1. 4.4 Prove Triangles Congruent by SAS and HL

2. 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.

3. 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

4. Extra Example 1

5. 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

6. 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

7. 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

8. 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- midpoint- divides a segment into two congruent parts
Parallel Lines- Look for alternate interior or other relationships we’ve discussed
Perpendicular- Automatically have right angles

9. 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.

10. 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.

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
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.

13. 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

14. 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?

15. 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, CBD
ANSWER
SAS Post.

16. 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, HJK
ANSWER
HL Thm.

17. 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 YZ, RS XY
ANSWER
S Y.

18. 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 Z, RT XZ
ANSWER
ST YZ .

19. Homework
4.4: 1, 2-18ev, 20 – 23, 25 – 29, 34 – 38