EXAMPLE 1
Use the SAS Congruence Postulate
Write a proof.
GIVEN
BC
DA, BC AD
ABC
PROVE
CDA
STATEMENTS
S
REASONS
1.
BC
DA
1. Given
2.
BC
AD
2. Given
A 3.
S 4.
BCA
AC
DAC
CA
3. Alternate Interior
Angles Theorem
4. Reflexive Property of
Congruence
EXAMPLE 1
Use the SAS Congruence Postulate
STATEMENTS
5.
ABC
CDA
REASONS
5. SAS Congruence
Postulate
EXAMPLE 2
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.
for Examples 1 and 2
GUIDED PRACTICE
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
1.
Prove that
SVR
UVR
STATEMENTS
REASONS
1.
1. Given
2.
3.
4.
SV
VU
SVR
RV
RVU
VR
SVR
UVR
2. Definition of
line
3. Reflexive Property of
Congruence
4. SAS Congruence
Postulate
GUIDED PRACTICE
2.
Prove that
for Examples 1 and 2
BSR
DUT
STATEMENTS
REASONS
1.
1. Given
2.
BS
RBS
3. RS
4.
DU
BSR
TDU
2. Definition of
line
3. Given
UT
DUT
4. SAS Congruence
Postulate
EXAMPLE 3
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.
EXAMPLE 3
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
EXAMPLE 4
Choose a postulate or theorem
Sign Making
You are making a canvas sign to hang on the triangular
wall over the door to the barn shown in the picture. You
think you can use two identical triangular sheets of
canvas. You know that RP QS and PQ
PS . What
postulate or theorem can you use to conclude that
PQR
PSR?
EXAMPLE 4
Choose a postulate or theorem
SOLUTION
You are given that PQ PS . By the Reflexive Property, RP
RP . By the definition of perpendicular lines, both
RPQ and RPS are right angles, so they are congruent.
So, two sides and their included angle are congruent.
ANSWER
You can use the SAS Congruence Postulate to conclude
PQR
PSR
that
.
GUIDED PRACTICE
for Examples 3 and 4
Use the diagram at the right.
3.
Redraw
ACB and
DBC side by
side with corresponding parts in the
same position.
GUIDED PRACTICE
for Examples 3 and 4
Use the diagram at the right.
4.
Use the information in the diagram to
ACB
DBC
prove that
STATEMENTS
H 1.
AB
2. AC
DC
BC, DB BC
B
REASONS
1. Given
2. Given
3. Definition of
lines
3.
C
4.
ACB and DBC are 4. Definition of a right
triangle
right triangles.
GUIDED PRACTICE
for Examples 3 and 4
STATEMENTS
REASONS
L 5. BC
5. Reflexive Property of
Congruence
6.
ACB
CB
DBC
6. HL Congruence
Theorem