1. PARALLEL LINES CUT BY A TRANSVERSAL

2. DEFINITIONS PARALLEL TRANSVERSAL ANGLE VERTICAL ANGLE
CORRESPONDING ANGLE
ALTERNATE INTERIOR ANGLE
ALTERNATE EXTERIOR ANGLE

3. DEFINITIONS
SUPPLEMENTARY ANGLE
COMPLEMENTARY ANGLE
CONGRUENT

4. Parallel lines cut by a transversal2 1 3 4 6 5 7 8

5. Parallel lines cut by a transversal2 1 3 4 6 5 7 8
< 1 and < 2 are called SUPPLEMENTARY ANGLES
DEFINITION:They form a straight angle measuring 180 degrees.

6. Parallel lines cut by a transversal2 1 3 4 6 5 7 8
< 2 and < 3
< 3 and < 4
< 4 and < 1
< 5 and < 6
Name other supplementary pairs:
< 6 and < 7
< 7 and < 8
< 8 and < 5

7. Parallel lines cut by a transversal2 1 3 4 6 5 7 8
< 1 and < 3 are called VERTICAL ANGLES
They are congruent m<1 = m<3
DEFINITION: The angles formed from two lines are crossing.

8. Parallel lines cut by a transversal2 1 3 4 6 5 7 8
< 2 and < 4
< 6 and < 8
Name other vertical pairs:
< 5 and < 7

9. Parallel lines cut by a transversal2 1 3 4 6 5 7 8
< 1 and < 5 are called CORRESPONDING ANGLES
They are congruent m<1 = m<5
DEFINITION: Corresponding angles occupy the same position on the top and bottom parallel lines.

10. Parallel lines cut by a transversal2 1 3 4 6 5 7 8
< 2 and < 6
< 3 and < 7
Name other corresponding pairs:
< 4 and < 8

11. Parallel lines cut by a transversal2 1 3 4 6 5 7 8
< 4 and < 6 are called ALTERNATE INTERIOR ANGLES
They are congruent m<4 = m<6
DEFINITION:Alternate Interior on the inside of the two parallel lines and on opposite sides of the transversal.

12. Parallel lines cut by a transversal2 1 3 4 6 5 7 8
< 3 and < 5
Name other alternate interior pairs:

13. Parallel lines cut by a transversal2 1 3 4 6 5 7 8
< 1 and < 7 are called ALTERNATE EXTERIOR ANGLES
They are congruent m<1 = m<7
Alternate Exterior on the outside of the two parallel lines and on opposite sides of the transversal.

14. Parallel lines cut by a transversal2 1 3 4 6 5 7 8
< 2 and < 8
< 1 and < 7
Name other alternate exterior pairs:

15. Parallel lines cut by a transversal2 1 3 4 6 5 7 8
< 4 and < 5 are called CONSECUTIVE INTERIOR ANGLES
The sum is m<4 = m<5
DEFINITION: Consecutive Interior on the inside of the two parallel lines and on same side of the transversal. Sum = 180

16. TRY IT OUT 2 1 3 4 6 5 7 8 120 degrees The m < 1 is 60 degrees.
What is the m<2 ?
120 degrees

17. TRY IT OUT 2 1 3 4 6 5 7 8 60 degrees The m < 1 is 60 degrees.
What is the m<5 ?
60 degrees

18. TRY IT OUT 2 1 3 4 6 5 7 8 60 degrees The m < 1 is 60 degrees.
What is the m<3 ?
60 degrees

19. TRY IT OUT
120 60 60
120
120 60 60
120

20. TRY IT OUT 2x + 20 x + 10 What do you know about the angles?
Write the equation.
Solve for x.
SUPPLEMENTARY
2x x + 10 = 180
3x + 30 = 180
3x = 150
x = 50

21. TRY IT OUT 3x - 120 2x - 60 What do you know about the angles?
Write the equation.
Solve for x.
ALTERNATE INTERIOR
3x = 2x - 60
x =
Subtract 2x from both sides
Add 120 to both sides

22. Warm Up
Identify each angle pair.
1. 1 and 3
2. 3 and 6
3. 4 and 5
4. 6 and 7
corr. s
alt. int. s
alt. ext. s
same-side int s

