Parallel and Perpendicular Lines

Geometry – Chapter 3

2.0 Students write geometric proofs 4.0 Students prove basic theorems involving

congruence 7.0 Students prove and use theorems

involving the properties of parallel lines cut by a transversal

12.0 Students find and use measures of interior and exterior angles of triangles and polygons to classify figures and solve problems.

Chapter 3 Standards

a transversal is a line that intersects two coplanar lines at two distinct points.

the diagram shows the eight angles formed by a transversal t and two lines, l and m.

3.1 Properties of Parallel LinesEQ: Identify different types of angles formed by parallel lines.

There are special names for certain angles in a 2 line and transversal relationship◦ alternate interior angles◦ same-side interior angles◦ corresponding angles◦ alternate exterior angles◦ same-side exterior angles

Draw a transversal using a ruler through your not-parallel lines. Discuss which angles you think are which and why.

3.1 Properties of Parallel LinesEQ: Identify different types of angles formed by parallel lines.

◦ alternate interior angles

◦ same-side interior angles

◦ corresponding angles

◦ alternate exterior angles

◦ same-side exterior angles

3.1 Properties of Parallel LinesEQ: Identify different types of angles formed by parallel lines.

Draw a transversal using a ruler through your parallel lines. Use a protractor to measure all of the angles. Discuss and draw conclusions about angle relationships when the two lines are parallel.◦ alternate interior angles◦ same-side interior angles◦ corresponding angles◦ alternate exterior angles◦ same-side exterior angles

3.1 Properties of Parallel LinesEQ: Identify different types of angles formed by parallel lines.

3.1 Properties of Parallel LinesEQ: Identify different types of angles formed by parallel lines.

Homework: page 132 (1-16) all

3.1 Properties of Parallel LinesEQ: Identify different types of angles formed by parallel lines.

What is the Corresponding Angles Postulate?

What is the converse to this?

3.2Proving Lines ParallelEQ: Prove the converses to the theorems of section 3.1

Everything goes back to either the Corresponding Angles Theorem or the Converse of the Corresponding Angles Theorem.

When you begin a proof involving parallel lines, you should ask yourself “How do I show that corresponding angles are congruent?”

3.2Proving Lines ParallelEQ: Prove the converses to the theorems of section 3.1

What is the converse to the Alternate Interior Angles Theorem?

If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel.

Proof:

3.2Proving Lines ParallelEQ: Prove the converses to the theorems of section 3.1

What is the converse to the Same-side Interior Angles Theorem?

If two lines and a transversal form same-side interior angles that are supplementary, then the two lines are parallel.

Proof:

3.2Proving Lines ParallelEQ: Prove the converses to the theorems of section 3.1

State the Converse to the Alternate Exterior Angle Theorem

If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel.

Proof:

3.2Proving Lines ParallelEQ: Prove the converses to the theorems of section 3.1

State the Converse to the Same-side Exterior Angle Theorem

If two lines and a transversal form same-side interior angles that are supplementary, then the two lines are parallel.

Proof:

3.2Proving Lines ParallelEQ: Prove the converses to the theorems of section 3.1

3-3 Parallel and Perpendicular LinesEQ: Use previously proven theorems to prove theorems about parallel and perpendicular lines.

Homework: page 137 (1-21) all page 143 (1-3)

In this drawing, line k is parallel to line j

1. Which angle is alternate interior with ∠4?2. Which angle is corresponding to ∠8?3. m∠3 = 37. What is m∠6?4. m∠1 = x+12 and m∠5 = 3x – 36. What is x?5. Given that k∥j, write a proof to show that ∠2

and ∠5 are supplementary.

Warm Up

3-4 The Triangle Angle Sum TheoremEQ: Determine the measures of interior and exterior angles of a triangle.

3-4 The Triangle Angle Sum TheoremEQ: Determine the measures of interior and exterior angles of a triangle.

3-4 The Triangle Angle Sum TheoremEQ: Determine the measures of interior and exterior angles of a triangle.

3-4 The Triangle Angle Sum TheoremEQ: Determine the measures of interior and exterior angles of a triangle.

3-4 The Polygon Angle Sum TheoremEQ: Determine the measures of interior and exterior angles of a polygon.

3-4 The Polygon Angle Sum TheoremEQ: Determine the measures of interior and exterior angles of a polygon.

3-4 The Polygon Angle Sum TheoremEQ: Determine the measures of interior and exterior angles of a polygon.

3-4 The Polygon Angle Sum TheoremEQ: Determine the measures of interior and exterior angles of a polygon.

3-4 The Polygon Angle Sum TheoremEQ: Determine the measures of interior and exterior angles of a polygon.

homework:

page 150 (1-6, 10-20) all page 161 (1-21) all