3-1 and 3-2: Parallel Lines and Transversals

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3-1 and 3-2: Parallel Lines and Transversals. Mr. Schaab’s Geometry Class Our Lady of Providence Jr.- Sr . High School 2014-2015. Identifying Pairs of Lines. Two lines are: Parallel if they do not intersect and are coplanar . Skew if they do not intersect and are not coplanar .

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### 3-1 and 3-2: Parallel Lines and Transversals

Mr. Schaab’s Geometry Class

Our Lady of Providence Jr.-Sr. High School

2014-2015

Identifying Pairs of Lines
• Two lines are:
• Parallel if they do not intersect and arecoplanar.
• Skew if they do not intersect and are not coplanar.
• Perpendicular if they intersect to form right angles.

(Note: ALL intersecting

lines are coplanar!)

Identifying Pairs of Lines - Example

In the cube below, identify the following:

• A pair of perpendicular lines:
• Line PS and Line PW
• A pair of parallel lines:
• Line PW and Line QX
• Line PW and Line RY
• Line QX and Line RY
• A pair of skew lines:
• Line PS and Line QX
• Line PS and Line RY
Identifying Pairs of Planes - Example

In the cube below, identify the following:

• A pair of perpendicular planes:
• Plane SRY and Plane PWZ
• Plane PQX and Plane QXY
• A pair of Parallel planes:
• Plane PQX and Plane SRY
• Plane PWZ and Plane QRY
• Plane PQR and WXY
• Pair of skew planes:
• None!
Angles and Transversals
• Transversal – a line that intersects two or more coplanarlines at different points.

In the diagram on the right, line tis a the transversal of lines L1andL2.

A transversal that intersects two lines forms 8 angles, all of which have special relationships.

Angle Relationships
• Corresponding Angles
• Two angles that are in matching locations on different intersections.
• ∠1 and ∠5 are corresponding angles.

1

5

Angle Relationships
• Alternate Interior Angles
• Two angles that lie between the two lines and on opposite sides of the transversal.
• ∠4 and ∠5 are alternate interior angles.

4

5

Angle Relationships
• Alternate Exterior Angles
• Two angles that lie outside the two lines and on opposite sides of the transversal.
• ∠2 and ∠7 are alternate interior angles.

2

7

Angle Relationships
• Consecutive Interior Angles
• Two angles that lie between the two lines and on the same side of the transversal. These are also called “Same-side interior angles.”
• ∠3 and ∠5 are consecutive interior angles.

3

5

Angles and Transversals - Example

Identify all pairs of angles of the given type:

• Corresponding:
• ∠1 & ∠5, ∠2 & ∠6, ∠3 & ∠7, ∠4 & ∠8
• Alternate Interior:
• ∠2 & ∠7, ∠4 & ∠5
• Alternate Exterior:
• ∠3 & ∠6, ∠1 & ∠8
• Consecutive Interior:
• ∠4 & ∠7, ∠2 & ∠5

6

5

8

7

2

1

4

3

Parallel Lines and Transversals
• Corresponding Angles Postulate:
• If two parallel lines are cut by a transversal, then all pairs of corresponding angles are congruent.
• ∠1 ≅ ∠5

1

5

Parallel Lines and Transversals
• Alternate Interior Angles Theorem:
• If two parallel lines are cut by a transversal, then all pairs of alternate interior angles are congruent.
• ∠4 ≅ ∠5

4

5

Parallel Lines and Transversals
• Alternate Exterior Angles Theorem:
• If two parallel lines are cut by a transversal, then all pairs of alternate exterior angles are congruent.
• ∠1 ≅ ∠8

1

8

Parallel Lines and Transversals
• Consecutive Interior Angles Theorem:
• If two parallel lines are cut by a transversal, then all pairs of consecutive interior angles are supplementary.
• m∠3 + m∠5 = 180°

3

5

Parallel Lines and Transversals
• If you’re angle 3, then you have a lot of relationships! The other angles must really like you.
• ∠3& ∠1 – Linear Pair (supplementary)
• ∠3& ∠2 – Vertical Angles (congruent)
• ∠3& ∠4 – Linear Pair (supplementary)
• ∠3& ∠5 – Consecutive Interior Angles (supplementary)
• ∠3& ∠6 – Alternate Interior Angles (congruent)
• ∠3& ∠7 – Corresponding Angles (congruent)
• ∠3& ∠8 – No relationship (∠8 is a jerk.)

1

2

3

4

5

6

7

8