23. Example 1: Using the Corresponding Angles Postulate
Find each angle measure.
A. mECF
x = 70
Corr. s Post.
mECF = 70°
B. mDCE
5x = 4x + 22
Corr. s Post.
x = 22
Subtract 4x from both sides.
mDCE = 5x
= 5(22)
Substitute 22 for x.
= 110°

24. Check It Out! Example 1
Find mQRS.
x = 118
Corr. s Post.
mQRS + x = 180°
Def. of Linear Pair
mQRS = 180° – x
Subtract x from both sides.
= 180° – 118°
Substitute 118° for x.
= 62°

25. WEBSITES FOR PRACTICE Ask Dr. Math: Corresponding /Alternate Angles
Project Interactive: Parallel Lines cut by Transversal

26. Triangle Sum Theorem
The sum of the angle measures in a triangle equal 180° 3 2 1
m<1 + m<2 + m<3 = 180°

27. Corollary
A corollary to a theorem is a statement that follows directly from that theorem

28. TRIANGLE ANGLE SUM THEOREM COROLLARIES
If 2 angles of 1 triangle are congruent to 2 angles of another triangle, then the 3rd angles are congruent
The acute angles of a right triangle are complementary
The measure of each angle of an equiangular triangle is 60o
A triangle can have at most 1 right or 1 obtuse angle

29. Exterior Angle Theorem (your new best friend)
The measure of an exterior angle in a triangle is the sum of the measures of the 2 remote interior angles
exterior angle
remote interior
angles 3 2 1 4
m<4 = m<1 + m<2

30. REMOTE INTERIOR ANGLE
In any polygon, a remote interior angle is an interior angle that is not adjacent to a given exterior angle A and B are remote to angle 1

31. Exterior Angle Theorem
interior
In the triangle below, recall that 1, 2, and 3 are _______ angles of ΔPQR.
Angle 4 is called an _______ angle of ΔPQR.
exterior
An exterior angle of a triangle is an angle that forms a _________ with one of
the angles of the triangle.
linear pair
In ΔPQR, 4 is an exterior angle at R because it forms a linear pair with 3.
____________________ of a triangle are the two angles that do not form
a linear pair with the exterior angle.
Remote interior angles
In ΔPQR, 1, and 2 are the remote interior angles with respect to 4. 1 2 3 4 P Q R

32. Exterior Angle Theorem
In the figure below, 2 and 3 are remote interior angles with respect to
what angle? 5 1 2 3 4 5

33. an example with numbers
find x & y
82°
x = 68°
y = 112°
30° x y
Determine the measure of <4,
If <3 = 50, <2 = 70
40x
10x2
30x
find all the angle measures
Do you hear the sirens?????
80°, 60°, 40°

34. Exterior Angle Theorem

35. Example 3: Applying the Exterior Angle Theorem
Find mB.
mA + mB = mBCD
Ext.  Thm.
Substitute 15 for mA, 2x + 3 for mB, and 5x – 60 for mBCD.
15 + 2x + 3 = 5x – 60
2x + 18 = 5x – 60
Simplify.
Subtract 2x and add 60 to both sides.
78 = 3x
26 = x
Divide by 3.
mB = 2x + 3 = 2(26) + 3 = 55°

37. There are several ways to prove certain triangles are similar
There are several ways to prove certain triangles are similar. The following postulate, as well as the SSS and SAS Similarity Theorems, will be used in proofs just as SSS, SAS, ASA, HL, and AAS were used to prove triangles congruent.

38. Example 1: Using the AA Similarity Postulate
Explain why the triangles are similar and write a similarity statement.
Since , B  E by the Alternate Interior Angles Theorem. Also, A  D by the Right Angle Congruence Theorem. Therefore ∆ABC ~ ∆DEC by AA~.

39. Check It Out! Example 1
Explain why the triangles
are similar and write a
similarity statement.
By the Triangle Sum Theorem, mC = 47°, so C  F. B  E by the Right Angle Congruence Theorem.
Therefore, ∆ABC ~ ∆DEF by AA ~.