1. Ways to prove Triangles Congruent (SSS), (SAS), (ASA)

2. EXAMPLE 4
Use the Third Angles Theorem
Find m BDC.
SOLUTION
A B and ADC BCD, so by the Third Angles Theorem, ACD BDC. By the Triangle Sum Theorem, m ACD = 180° – 45° – 30° = 105° .
So, m ACD = m BDC = 105° by the definition of congruent angles.
ANSWER

3. EXAMPLE 5
Prove that triangles are congruent
Write a proof.
GIVEN
AD CB, DC AB
ACD CAB,
CAD ACB
PROVE
ACD CAB
Plan for Proof
AC AC.
Use the Reflexive Property to show that
Use the Third Angles Theorem to show that
B D

4. Prove that triangles are congruent
EXAMPLE 5
Prove that triangles are congruent
Plan in Action
STATEMENTS
REASONS
AD CB
, DC BA
Given
AC AC.
Reflexive Property of Congruence
ACD CAB,
CAD ACB
Given
B D
Third Angles Theorem
ACD CAB
Definition of

5. GUIDED PRACTICE
for Examples 4 and 5
DCN.
In the diagram, what is m
SOLUTION
CDN NSR, DNC SNR then the third angles are also congruent NRS DCN = 75°

6. GUIDED PRACTICE
for Examples 4 and 5
By the definition of congruence, what
additional information is needed to
know that
NDC NSR.
SOLUTION
CN NR, CDN NSR,
DCN NRS
Given :
(Proved from above sum)
NDC NSR.
Proved :

7. GUIDED PRACTICE for Examples 4 and 5 STATEMENT REASON CDN NSR DCN NRS
Given
DCN NRS
Given
Therefore DC RS, DN SN as angles are congruent their sides are congruent.

8. EXAMPLE 1
Identify congruent parts
Write a congruence statement for the triangles. Identify all pairs of congruent corresponding parts.
SOLUTION
The diagram indicates that
JKL TSR.
Corresponding angles
J T, ∠ K S, L R
Corresponding sides
JK TS, KL SR, LJ RT

9. EXAMPLE 2
Use properties of congruent figures
In the diagram, DEFG SPQR.
Find the value of x.
Find the value of y.
SOLUTION
You know that FG QR. FG
= QR 12
= 2x – 4 16
= 2x 8
= x

10. Use properties of congruent figures
EXAMPLE 2
Use properties of congruent figures
You know that ∠ F Q.
m F
= m Q 68 o
= (6y + x) 68
= 6y + 8 10
= y

11. EXAMPLE 3
Show that figures are congruent
PAINTING
If you divide the wall into orange and blue sections along JK , will the sections of the wall be the same size and shape?Explain.
SOLUTION
From the diagram, A C and D B because all right angles are congruent. Also, by the Lines Perpendicular to a Transversal Theorem, AB DC .

12. EXAMPLE 3
Show that figures are congruent
Then, and by the Alternate Interior Angles Theorem. So, all pairs of corresponding angles are congruent.
The diagram shows AJ CK , KD JB , and DA BC . By the Reflexive Property, JK KJ . All corresponding parts are congruent, so AJKD CKJB.

13. GUIDED PRACTICE
for Examples 1, 2, and 3
In the diagram at the right, ABGH CDEF.
Identify all pairs of congruent corresponding parts.
SOLUTION
Corresponding sides:
AB CD, BG DE,
GH FE, HA FC
Corresponding angles:
A C, B D,
G E, H F.

14. GUIDED PRACTICE
for Examples 1, 2, and 3
In the diagram at the right, ABGH CDEF. 2.
Find the value of x and find m H.
SOLUTION
(a)
You know that H F
(4x+ 5)° = 105°
4x = 100
x = 25
(b)
You know that H F
m H m F =105°

15. GUIDED PRACTICE
for Examples 1, 2, and 3
In the diagram at the right, ABGH CDEF. 3.
Show that PTS RTQ.
SOLUTION
In the given diagram
PS QR, PT TR, ST TQ and
Similarly all angles are to each other, therefore all of the corresponding points of PTS are congruent to those of RTQ by the indicated markings, the Vertical Angle Theorem and the Alternate Interior Angle theorem.

16. EXAMPLE 1
Use the SSS Congruence Postulate
Write a proof.
GIVEN
KL NL, KM NM
PROVE
KLM NLM
Proof
It is given that
KL NL and KM NM
By the Reflexive Property,
LM LN.
So, by the SSS Congruence Postulate,
KLM NLM

17. GUIDED PRACTICE
for Example 1
Decide whether the congruence statement is true. Explain your reasoning.
DFG HJK
SOLUTION
Three sides of one triangle are congruent to three sides of second triangle then the two triangle are congruent.
Side DG HK, Side DF JH,and Side FG JK.
So by the SSS Congruence postulate, DFG HJK.
Yes. The statement is true.

18. GUIDED PRACTICE
for Example 1
Decide whether the congruence statement is true. Explain your reasoning. 2.
ACB CAD
SOLUTION
BC AD
GIVEN :
PROVE :
ACB CAD
PROOF:
It is given that BC AD By Reflexive property
AC AC, But AB is not congruent CD.

19. GUIDED PRACTICE
for Example 1
Therefore the given statement is false and ABC is not
Congruent to CAD because corresponding sides
are not congruent

20. GUIDED PRACTICE
for Example 1
Decide whether the congruence statement is true. Explain your reasoning.
QPT RST 3.
SOLUTION
QT TR , PQ SR, PT TS
GIVEN :
PROVE :
QPT RST
PROOF:
It is given that QT TR, PQ SR, PT TS. So by
SSS congruence postulate, QPT RST. Yes the statement is true

21. 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

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

23. 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

24. 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

25. 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

26. 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.

27. 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

28. 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?

29. 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

30. 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.

31. 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
AB DC
Given H
AC BC, DB BC
Given
Definition of lines
C B
Definition of a right triangle
ACB and DBC are right triangles.

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