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Warm-Up Exercises
Given: DF bisects CE, DC
Prove: ∆CDF
∆EDF
DE
C
F
D
E
ANSWER
It is given that DC DE and
DF bisects
CE. CF
EF by the def. of bisector.
DF DF by the Refl. Prop. of
Segs.
So ∆CDF ∆ EDF by the SSS Post.
Warm-Up Exercises
Target
Proving triangles
congruent.
GOAL:
4.5 Use sides and angles to prove
triangles congruent.
Warm-Up Exercises
Vocabulary
• included angle – the angle between two identified sides
• SAS (Side-Angle-Side) Congruence Postulate
If two sides and the included angle of one triangle
are congruent to two sides and the included angle
of a second triangle,
then the two triangles are congruent.
• HL (Hypotenuse-Leg) Congruence Theorem
If the hypotenuse and leg of one right triangle
are congruent to the hypotenuse and leg of a
second right triangle,
then the two triangles are congruent.
Warm-Up1Exercises
EXAMPLE
Use the SAS Congruence Postulate
Write a proof.
GIVEN BC
DA, BC AD
ABC
CDA
PROVE
STATEMENTS
REASONS
S 1.
BC
2.
A 3.
BC AD
BCA
1. Given
2. Given
3. Alternate Interior
Angles Theorem
S 4.
AC
5.
ABC
DA
DAC
4. Reflexive Property of
Congruence
CA
CDA
5. SAS Congruence
Postulate
Warm-Up2Exercises
EXAMPLE
Use SAS and properties of shapes
In the diagram, QS and RP pass through
the center M of the circle. What can you
conclude about
MRS and
MPQ?
SOLUTION
Because they are vertical angles, PMQ
RMS. All
points on a circle are the same distance from the center,
so MP, MQ, MR, and MS are all equal.
ANSWER
MRS and
MPQ are congruent by the SAS
Congruence Postulate.
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 1 and 2
In the diagram, ABCD is a square with four
congruent sides and four right angles. R,
S, T, and U are the midpoints of the sides
of ABCD. Also, RT SU and SV VU .
Prove that
1.
SVR
STATEMENTS
REASONS
A
SV VU
RT SU
SVR
RVU
S
RV
S
VR
SVR
UVR
UVR
Given
Given
Def. of lines; Right
Ref. Prop. of Congruence
SAS Congruence Postulate
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 1 and 2
In the diagram, ABCD is a square with four
congruent sides and four right angles. R,
S, T, and U are the midpoints of the sides
VU .
of ABCD. Also, RT SU and SU
2.
Prove that
BSR
STATEMENTS
S
A
S
BC
DA
BS
DU
RBS
RS
BSR
DUT
REASONS
Given
Def. Midpoint, Trans.
TDU
Given; Right
CPCTC (previous proof)
UT
DUT
SAS Congruence Postulate
Warm-Up3Exercises
EXAMPLE
Use the Hypotenuse-Leg Congruence Theorem
Write a proof.
GIVEN
PROVE
WY
XZ, WZ ZY, XY ZY
WYZ
XZY
SOLUTION
Redraw the triangles so
they are side by side with
corresponding parts in the
same position. Mark the
given information in the
diagram.
Warm-Up3Exercises
EXAMPLE
Use the Hypotenuse-Leg Congruence Theorem
STATEMENTS
H 1.
WY
4.
ZY
2. Given
3. Definition of
Z and Y are
lines
right angles
WYZ and XZY are 4. Definition of a right
triangle
right triangles.
L 5. ZY
6.
1. Given
XZ
2. WZ ZY, XY
3.
REASONS
WYZ
YZ
5. Reflexive Property of
Congruence
XZY
6. HL Congruence
Theorem
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 3 and 4
Use the diagram at the right.
Redraw
ACB and
DBC side by
side with corresponding parts in the
same position.
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 3 and 4
Use the information in the diagram to
ACB
DBC
prove that
STATEMENTS
REASONS
H 1.
1. Given
2. Given
3. Definition of lines
4. Definition of a right
triangle
AC DB
2. AB BC, CD BC
3.
C, B are right
4.
ACB and DBC are
right triangles.
CB
L 5. BC
6.
ACB
DBC
5. Reflexive Property of
Congruence
6. HL Congruence
Theorem