1. Proving Triangles are Congruent: SSS, SAS
Section 5.2

2. Objectives:
Show triangles are congruent using SSS and SAS.

3. Key Vocabulary
Included angle
proof

4. Postulates
12 SSS Congruence Postulate
13 SAS Congruence Postulate

5. Review Congruent Triangles
Triangles that are the same shape and size are congruent.
Each triangle has three sides and three angles.
If all 6 parts (3 corresponding sides and 3 corresponding angles) are congruent….then the triangles are congruent.

6. Review
CPCTC –
Corresponding Parts of Congruent Triangles are Congruent
Be sure to label  Δs with proper markings (i.e. if D  L, V  P, W  M, DV  LP, VW  PM, and WD  ML then we must write ΔDVW  ΔLPM)

7. So, to prove Δs  we must prove ALL sides & ALL s are  ?
Fortunately, NO!
There are some shortcuts…
Two of which follow!!!

8. Postulate 12: Side-Side-Side Congruence Postulate
SSS Congruence - If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. If
then,

9. Adjacent triangles share a side, so you can apply the Reflexive Property to get a pair of congruent parts.
Remember!
∆PMN is adjacent to ∆PON, PN≅PN by the Reflexive Property.

10. Example 1: Using SSS to Prove Triangle Congruence
Use SSS to explain why ∆ABC  ∆DBC.
It is given that AC  DC and that AB  DB. By the Reflexive Property of Congruence, BC  BC. Therefore ∆ABC  ∆DBC by SSS.

11. Your Turn Use SSS to explain why ∆ABC  ∆CDA.
It is given that AB  CD and BC  DA.
By the Reflexive Property of Congruence, AC  CA.
So ∆ABC  ∆CDA by SSS.

12. Example #2 – SSS – Coordinate Geometry
Use the SSS Postulate to show the two triangles are congruent. Find the length of each side.
AC = 5
BC = 7
AB =
MO = 5
NO = 7
MN =

13. Example 3 From the diagram you know that HJ  LJ and HKLK.
Does the diagram give enough
information to show that the
triangles are congruent? Explain.
SOLUTION
From the diagram you know that HJ  LJ and HKLK.
By the Reflexive Property, you know that JK  JK.
ANSWER
Yes, enough information is given. Because corresponding sides are congruent, you can use the SSS Congruence Postulate to conclude that ∆HJK  ∆LJK.

14. Definition: Included Angle
The angle between two sides in a figure.
∠B is included between:

15. Angle in between two consecutive sides
INCLUDED ANGLE X Z Y A B C
Angle in between two consecutive sides

16. Use the diagram. Name the included angle between the pair of sides given.
∠MTR
∠RTQ
∠MRT ∠Q

17. The 2nd Congruence Short Cut
It can also be shown that only two pairs of congruent corresponding sides are needed to prove the congruence of two triangles if the included angles are also congruent.
SAS

18. Postulate 13: Side-Angle-Side Congruence Postulate
SAS 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. If
then,

19. The letters SAS are written in that order because the congruent angles must be between pairs of congruent corresponding sides.
Caution S A S

20. Example 4: Engineering Application
The diagram shows part of the support structure for a tower. Use SAS to explain why ∆XYZ  ∆VWZ.
It is given that XZ  VZ and that YZ  WZ. By the Vertical s Theorem. XZY  VZW. Therefore ∆XYZ  ∆VWZ by SAS.

21. Example 5
Does the diagram give enough information to use the SAS Congruence Postulate? Explain your reasoning. a. b.
SOLUTION a.
From the diagram you know that AB  CB and DB  DB.
The angle included between AB and DB is ABD.
The angle included between CB and DB is CBD.
Because the included angles are congruent, you can use the SAS Congruence Postulate to conclude that ∆ABD  ∆CBD.

22. Example 5 b.
You know that GF  GH and GE  GE. However, the congruent angles are not included between the congruent sides, so you cannot use the SAS Congruence Postulate.

23. Your Turn Use SAS to explain why ∆ABC  ∆DBC.
It is given that BA  BD and ABC  DBC. By the Reflexive Property of , BC  BC. So ∆ABC  ∆DBC by SAS.

