1. Congruent Triangles Part 2
Unit 4
SOL G.5 & G.6
Sections 4.5, 4.6, 4.7 Review & Proofs
Resource: Henrico Geometry

2. Parts of An Isosceles Triangle
An isosceles triangle is a triangle with at least two congruent sides.
The congruent sides are called legs and the third side is called the base.
The angles adjacent to the base are called base angles.
The angle opposite the base is called the vertex angle.
and are legs.
is the base.
1 and 2 are base angles.
3 is the vertex angle.

3. Isosceles Triangle Theorems
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

4. Example By the Isosceles Triangle Theorem,
the third angle must also be x.
Therefore,
x + x + 50 = 180
2x + 50 = 180
2x = 130
x = 65 x

5. More Examples Since two angles are congruent, the
sides opposite these angles must be
congruent.
3x – 7 = x + 15
2x = 22
x = 11

6. Equilateral and Equiangular Triangles
Corollary: If a triangle is equilateral, then …
it is equiangular. .

7. What if all the ANGLES in a triangle are DIFFERENT?
Then the triangle is SCALENE
because all of the sides are DIFFERENT too!

8. The triangles are congruent by HL.
Hypotenuse-Leg (HL)
If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.
Right Triangle
Leg
Hypotenuse
AB  HL
CB  GL
C and G are rt.  ‘s
ABC   DEF
The triangles are congruent by HL.

9. Reflexive Sides and Angles
Vertical Angles,
Reflexive Sides and Angles
When two triangles touch, there may be additional congruent parts.
Vertical Angles
Reflexive Side
side shared by two
triangles

10. Name That Postulate SAS SAS SSA AAS Vertical Angles Reflexive Property
(when possible)
Vertical Angles
Reflexive Property
SAS
SAS
Vertical Angles
Reflexive Property
SSA
AAS
Not enough info!

11. Reflexive Sides and Angles
When two triangles overlap, there may be additional congruent parts.
Reflexive Side
side shared by two
triangles
Reflexive Angle
angle shared by two

12. Let’s Practice B  D AC  FE A  F
Indicate the additional information needed to enable us to apply the specified congruence postulate.
For ASA:
B  D
For SAS:
AC  FE
A  F
For AAS:

13. Problem #1

14. Step 1: Mark the Given

15. Step 3: Choose a Method
SSS
SAS
ASA
AAS HL

16. Step 4: List the Parts S S S … in the order of the Method STATEMENTS
REASONS S S S
… in the order of the Method

17. Step 5: Fill in the Reasons
STATEMENTS
REASONS S S S
(Why did you mark those parts?)

18. Step 6: Is there more? S S S The “Prove” Statement is always last !
STATEMENTS
REASONS S S S

19. Problem #2

20. Step 1: Mark the Given

21. Step 3: Choose a Method
SSS
SAS
ASA
AAS HL

22. Step 4: List the Parts S A S … in the order of the Method STATEMENTS
REASONS S A S
… in the order of the Method

23. Step 5: Fill in the Reasons
STATEMENTS
REASONS S A S
(Why did you mark those parts?)

24. Step 6: Is there more? S A S The “Prove” Statement is always last !
STATEMENTS
REASONS S A S

25. Problem #3

26. Step 1: Mark the Given

27. Step 3: Choose a Method
SSS
SAS
ASA
AAS HL

28. Step 4: List the Parts A S A … in the order of the Method STATEMENTS
REASONS A S A
… in the order of the Method

29. Step 6: Is there more? A S A The “Prove” Statement is always last !
STATEMENTS
REASONS A S A

30. Problem #4 AAS Statements Reasons Given Given AAS Postulate
Vertical Angles Thm
Given
AAS Postulate

31. Problem #5 HL Statements Reasons Given Given Reflexive Property
Given ABC, ADC right s,
Prove:
Statements
Reasons
Given
1. ABC, ADC right s
Given
Reflexive Property
HL Postulate

32. Congruence Proofs 1. Mark the Given. 2. Mark …
Reflexive Sides or Angles / Vertical Angles
Also: mark info implied by given info.
3. Choose a Method. (SSS , SAS, ASA)
4. List the Parts …
in the order of the method.
5. Fill in the Reasons …
why you marked the parts.
6. Is there more?

33. Given implies Congruent Parts
segments
midpoint
angles
parallel
segments
segment bisector
angles
angle bisector
angles
perpendicular

34. Example Problem

35. … and what it implies
Step 1: Mark the Given

36. Reflexive Sides
Vertical Angles
Step 2: Mark . . .
… if they exist.

37. Step 3: Choose a Method
SSS
SAS
ASA
AAS HL

38. Step 4: List the Parts S A … in the order of the Method STATEMENTS
REASONS S A
… in the order of the Method

39. Step 5: Fill in the Reasons
STATEMENTS
REASONS S A S
(Why did you mark those parts?)

40. Step 6: Is there more?
STATEMENTS
REASONS S 1. 2. 3. 4. 5. A S

41. Midpoint implies segments.
Back
Midpoint implies segments.
STATEMENTS
REASONS S … 3.
3. Given

42. Parallel implies angles.
Back
Parallel implies angles.
STATEMENTS
REASONS A A

43. Seg. bisector implies segments.
Back
Seg. bisector implies segments.
STATEMENTS
REASONS S … S

44. Angle bisector implies angles.
Back
Angle bisector implies angles.
STATEMENTS
REASONS A …

45. implies right ( ) angles.
Back
STATEMENTS
REASONS A … S 4.
4. Given

46. Congruent Triangles Proofs
1. Mark the Given and what it implies.
2. Mark … Reflexive Sides / Vertical Angles
3. Choose a Method. (SSS , SAS, ASA)
4. List the Parts …
in the order of the method.
5. Fill in the Reasons …
why you marked the parts.
6. Is there more?

47. Using CPCTC in Proofs
According to the definition of congruence, if two triangles are congruent, their corresponding parts (sides and angles) are also congruent.
This means that two sides or angles that are not marked as congruent can be proven to be congruent if they are part of two congruent triangles.
This reasoning, when used to prove congruence, is abbreviated CPCTC, which stands for Corresponding Parts of Congruent Triangles are Congruent.

48. Corresponding Parts of Congruent Triangles
For example, can you prove that sides AD and BC are congruent in the figure at right?
The sides will be congruent if triangle ADM is congruent to triangle BCM.
Angles A and B are congruent because they are marked.
Sides MA and MB are congruent because they are marked.
Angles 1 and 2 are congruent because they are vertical angles.
So triangle ADM is congruent to triangle BCM by ASA.
This means sides AD and BC are congruent by CPCTC.

49. Corresponding Parts of Congruent Triangles
A two column proof that sides AD and BC are congruent in the figure at right is shown below:
Statement
Reason MB
Given ÐB Ð2
Vertical angles
DBCM
ASA BC
CPCTC

50. Corresponding Parts of Congruent Triangles
A two column proof that sides AD and BC are congruent in the figure at right is shown below:
Statement
Reason MB
Given ÐB Ð2
Vertical angles
DBCM
ASA BC
CPCTC

51. Corresponding Parts of Congruent Triangles
Sometimes it is necessary to add an auxiliary line in order to complete a proof
For example, to prove ÐO in this picture
Statement
Reason FO
Given OU UF
reflexive prop.
DFOU
SSS ÐO
CPCTC