Geometry Chapter 3: Parallel and Perpendicular Lines 3-1: Lines and Angles Define each term and give an example of each one using the figure: Parallel lines

Perpendicular lines

Skew lines

Parallel planes

1. Use the diagram to identify each of the following. A. a pair of parallel segments

B. a pair of skew segments

C. a pair of perpendicular segments

D. a pair of parallel planes

2. Identify each of the following: a) A pair of parallel segments: b) A pair of skew segments: c) A pair of perpendicular segments: d) A pair of parallel planes:

Name __________________

Define: Transversal

Corresponding angles

Alternate interior angles

Alternate exterior angles

Same-side interior angles

3. Using the diagram, answer the following questions: A. Three pairs of corresponding angles.

B. Two pairs of alternate interior angles.

C. Two pairs of alternate exterior angles.

D. Two pairs of same-side interior angles.

E. Two pairs of same-side exterior angles. 4.

Identifying Angle Pairs and Transversals Helpful Hint: To determine which line is the ____________ for a given angle pair, locate the ________that connects the ___________ of the angles.

5. Use the diagram to identify the transversal and classify each angle pair. A. ∠1 and ∠3 Transversal: Angle pair: B. ∠2 and ∠6 Transversal: Angle pair: C. ∠4 and ∠6 Transversal: Angle pair:

6. Identify each of the following. a. a pair of parallel segments

b. a pair of skew segments

c. a pair of perpendicular segments

d. a pair of parallel planes

3-2: Angles Formed by Parallel Lines and Transversals

1. Find each angle measure. A. m∠ECF

B. m∠DCE

2. Find m∠QRS

Helpful Hint: If a transversal is ________________to two parallel lines, all eight angles are __________________.

Remember that postulates are statements that are accepted without proof. Since the Corresponding Angles Postulate is given as a postulate, it can be used to prove the next three theorems.

3. Find each angle measure A. m ∠EDG

B. m ∠BDG

4. Find x and y in the diagram.

x=________ y=________

3-3: Proving Lines Parallel

Ex 1: Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m. ∠4 ≅ ∠8

Ex 2: Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m. m∠3 = (4x – 80) m∠7 = (3x – 50), x = 30.

Ex 3: Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m. m∠7 = (4x + 25)°, m∠5 = (5x + 12)°, x = 13

Determining whether lines are parallel Ex 4: Use the given information and the theorems you know/learned to show that r ll s. ∠4 ≅ ∠8

Ex 5: Use the given information and the theorems you know/learned to show that r ll s. m∠2 = (10x + 8) ˚ m∠3 = (25x – 3)˚; x = 5

Ex 6: Proving Lines Parallel Given: p || r , ∠1 ≅ ∠3 Prove: ℓ || m

Ex 7: What if…? Suppose the corresponding angles on the opposite side of the boat measure (4y – 2)° and (3y + 6)°, where y = 8. Show that the oars are parallel.

3-4: Perpendicular Lines Define Perpendicular Bisector:

Distance from a point to a line:

Ex 1: A. Name the shortest segment from point A to BC .

B. Write and solve an inequality for x.

Ex 2: A. Name the shortest segment from point A to BC .

B. Write and solve an inequality for x.

x

3-5 Slopes of Lines Define Slope:

Four types of slope:

Ex 1: Find the slope of AB

Ex 2: Find the slope of AC

Ex 3: Find the slope of AD

Ex 4: Find the slope of CD

Ex 5: Use the slope formula to determine the slope of JK through J(3,1) and K(2, -1)

Ex 6: Justin is driving from home to his college dormitory. At 4:00 p.m., he is 260 miles from home. At 7:00 p.m., he is 455 miles from home. Graph the line that represents Justin’s distance from home at a given time. Find and interpret the slope of the line.

Opposite reciprocals:

Ex 7: Graph each pair of lines. Use their slopes to determine whether they are parallel, perpendicular, or neither. UV and XY for U(0, 2), V(–1, –1), X(3, 1), and Y(–3, 3)

Ex 8: Graph each pair of lines. Use their slopes to determine whether they are parallel, perpendicular, or neither. GH and IJ for G(–3, –2), H(1, 2), I(–2, 4), and J(2, –4)

Ex 9: Graph each pair of lines. Use their slopes to determine whether they are parallel, perpendicular, or neither CD and EF for C(–1, –3), D(1, 1), E(–1, 1), and F(0, 3)

Ex 10: Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular or neither. WX and YZ W(3, 1), X(3, -2), Y(-2, 3) and Z(4, 3).

3-6: Lines in the Coordinate Plane Point-slope form:

Slope-intercept form:

Vertical line:

Horizontal line:

1. Write an equation of line with slope of six and through (3, -4) in point-slope form.

2. Write the equation of the line in slope intercept form through (-1, 0) and (1, 2)

3. Write the equation in point-slope form with a x-intercept of 3 and y-intercept of -5.

4. Write the equation in point slope form through (-3, 2) and (1, 2).

Graph the following: 1 5. y = 2 x + 1

6. y – 3 = -2(x + 4)

A system of two linear equations in two variables represents two lines. The lines can be parallel, intersecting, or coinciding. Lines that coincide are the same line, but the equations may be written in different forms.

7. Determine whether the lines are parallel, intersecting or coinciding. y = 3x + 7, y = -3x – 4

8. Determine whether the lines are parallel, intersecting or coinciding. 1 y =- 3 x + 5, 6y = -2x + 12

9. Determine whether the lines are parallel, intersecting or coinciding. 2y – 4x = 16, y – 10 = 2(x - 1)