1. Warm Up Lesson Presentation Lesson Quiz
Prove Triangles Congruent by SAS and HL
Warm Up
Lesson Presentation
Lesson Quiz

2. Warm-Up
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.

3. Example 1
Write a proof.
GIVEN:
BC DA, BC AD
PROVE:
ABC CDA
STATEMENTS
REASONS
BC DA S
Given
BC AD
Given
BCA DAC A
Alternate Interior Angles Theorem
AC CA S
Reflexive Property of Congruence 5.
ABC CDA 5.
SAS Congruence Postulate

4. Example 2
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
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
ANSWER

6. 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 of ABCD. Also, RT SU and
SU VU
Prove that
BSR DUT
ANSWER

7. Guided Practice

8. Example 3
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.

9. Example 3
STATEMENTS
REASONS
WY XZ H
Given
WZ ZY, XY ZY
Given
Z and Y are right angles
Definition of lines
WYZ and XZY are right triangles.
Definition of a right triangle L
ZY YZ
Reflexive Property of Congruence
WYZ XZY
HL Congruence Theorem

10. Example 4
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?

11. Example 4
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

12. Guided Practice
Use the diagram at the right.
Redraw ACB and DBC side by side with corresponding parts in the same position.
ANSWER

13. Guided Practice
Use the diagram at the right.
Use the information in the diagram to prove that
ACB DBC
ANSWER

14. Lesson 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. Lesson 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. Lesson 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. Lesson 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 .