1. Properties of Equality and Congruence, and Proving Lines Parallel
Objectives: Students will be able to…
Use properties of equality and congruence
Justify each step in solving equations
Identify angles formed by two lines and a transversal
Prove and use properties of parallel lines
Use a transversal to prove two lines parallel

2. Remember…
In Geometry, we cannot assume anything is necessarily true (unless it is a theorem or postulate). We can’t say “That looks like an acute angle, so it is an acute angle”
It needs to be told to us, marked on a diagram or we have to use logic to prove it
THEOREMS have already been proven true for us
POSTULATES are Geometric statements that are assumed to be true

3. Drawing Conclusions There are conclusions you can make from a diagram.
You can assume that angles are…
Adjacent angles
Adjacent supplementary angles
Vertical angles
Unless it is marked or you are told, you cannot assume…
Angles or segments are congruent
An angle is a right angle
Lines are parallel or perpendicular

4. To justify what we do in Geometry, we can use:
Properties
Postulates
Theorems
Definitions (ex. Definition of a right angle, definition of an angle bisector, etc…)
In summary, we must justify everything we do. This helps us with logical thinking!!!!

5. Properties of equality (use with numbers)
If a = b then a + c = b + c Addition Property of Equality
If a = b then a - c = b – c Subtraction Property of Equality
If a = b, then a ● c = b ● c Multiplication Property of Equality
If a = b, then , c ≠ Division Property of Equality
a = a Reflexive Property of Equality
If a = b, then b = a Symmetric Property of Equality
If a = b and b = c, then a = c Transitive Property of Equality

6. More properties of equality
Substitution Property:
If a = b, then b can replace a in any expression
The Distributive Property:
a(b + c) = ab + bc

7. Properties of congruence
Reflexive Property: , A A
Symmetric Property: If , then
If A B, then B A
Transitive Property: If and , then
If A B and B C, then A C

8. Using Properties of equality and congruence
Name the property that justifies each statement.
If x = y and y + 4 = 3x, then x + 4 = 3x
If x + 4 = 3x, then 4 = 2x If

9. Justify each step of solving the following problem…
2 (3x + 7) = 26

10. Two-Column Proof Displays steps that prove a statement
Statements on left; Reasons/justifications on right
Use theorems, postulates, definition, properties and given statements for justifications
GIVEN: (what you know)
PROVE: (what you must show)
Statements: Reasons:

11. Don’t forget…
Vertical Angles are congruent

12. Also don’t forget…
Angles that form a line (straight angle) add up to 180°

13. Solve for x. Justify each step.

14. Solve for x. Justify each step.

15. Extra examples, if necessary. Find the value of the variables
Extra examples, if necessary. Find the value of the variables. Justify each step. 1. 2.
(8t)°
126°
(10t)°
(3x)°
(6x-54)°

16. TRANSVERSAL: Transversal n intersects line l and m
Line intersects 2 lines in 2 distinct points
The intersection of a transversal and the 2 lines form 8 angles
Transversal n intersects line l and m

17. The angles formed when a transversal intersects 2 lines depends on their position
ALTERNATE INTERIOR ANGLES:
Non-adjacent
Lie on opposite sides of the transversal in between the 2 lines it intersects

18. Alternate Exterior Angles
Lie outside the 2 lines on opposite sides of the transversal

19. Same-Side Interior Angles (Co-interior)
Lie on the same side of the transversal between the two lines

20. Same-Side Exterior Angles
Lie outside the 2 lines on same side of transversal

21. Corresponding Angles Lie on the same side of the transversal
In corresponding positions

23. Using a protractor, measure the following:
a.) 1 pair of corresponding angles
b.) 1 pair of alternate interior angles
c.) 1 pair of same side interior angles

24. If a transversal intersects 2 parallel lines:
Corresponding angles are congruent
Alternate interior angles are congruent
Same side interior angles are supplementary
Alternate exterior angles are congruent
Same side exterior angles are supplementary

25. Finding Measures of Angles
a b c d Find the measure of each angle. Justify your answer. 8 7 6 2
50° 5 4 1 3

26. Find the values of x and y. Then find the measures of the angles.

27. What is the converse of this statement
What is the converse of this statement? (The converse of a statement switches the hypothesis and the conclusion)
If a transversal intersects 2 parallel lines, then corresponding angles are congruent.

28. Converse of the Corresponding Angles Postulate
If two lines and a transversal form corresponding angles that are congruent, then the two lines are parallel.

29. Write the converse:
If a transversal intersects 2 parallel lines, then alternate interior angles are congruent.

30. Converse of the Alternate Interior Angle Theorem
If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel.
If , then line 1 ll line 2

31. Write the converse:
If a transversal intersects 2 parallel lines, then same side interior angles are supplementary.

32. Converse of the Same-Side Interior Angles Theorem
If two lines and a transversal form same-side interior angles that are supplementary, then the two lines are parallel.
If are supplementary, then m ll n

33. Which lines or segments are parallel? How do you know??1. 2. C H e
45°
45° M A R g c b

34. Which segments are parallel?J K L O N M

35. Theorem 3-5
If two lines are parallel to the same line, then they are parallel to each other.
a ll b
* Lines can be coplanar or noncoplanar

36. Theorem
In a plane, if two lines are perpendicular to the same line, then they are parallel to each other.
k ll l

37. In City Hall, Corridor 1 and Corridor 2 are both perpendicular to Corridor 3. What can you say about corridor 1 and corridor 2?

38. Solve for x and then solve for each angle such that n ll m
5x m

39. Find the value of x so that m||n.
7x n

40. Proof: Given Prove: n ll m Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. n ll m 6. 7