24. Example 6A: Verifying Triangle Congruence
Show that the triangles are congruent for the given value of the variable.
∆MNO  ∆PQR, when x = 5. PQ
= x + 2
= = 7 QR
= x = 5 PR
= 3x – 9
= 3(5) – 9 = 6
PQ  MN, QR  NO, PR  MO
∆MNO  ∆PQR by SSS.

25. Example 6B: Verifying Triangle Congruence
Show that the triangles are congruent for the given value of the variable.
∆STU  ∆VWX, when y = 4. ST
= 2y + 3
= 2(4) + 3 = 11 TU
= y + 3
= = 7
mT
= 20y + 12
= 20(4)+12 = 92°
ST  VW, TU  WX, and T  W.
∆STU  ∆VWX by SAS.

26. Your Turn Show that ∆ADB  ∆CDB, t = 4. DA = 3t + 1 = 3(4) + 1 = 13 DC
= 4(4) – 3 = 13
mD
= 2t2
= 2(16)= 32°
ADB  CDB Def. of .
DB  DB Reflexive Prop. of .
∆ADB  ∆CDB by SAS.

27. Practice Determining Triangle Congruence Using SSS and SAS

28. #1
Determine if whether the triangles are congruent. If they are, write a congruency statement explaining why they are congruent. R T S Y X Z
ΔRST  ΔYZX by SSS

29. #2
Determine if whether the triangles are congruent. If they are, write a congruency statement explaining why they are congruent. R P S Q
ΔPQS  ΔPRS by SAS

30. Not enough Information to Tell#3
Determine if whether the triangles are congruent. If they are, write a congruency statement explaining why they are congruent. R T S B A C
Not congruent.
Not enough Information to Tell

31. ΔPQR  ΔSTU by SSS #4 P S U Q R T
Determine if whether the triangles are congruent. If they are, write a congruency statement explaining why they are congruent. R P Q S T U
ΔPQR  ΔSTU by SSS

32. Not enough Information to Tell#5
Determine if whether the triangles are congruent. If they are, write a congruency statement explaining why they are congruent. N M R Q P
Not congruent.
Not enough Information to Tell

33. Proof
A proof is an argument that uses logic, definitions, properties, and previously proven statements to show that a conclusion is true.
An important part of writing a proof is giving justifications to show that every step is valid.

34. Two-Column Proof
Two-Column Proof – A proof format used in geometry in which an argument is presented with two columns, statements and reasons, to prove conjectures and theorems are true. Also referred to as a formal proof.
Two- Column Proof is a formal proof because it has a specific format.
Two-Column Proof
Left column – statements in a logical progression.
Right column – Reason for each statement (definition, postulate, theorem or property).

35. Two-Column Proof
Statements Reasons
Given:
Prove:
Proof:

36. How to Write A Proof List the given information first
Use information from the diagram
Give a reason for every statement
Use given information, definitions, postulates, and theorems as reasons
List statements in order. If a statement relies on another statement, list it later than the statement it relies on
End the proof with the statement you are trying to prove

37. Geometric Proof
Since geometry also uses variables, numbers, and operations, many of the algebraic properties of equality are true in geometry. For example:
These properties can be used to write geometric proofs.
Property
Segments
Angles
Reflexive
AB = AB
m ∠1 = m ∠1
Symmetric
If AB = CD, then CD = AB.
If m ∠1 = m ∠2,
then m ∠2 = m ∠1.
Transitive
If AB = CD and CD = EF, then AB = EF.
If m ∠1 = m ∠2 and m ∠2 = m ∠3, then m ∠1 = m ∠3.

38. Numbers are equal (=) and figures are congruent ().
Remember!

39. Example 7 Write a two-column proof that shows ∆JKL  ∆NML. JL  NL
L is the midpoint of KM.
SOLUTION
The proof can be set up in two columns. The proof begins with the given information and ends with the statement you are trying to prove.

