Corresponding Parts of Congruent
Triangles
Given: ∆ABC
∆DBC.
Find m DBC.
m ABC + m BCA + m A = 180° ∆ Sum Thm.
Substitute values for m BCA and
m ABC + 90 + 49.3 = 180
m A.
m ABC + 139.3 = 180 Simplify.
m ABC = 40.7
DBC
ABC
Subtract 139.3 from both
sides.
Corr. s of ∆s are .
m DBC = m ABC Def. of
m DBC
Holt McDougal Geometry
40.7°
s.
Trans. Prop. of =

13. 4-3 Congruent Triangles
Check It Out! Example 2a
Given: ∆ABC
∆DEF
Find the value of x.
AB
DE
AB = DE
2x – 2 = 6
2x = 8
x=4
Holt McDougal Geometry
Corr. sides of
Def. of
∆s are .
parts.
Substitute values for AB and DE.
Add 2 to both sides.
Divide both sides by 2.

14. 4-3 Congruent Triangles
Check It Out! Example 2b
Given: ∆ABC
∆DEF
Find m F.
m EFD + m DEF + m FDE = 180°
ABC
DEF
∆ Sum Thm.
Corr.
s of
∆ are .
m ABC = m DEF
Def. of
m DEF = 53°
Transitive Prop. of =.
m EFD + 53 + 90 = 180
m F + 143 = 180
m F = 37°
Holt McDougal Geometry
s.
Substitute values for m DEF
and m FDE.
Simplify.
Subtract 143 from both sides.

15. 4-3 Congruent Triangles
Example 3: Proving Triangles Congruent
Given:
YWX and
YWZ are right angles.
YW bisects XYZ. W is the midpoint of XZ. XY
Prove: ∆XYW ∆ZYW
Holt McDougal Geometry
YZ.

16. 4-3 Congruent Triangles
Statements
1.
YWX and
2.
YWX
YWZ
3. YW bisects
4.
YWZ are rt.
XYW
XYZ
ZYW
Reasons
s.
1. Given
2. All right
’s are
3. Given
4. Def. of bisector
5. W is mdpt. of XZ
5. Given
6. XW
ZW
6. Def. of mdpt.
7. YW
YW
7. Reflex. Prop. of
8.
X
9. XY
Z
8. Third
YZ
10. ∆XYW
s Thm.
9. Given
∆ZYW
Holt McDougal Geometry
10. Def. of
∆

17. 4-3 Congruent Triangles
Check It Out! Example 3
Given: AD bisects BE.
BE bisects AD.
AB DE, A
D
Prove: ∆ABC ∆DEC
Holt McDougal Geometry

18. 4-3 Congruent Triangles
Statements
1.
A
D
Reasons
1. Given
2.
BCA
DCE
2. Vertical
3.
ABC
DEC
3. Third
4. AB
DE
s are .
s Thm.
4. Given
5. Given
5. AD bisects BE,
BE bisects AD
6. BC
EC, AC
7. ∆ABC
∆DEC
Holt McDougal Geometry
DC
6. Def. of bisector
7. Def. of
∆s

19. 4-3 Congruent Triangles
Example 4: Engineering Application
The diagonal bars across a gate give it
support. Since the angle measures and the
lengths of the corresponding sides are the
same, the triangles are congruent.
Given: PR and QT bisect each other.
PQS
RTS, QP RT
Prove: ∆QPS
Holt McDougal Geometry
∆TRS

20. 4-3 Congruent Triangles
Example 4 Continued
Statements
1. QP RT
2. PQS
RTS
3.
4.
5.
6.
7.
1.
2.
PR and QT bisect each other. 3.
QS TS, PS RS
4.
QSP
TSR
5.
QSP
TRS
6.
∆QPS ∆TRS
7.
Holt McDougal Geometry
Reasons
Given
Given
Given
Def. of bisector
Vert. s Thm.
Third s Thm.
Def. of ∆s