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Warm Up

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State the converse of (1) and (2):

1. If ∠1 is a right angle, then m∠1 = 90.

2. If m∠1 + m∠2 = 180, then ∠1 and ∠2 are supplementary.

State the hypothesis and conclusion of theorem 3.1.

41. If I go fishing, I will not have to work.

42. 67

43. 21

44. 70

33. 130

34. 107

35. 79

36. 73

37. 69

38. 62

39. If an angle is acute, then its measure is 19°.

40. If I go to the park, then you will go with me.

C

B

A

D

F

E

One Strategy:

By the LP postulate, the angles are supplementary and therefore add up to 180.

If the angles are equal, they each must be 90.

By the vertical angles theorem, m∠BFD must be 90.

By the angle addition postulate (and substitution), m∠BFC + m∠DFC = 90, and so the angles are complementary.

C

B

A

D

F

E

Another Strategy:

If the angles are a linear pair and congruent, then line AD ⊥ line BE (this is theorem 3.1, although most of you took a more roundabout route).

Then all four angles are right angles, by theorem 3.3.

If ∠BFD is a right angle, then its measure is 90.

By angle addition, m∠BFC + m∠DFC = 90, so the two angles are complementary.

C

B

A

D

F

E

1. ∠AFE and ∠DFE are a linear pair of congruent angles. (Given)

2. Line BE is perpendicular to line AD. (Theorem 3.1)

3. ∠BFC and ∠DFC are complementary. (Theorem 3.2)

UnderstoodConcepts/Theorems/Postulates/Definitions

5 points

Attention to Detail

5 points

Got from the Givens to the Conclusion

5 points

Linear Pair Postulate/Supplementary Angles:

The LP postulate says that if two angles form a linear pair, then they are supplementary. It does not say that their measures add up to 180. For this you need a second step. For the second step, your reason will be “definition of supplementary angles.”

Angle Addition Postulate:

You need this to conclude that the two component angle measures add up to the measure of the whole. It isn't simply “given” by the picture.

Vertical Angles Theorem:

It can be given by a picture that two angles are vertical angles. But the conclusion that they are congruent requires the vertical angles theorem. Also, the theorem says they are congruent, not that their measures are the same. For this, you need a second step (with “definition of congruence” as your reason).

1. Missing m's. Again, there's a difference between an angle and the measure of the angle.

2. Missing Steps: LP postulate, congruence of angles vs. equal measures.

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Corresponding Angles Postulate

Transversal

1

2

3

4

Parallel Lines

5

6

7

8

Corresponding Angles: (4 and 8), (2 and 6), (5 and 1), (7 and 3)

Corresponding Angles Postulate: If two parallel lines are cut a by a transversal, then corresponding angles are congruent.

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Theorems

Transversal

1

2

3

4

Parallel Lines

5

6

7

8

What do you notice about angles 2 and 4?

So what must be true of angles 6 and 4?

Consecutive Interior Angles Theorem: If parallel lines are cut by a transversal, then consecutive interior angles are supplementary.

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Theorems

Transversal

1

2

3

4

Parallel Lines

5

6

7

8

We know that angles 6 and 2 are congruent. What do we know about angles 2 and 3?

So what must be true of angles 6 and 3?

Alternative Interior Angles Theorem: If parallel lines are cut by a transversal, then alternative interior angles are congruent.

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Theorems

Transversal

1

2

3

4

Parallel Lines

5

6

7

8

What do we know about 6 and 8?

So what must be true of angles 4 and 8?

What do we know about 4 and 1?

What do we know about 8 and 1?

Alternative Exterior Angles Theorem: If parallel lines are cut by a transversal, then alternative exterior angles are congruent.

Starting with the Corresponding Angles Postulate, we proved:

1. The Consecutive Interior Angles Theorem

2. The Alternative Interior Angles Theorem

3. The Alternative Exterior Angles Theorem

Suppose we only knew that the Alternative Exterior Angles theorem was true.

Could we prove the Corresponding Angles Postulate?

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Transversal

1

2

3

4

Parallel Lines

5

6

7

8

So we know 1 and 8 are congruent. (Alternative Exterior Angles Theorem)

We know 2 is supplementary to 1, and that 6 is supplementary to 8. (Linear pair postulate)

Therefore, 2 and 6 are congruent. (Congruent Supplements Theorem)

And that's the Corresponding Angles Postulate.

Starting with the Corresponding Angles Postulate, we proved:

1. The Consecutive Interior Angles Theorem

2. The Alternative Interior Angles Theorem

3. The Alternative Exterior Angles Theorem

Suppose we only knew that the Alternative Exterior Angles theorem was true.

Could we prove the Corresponding Angles Postulate?

So, yes, we can prove the Corresponding Angles Postulate from the Alternative Exterior Angles Theorem. What we have is a circle of statements. If we assume one of them (as a postulate), we get all the others for free.

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One More Theorem

Transversal

Parallel Lines

Perpendicular Transversal Theorem: If a transversal is perpendicular to one of two parallel lines, then it it perpendicular to the other parallel line.

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Transversal

f

x

e

d

Parallel Lines

b

c

a

120°

Find x. Explain your reasoning (no proof necessary).