40. Example 7 Statements Reasons Given 1. JL  NL Given 2. L is the
These are the given
statements.
Statements
Reasons
Given 1.
JL  NL
Given 2.
L is the
midpoint of KM.
This information is from the diagram.
JKL  NML 3.
Vertical
Angles
Theorem

41. Example 7 Definition of midpoint 4. KL  ML Statements Reasons
Statement 4 follows from Statement 2.
Definition of
midpoint 4.
KL  ML
Statements
Reasons
∆JKL  ∆NML 5.
Statement 5 follows from the congruences of Statements 1, 3, and 4.
SAS Congruence
Postulate

42. Example 8 A D G R  RG RA AG DR
You are making a model of the window shown in the figure. You know that
and 
proof to show that ∆DRA  ∆DRG.
. Write a
SOLUTION 1.
Make a diagram and label it with
the given information.

43. Example 8 2.
Write the given information and the statement you need to prove.
∆DRA  ∆DRG
DR AG, RA  RG 
Write a two-column proof. List the given statements first. 3. 43

44. Example 8 Statements Reasons 1. RA  RG Given 2. DR AG  Given 3.
DRA and DRG are right angles.
lines form right angles.  4.
DRA  DRG
Right angles are congruent. 5.
DR  DR
Reflexive Property of Congruence 6.
∆DRA  ∆DRG
SAS Congruence Postulate 44

45. Your Turn: Fill in the missing statements and reasons. 1. ∆BCA  ∆ECDDC
AC 
CB  CE ,
CB  CE
Statements
Reasons 1. ?
_____
Given
ANSWER
Given 2. ?
_____ DC
AC 
ANSWER
BCA  ECD 3. ?
_____
Vertical Angles
Theorem
ANSWER
SAS
Congruence
Postulate
ANSWER
∆BCA  ∆ECD 4. ?
_____

46. Given: QR  UT, RS  TS, QS = 10, US = 10 Prove: ΔQRS  ΔUTS
Example 9:
Given: QR  UT, RS  TS, QS = 10, US = 10 Prove: ΔQRS  ΔUTS Q U R S T 10 U Q 10 10 R S T

47. Example 9: Statements Reasons________ 1. QR  UT, RS  TS, 1. Given
Given: QR  UT, RS  TS, QS = 10, US = 10 Prove: ΔQRS  ΔUTS Q U R S T 10
Statements Reasons________
1. QR  UT, RS  TS, Given
QS=10, US=10
2. QS = US Substitution
3. QS  US Def of  segs.
4. ΔQRS  ΔUTS SSS Postulate

48. Given: Prove: Your turn: Proof by SSS Statement Reason 1) 2) 3) 4) 5)X
Given:
Prove:
Your turn: Proof by SSS
Statement
Reason 1) 2) 3) 4) 5) 6)
Given
Given
Def. line bisector
Def. midpoint
Reflexive Postulate
SSS Postulate

49. Example 10: Proving Triangles Congruent
Given: BC ║ AD, BC  AD
Prove: ∆ABD  ∆CDB
Statements
Reasons
1. BC || AD
1. Given
2. CBD  ABD
2. Alt. Int. s Thm.
3. BC  AD
3. Given
4. BD  BD
4. Reflex. Prop. of 
5. ∆ABD  ∆ CDB
5. SAS

50. Your Turn Given: QP bisects RQS. QR  QS Prove: ∆RQP  ∆SQP
Statements
Reasons
1. QR  QS
1. Given
2. QP bisects RQS
2. Given
3. RQP  SQP
3. Def. of bisector
4. QP  QP
4. Reflex. Prop. of 
5. ∆RQP  ∆SQP
5. SAS

51. Summary – SSS & SAS

53. Hints to Proofs!
If two triangles share a side, in the statement column state that the side is congruent to itself. The reason is the Reflexive Property.
If two triangles share a vertex, in the statement column state that the angles are congruent. The reason is the Vertical Angles Theorem.

54. Hints to Proofs!
If a vocabulary word is in the given, you will need its definition as a reason somewhere in the proof, otherwise you wouldn’t need to know that piece of information.
If two lines are parallel, look for congruent angles pairs, such as Alternate Interior Angles, Corresponding Angles, or Alternate Exterior Angles.

56. Assignment
Pg #1 – 25 odd, 29 – 37 odd