NAME: _______________________________ ANGLES FORMED BY TRANSVERSALS INVESTIGATION PARALLEL LINES Step 1:

Construct a pair of parallel lines on a piece of patty paper. Draw a third line called a transversal that intersects the parallel lines and is not perpendicular to the parallel lines. A transversal is a line that intersects two or more coplanar lines at different points. Label the angles as shown.

t

1

5 2

6

3

7 4

8

Step 2:

Use a second piece of patty paper and trace ∠1 with a ruler.

Step 3:

Take the piece of patty paper with the traced ∠1 on it and place it over each of the other angles.

What angles have the same measure as ∠1?

________________________________

1

Step 4:

What angle is congruent with ∠1 after a translation of ∠1?

________________

Step 5:

What angle is congruent with ∠1 and ∠3 after a 180˚ rotation of ∠1? _______________

Step 6:

What angle is congruent with ∠1, ∠3, and ∠8 after a translation of ∠1? _____________

Step 7:

From the current position of ∠1, what transformation would you need to perform to get ∠1 back to its original position? ______________________________________________________________________

Step 8:

Using a third piece of patty paper, trace ∠5 with a ruler.

Step 9:

Take the piece of patty paper with the traced ∠5 on it and place it over each of the other angles.

What angles have the same measure as ∠5?

________________________________

Step 10:

What angle is congruent with ∠5 after a translation of ∠5?

________________

Step 11:

What angle is congruent with ∠5 and ∠7 after a 180˚ rotation of ∠5? ________________

Step 12:

What angle is congruent with ∠5, ∠7, and ∠4 after a translation of ∠5? _____________

Step 13:

From the current position of ∠5, what transformation would you need to perform to get ∠5 back to its original position? ______________________________________________________________________

2

MEASURES OF ANGLES WHEN TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL

t

1

5 2

6

3

7 4

Identify the Corresponding Angles:

8

________________________________ ________________________________ ________________________________ ________________________________

What do you notice about the pairs of corresponding angles? ______________________________________________________________________

Identify the Alternate Interior Angles:

________________________________ ________________________________

What do you notice about the pairs of alternate interior angles? ______________________________________________________________________

3

MEASURES OF ANGLES WHEN TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL

t

1

5 2

6

3

7 4

Identify the Alternate Exterior Angles:

8

________________________________ ________________________________

What do you notice about the pairs of alternate exterior angles? ______________________________________________________________________

Identify the Consecutive Interior Angles:

________________________________ ________________________________

Take a protractor and measure each pair of the consecutive interior angles.

What do you notice about the sum of the angles of each pair of consecutive interior angles? ______________________________________________________________________

4

The measure of angle 1 is 125˚. Find the measures of the other angles WITHOUT USING A PROTRACTOR! Use the angle relationships you just discovered.

t

125° 4

1 3

2

5

8 7

6

m∠2 ____________ m∠3 ____________ m∠4 ____________ m∠5 ____________ m∠6 ____________ m∠7 ____________ m∠8 ____________

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POSTULATES, THEOREMS, AND DEFINITIONS 1.)

Postulate (or axiom): In Geometry, a rule that is accepted without proof.

2.)

Theorem: In Geometry, a rule that can be proved.

3.)

Parallel Lines: Two lines are parallel lines if they do not intersect and are coplanar.

4.)

Transversals: A transversal is a line that intersects two or more coplanar lines at different points.

5.)

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

6.)

Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.

7.)

Alternate Exterior Angles Theorem: If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.

8.)

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

9.)

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

10.)

Alternate Interior Angles Converse Theorem: If two lines are cut by a transversal so the alternate interior angles are congruent, then the lines are parallel.

11.)

Alternate Exterior Angles Converse Theorem: If two lines are cut by a transversal so the alternate exterior angles are congruent, then the lines are parallel.

12.)

Consecutive Interior Angles Converse Theorem: If two lines are cut by a transversal so the consecutive interior angles are supplementary, then the lines are parallel. 6