1. Prove: ∆CDF ∆EDF
Given: DF bisects CE, DC DE C F E D
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.

2. EXAMPLE 1
Use the SAS Congruence Postulate
Write a proof.
GIVEN
BC DA, BC AD
PROVE
ABC CDA
STATEMENTS
REASONS
Given
BC DA S
Given
BC AD
BCA DAC
Alternate Interior Angles Theorem A
AC CA
Reflexive Property of Congruence S

3. EXAMPLE 1
Use the SAS Congruence Postulate
STATEMENTS
REASONS
ABC CDA
SAS Congruence Postulate

4. 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.
MRS and MPQ are congruent by the SAS Congruence Postulate.
ANSWER

5. 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
SU VU
Prove that
SVR UVR
STATEMENTS
REASONS
SV VU
Given
SVR RVU
Definition of line
RV VR
Reflexive Property of Congruence
SVR UVR
SAS Congruence Postulate

6. GUIDED PRACTICE
for Examples 1 and 2
Prove that
BSR DUT
STATEMENTS
REASONS
Given
BS DU
RBS TDU
Definition of line
RS UT
Given
BSR DUT
SAS Congruence Postulate

7. EXAMPLE 3
Use the Hypotenuse-Leg Congruence Theorem
Write a proof.
GIVEN
WY XZ, WZ ZY, XY ZY
PROVE
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.

8. EXAMPLE 3
Use the Hypotenuse-Leg Congruence Theorem
STATEMENTS
REASONS
WY XZ
Given
WZ ZY, XY ZY
Given
Definition of lines
Z and Y are right angles
Definition of a right triangle
WYZ and XZY are right triangles. L
ZY YZ
Reflexive Property of Congruence
WYZ XZY
HL Congruence Theorem

9. 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?

10. EXAMPLE 4
Choose a postulate or theorem
SOLUTION
RPQ and RPS are right angles, so they are congruent. So, two sides and their included angle are congruent.
You are given that PQ PS . By the Reflexive Property, RP RP . By the definition of perpendicular lines, both
You can use the SAS Congruence Postulate to conclude that
PQR PSR
ANSWER

11. 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.

12. GUIDED PRACTICE
for Examples 3 and 4
STATEMENTS
REASONS L
BC CB
Reflexive Property of Congruence
ACB DBC
HL Congruence Theorem

13. GUIDED PRACTICE
for Examples 3 and 4
Use the diagram at the right.
Use the information in the diagram to prove that
ACB DBC
STATEMENTS
REASONS
AC DB
Given
AB BC, CD BC
Given
Definition of lines
C B
Definition of a right triangle
ACB and DBC are right triangles.

14. Daily Homework Quiz
Is there enough given information to prove the triangles congruent? If there is, state the postulate or theorem. 1.
ABE, CBD
ANSWER
SAS Post.

15. Daily Homework Quiz
Is there enough given information to prove the triangles congruent? If there is, state the postulate or theorem. 2.
FGH, HJK
ANSWER
HL Thm.

16. Daily Homework Quiz
State a third congruence that would allow you to prove RST XYZ by the SAS Congruence postulate. 3.
ST YZ, RS XY
ANSWER
S Y.

17. Daily Homework Quiz
State a third congruence that would allow you to prove RST XYZ by the SAS Congruence postulate. 4.
T Z, RT XZ
ANSWER
ST YZ .