Geometry

Unit II3.3 Part 2

Proofs Involving Parallel Lines

We will do proofs in two columns. In the left-hand column, we will write statements which will lead from

the given information (the given information is always listed as the first statement) down to what we

need to prove (what we need to prove will always be the last statement). In the right-hand column, we

must give a reason why each statement is true. The reason for the first statement will always be

____________, and the reason for each of the other statements must be a __________________,

_________________ or ________________.

given postulate

Theorem definition

Definitions:

Vertical angles Linear Pairs Corresponding Angles Alternate interior angles Alternate exterior angles Same-side interior angles

Use these definitions when you make a statement that a pair of angles in the figure is one of these special angle pairs.

Linear pairs

Angle bisector: An angles bisector is _______________________________________________________

____________________________________

Supplementary angles: Two angles are supplementary if ______________________________________

A ray that divides an angle into congruent angles

add up to 180 degrees

Postulates:

Corresponding Angles Postulate (CAP): If _____________________________ are cut by a transversal,

then ________________________________________

Linear Pair Postulate (LPP): If two angles form a linear pair, then ________________________________

*Substitution: _________________________________________________________________________

Two parallel linesThe pairs of CORRSP. angles are congruent

= 180 degreesReplacing one quantity with an equivalent one.

Theorems:

Vertical Angle Theorem (VAT): If two angles are vertical angles, then ____________________________

Alternate Interior Angle Theorem (AIAT): If two parallel lines are cut by a transversal, then ___________

______________________________________________

Alternate Exterior Angle Theorem (AEAT): If two parallel lines are cut by a transversal, then __________

______________________________________________

Same-side Interior Angle Theorem (SSIAT): If two parallel lines are cut by a transversal, then _________

___________________________________________________

They are congruent

AIA are congruent

AEA are congruent

SSIA are supplementary.

Given: dbca //,// Prove: 111

b

c

d

9 10 11 12

8 a 1 2

3 4 5 6

7

1. a II c, b II d Given2.

Are corresponding angle Def. of corrs. Angles3. _______ CAP4. _____ Def alt. ext angles

Alternative Exterior Angles5. ________ AEAT

6. _________ Substitution

51 and

51 115 and

111

115

Given: dbca //,// Prove: 1 and 10 are supplementary

b

c

d

9 10 11 12

8 a 1 2

3 4 5 6

7

1. a II c, b II d Given

2. Are same side interior angles Def. same side INT Angle 10&7

3. Are supplementary SSIAT10&7

4. Def. of supplementary angles 180107 mm

7&1

71

180101 mm

10&1

5. Are Alt. ext. angles Def. of Alt. Ext. Angles

6. AEAT

7. Substitution

8. Are supplementary Def. of supplementary

Given: LDBE // , BE bisects ABD Prove: DABE

L

B

A

E

D

1. BE II LD, BE bisects angle ABD Given

2. Def of Ang. Bisector EBDABE

3. Are Alt. Int. Angle Def. Alt. IntDEBD &

4. AIATDEBD

5. Substitution DABE