Chapter 7

Parallel Lines

7.1 Transversals and Angles Objective: To identify the angles formed when two lines are cut by a transversal. Transversal: Interior angles: Exterior angles: Alternate Interior Angles: 1) 2) 3) Corresponding Angles: 1) 2) 3)

Note: Remember when naming angles to use the ∠ symbol. Lines are usually names by a lower case letter located by one of the arrows.

Other terms that you may see: Same-side Interior angles:

Same-side Exterior angles:

Vertical angles:

Alternate exterior angles:

Example One: From the list below, choose the correct classification for each pair of angles. Write the letter of the answer in the space provided.

a. alternate interior angles b. same-side interior angles c. alternate exterior angles d. same-side exterior angles e. corresponding angles f. vertical angles g. none of these

1. ∠11, ∠14____________ 2. ∠4, ∠12___________ 3. ∠1, ∠6 ____________

4. ∠1, ∠13 ____________ 5. ∠11, ∠12 _________ 6. ∠6, ∠14___________

7. ∠2, ∠3 _____________ 8. ∠9, ∠16 __________ 9. ∠5, ∠7 ____________

10. ∠7, ∠12 ___________ 11. ∠4, ∠16 __________ 12. ∠10, ∠13 __________

Example Two: Use the figure above to answer the following questions. 13. Name two angles that form alternate interior angles with ∠6: __________and ___________ 14. Name two angles that form corresponding angles with ∠8. ________ and ________ Name two angles that form alternate exterior angles with ∠1. ________and ________ Example three: Draw lines r and s and transversal t. Number exterior angles formed.

7.2 Parallel Lines and Interior Angles Objective: to use relationships between interior angles and parallel lines.

Given d//e

Given: d // c

Postulate, Theorem Name/Term Definition Example Postulate 9 Alternate Interior

Angle If two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent.

Picture A ∠4 and ∠6 ∠3 and ∠5

Both Pairs are Alternate interior angles

If m∠5 is 720, then the

m∠3 is ________.

Theorem 7.1 Same-Side Interior Angles

If two parallel lines are cut by a transversal, then each pair of same-side interior angles are supplementary

Picture A ∠4 and ∠5 ∠3 and ∠6

Both are Same-side interior Angles

If m∠ 3 is 720, then the

m∠6 is ______________

Postulate 10 (converse of Postulate 9)

If two lines are cut by a transversal so that one pair of alternate interior angles are congruent, then the lines are parallel.

Picture B ∠YNM and ∠TYR are Alternate interior angles and are both 68o∴ lines d and c are __________.

Theorem 7.2 (converse of Theorem 7.1)

If two lines are cute by a transversal so that one pair same-side interior angles are supplementary, then the lines are

Picture B ∠SNP and ∠TYR add to be 180o and are same-side interior angles ∴ Lines d and c are _______________.

Example One. Find the measure of each angle. 1. ∠1 ________

2. ∠2 ________

3.∠3 ________

4. ∠4 ________

5. ∠6 ________

6. ∠5 ________

7. ∠10 _______

8. ∠11 _______

9. ∠13 _______

Example Two. Refer to the figure at the right.

If ∠6≅ ∠7, name two lines that must be parallel.

If ∠2 and ∠3 are supplementary, name two lines that must be parallel. Complete: if ∠6 + ________ =180, then j // k. Example Three. Find the measure of each angle. ABCD is a Parallelogram a. ∠DCB __________

b. ∠1 ____________

c. ∠2 ____________

d. ∠3 ____________

e. ∠4 ____________

7.3 Parallel Lines and Corresponding Angles Objective: To use facts about parallel lines cut by a transversal and pairs of corresponding angles. Corollary Theorem 7.3 (Corresponding Angles Theorem): If two parallel lines are cut by a transversal, then each pair of corresponding angles are congruent. Example One

Name the corresponding angle(s) congruent to the given angle. 1. ∠ 10 _________ 2. ∠ 1 __________ 3. ∠15 __________

Theorem 7.4: If two lines are cut by a transversal so that one pair of corresponding angles are congruent, then the lines are parallel. Corollary: If two coplanar lines are perpendicular to a third line, then two

lines are parallel to each other.

Example two Name the line(s) that must be parallel.

7.4 Constructing Parallel Lines Objective: To construct the line parallel to a given line through a point not on a line. Construction 11: Construct the line through P that is parallel to line k. 1 Begin with point P and line k. 2 Draw an random line through

point P, intersecting line k. Call the intersection point Q. Now the goal is to construct an angle with vertex P, congruent to the angle of intersection.

3 Place the compass at point Q and draw an arc intersecting both lines. Without changing the radius of the compass, center it at point P and draw another arc.

4 Set the compass radius to the distance between the two intersection points of the first arc. Now center the compass at the point where the second arc intersects line PQ. Mark the arc intersection point R.

5 Line PR is parallel to line k. Label the lines parallel.

Using only the given information, can you conclude that a // b?

7.5 Interior Angle Measures of a Triangle Objective; To use the sum of the interior angles measures of a triangle. Theorem 7.5 (Triangle Angle-Sum Theory): The sum of the measures of the interior angles of any triangle is 1800. Find the measure of each angle. 1. 2.

3. Corollary 1: The acute angles of any right triangle are complementary. Find the measure of each angle. 4. 5. Corollary 2: The Measure of each interior angle of an equilateral triangle is 600. 6.

Extra Examples: Find the angle of each angle. 7. 8.

7.6 Exterior Angle Measures of a Triangle Objective: To use the relationship between the exterior angle of a triangle and the remote angles. Exterior Angle of a Polygon: **** Each exterior angle has 1 adjacent interior angle and two remote angles.

Theorem 7.6: In a triangle, the measure of each exterior angle is equal to the sum of the measures of its two remote interior angles. Corollary: In a triangle, the measure of each exterior angle is greater than the measure of either of its remote interior angles. Examples: Find the measure of each angle.

1. ________________ 2. ________________ 3. _______________

4. _________________ 6. ______________ 7. _______________

5. _________________ 8. _______________

7.7 Angle Measures of a Polygon Objective: To find the sum of angle measures of any polygon. Theorem 7.7: The sum of the measures of the interior angles of any quadrilateral is 3600. Example one: Find the measure of angle one.

Polygon Sides Figure # of Triangles Formed

Sum of Interior Angle Measures

Triangle

Quadrilateral

Pentagon

Hexagon

Theorem 7.8 (Interior Angle Sum Theorem): The sum of the measures of the interior angles of a polygon with n sides is (n – 2)1800. Example two: The number of sides of a polygon is given. Find the sum of the interior angle measures. 1. 5: ____________ 2. 13: ________________ 3. 7: _______________

The sum of the measures of the interior angles of a polygon is given. Find the number of sides. 4. 1,6200: ____________ 5. 1,8000: ______________ What is the measure of each interior angle of a regular octagon? _____________________ Each interior angle of a regular polygon measures 1790

a. Find the measure of each exterior angle. ________________

b. Find the number of sides in the polygon. ___________ Find the measure of each angle. Theorem 7.9 (Exterior Angle Sum Theorem): The sum of the measures of the exterior angles, one at each vertex, of a polygon is 3600. Example three: Find the measure of each angle.

Each exterior angle of a regular polygon measures 200. Find the number of sides. ___________