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. Proving triangles congruent.
Target
Proving triangles congruent.
GOAL:
4.5 Use sides and angles to prove triangles congruent.

3. Vocabulary
included angle – the angle between two identified sides
SAS (Side-Angle-Side) Congruence Postulate
If two sides and the included angle of one triangle are congruent to two sides and the included angle
of a second triangle,
then the two triangles are congruent.
HL (Hypotenuse-Leg) Congruence Theorem
If the hypotenuse and leg of one right triangle are congruent to the hypotenuse and leg of a
second right triangle,
then the two triangles are congruent.

4. 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
ABC CDA
SAS Congruence Postulate

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
SV VU
Prove that
SVR UVR
STATEMENTS
REASONS S A
SV VU
Given
RT SU Given
SVR RVU
Def. of lines; Right
RV VR
Ref. Prop. of Congruence
SVR UVR
SAS Congruence Postulate

7. 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
BSR DUT
STATEMENTS
REASONS
Given
BC DA S A
Def. Midpoint, Trans.
BS DU
RBS TDU
Given; Right
RS UT
CPCTC (previous proof)
BSR DUT
SAS Congruence Postulate

8. 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.

9. EXAMPLE 3
Use the Hypotenuse-Leg Congruence Theorem
STATEMENTS
REASONS
WY XZ
Given H
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

10. 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.

11. GUIDED PRACTICE
for Examples 3 and 4
Use the information in the diagram to prove that
ACB DBC
STATEMENTS
REASONS
AC DB
Given H
AB BC, CD BC
Given
Definition of lines
C, B are right
Definition of a right triangle
ACB and DBC are right triangles. L
BC CB
Reflexive Property of Congruence
ACB DBC
HL Congruence Theorem