1. Parallel and Perpendicular Lines
Geometry – Chapter 3

2. Chapter 3 Standards 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.

3. 3.1 Properties of Parallel Lines EQ: Identify different types of angles formed by parallel lines.
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.

4. 3.1 Properties of Parallel Lines EQ: 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.

5. 3.1 Properties of Parallel Lines EQ: 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

6. 3.1 Properties of Parallel Lines EQ: 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

7. 3.1 Properties of Parallel Lines EQ: Identify different types of angles formed by parallel lines.

8. 3.1 Properties of Parallel Lines EQ: Identify different types of angles formed by parallel lines.
Homework: page 132 (1-16) all

9. 3.2Proving Lines Parallel EQ: Prove the converses to the theorems of section 3.1
What is the Corresponding Angles Postulate?
What is the converse to this?

10. 3.2Proving Lines Parallel EQ: 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?”

11. 3.2Proving Lines Parallel EQ: 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:

12. 3.2Proving Lines Parallel EQ: 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:

13. 3.2Proving Lines Parallel EQ: 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:

14. 3.2Proving Lines Parallel EQ: 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:

15. 3-3 Parallel and Perpendicular Lines EQ: Use previously proven theorems to prove theorems about parallel and perpendicular lines.

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

17. Warm Up In this drawing, line k is parallel to line j
Which angle is alternate interior with ∠4?
Which angle is corresponding to ∠8?
m∠3 = 37. What is m∠6?
m∠1 = x+12 and m∠5 = 3x – 36. What is x?
Given that k∥j, write a proof to show that ∠2 and ∠5 are supplementary.

18. 3-4 The Triangle Angle Sum Theorem EQ: Determine the measures of interior and exterior angles of a triangle.

19. 3-4 The Triangle Angle Sum Theorem EQ: Determine the measures of interior and exterior angles of a triangle.

20. 3-4 The Triangle Angle Sum Theorem EQ: Determine the measures of interior and exterior angles of a triangle.

21. 3-4 The Triangle Angle Sum Theorem EQ: Determine the measures of interior and exterior angles of a triangle.

22. 3-4 The Polygon Angle Sum Theorem EQ: Determine the measures of interior and exterior angles of a polygon.

23. 3-4 The Polygon Angle Sum Theorem EQ: Determine the measures of interior and exterior angles of a polygon.

24. 3-4 The Polygon Angle Sum Theorem EQ: Determine the measures of interior and exterior angles of a polygon.

25. 3-4 The Polygon Angle Sum Theorem EQ: Determine the measures of interior and exterior angles of a polygon.

26. 3-4 The Polygon Angle Sum Theorem EQ: Determine the measures of interior and exterior angles of a polygon.
homework:
page 150 (1-6, 10-20) all
page 161 (1-21) all