4.4 Prove Triangles Congruent by
SAS and HL
Side-Angle-Side (SAS) Congruence
Postulate
• If two sides and the included angle are
congruent to the corresponding sides and
angles on another triangle, then the triangles
are congruent.
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
CA
DAC
3. Alternate Interior
Angles Theorem
4. Reflexive Property of
Congruence
Extra Example 1
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
Postulate.
MPQ are congruent by the SAS Congruence
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
S. U
VU
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.
for Examples 1 and 2
Prove that
BSR
DUT
STATEMENTS
REASONS
1.
1. Given
2.
BS
RBS
3. RS
4.
DU
TDU
line
3. Given
UT
BSR
2. Definition of
DUT
4. SAS Congruence
Postulate
Key Ideas for Proving SAS
• Use the given- often sides are already given in
a direct (BC is congruent to DA) or in an
indirect way (ABCD is a square)
• Think about what the given means- midpointdivides a segment into two congruent parts
Parallel Lines- Look for alternate interior or other
relationships we’ve discussed
Perpendicular- Automatically have right angles
HL
• All right triangles have two legs and one
hypotenuse.
• To prove these triangles congruent- the
hypotenuse and a leg for two different
triangles have to be congruent to each other.
EXAMPLE 3
Use the Hypotenuse-Leg Congruence Theorem
Write a proof.
GIVEN
WY
PROVE
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.
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
prove that
ACB
DBC
STATEMENTS
1.
AC
2. AB
REASONS
1. Given
DB
BC, CD
BC
3.
C
B
4.
ACB and DBC are
right triangles.
2. Given
3. Definition of
lines
4. Definition of a right
triangle
Questions to ask When Deciding Which
Postulate to Use
• Can I see that all sides are going to be
congruent? SSS
• Do I have congruent hypotenuses- have to
have a right angle to have a hypotenuse- can I
show the legs congruent? HL
• Do I have an angle congruent in both? Is it in
between two sides that are congruent or I can
show they’re congruent? SAS
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?
Daily Homework Quiz
For use after Lesson 4.4
Is there enough given information to prove the triangles
congruent? If there is, state the postulate or theorem.
1.
ABE,
ANSWER
CBD
SAS
Post.
Daily Homework Quiz
For use after Lesson 4.4
Is there enough given information to prove the triangles
congruent? If there is, state the postulate or theorem.
2.
FGH,
ANSWER
HJK
HL
Thm.
Daily Homework Quiz
For use after Lesson 4.4
State a third congruence that would allow you to prove
RST
XYZ by the SAS Congruence postulate.
3.
ST
ANSWER
YZ, RS
XY
S
Y.
Daily Homework Quiz
For use after Lesson 4.4
State a third congruence that would allow you to prove
RST
XYZ by the SAS Congruence postulate.
4.
T
ANSWER
Z, RT
XZ
ST
YZ .
Homework
• 4.4: 1, 2-18ev, 20 – 23, 25 – 29, 34 – 38