Geometry Honors - Chapter 3 – Parallel and Perpendicular LinesSection 1 – Parallel Lines and Transversals

I can precisely define line segments, rays, parallel lines, perpendicular lines, and skew lines and describe their characteristics.

I can identify and name angle pairs formed by parallel lines and transversals (corresponding, alternate interior, alternate exterior, and consecutive interior).

Parallel Lines – coplanar lines that do not intersect or touch. Skew Lines – are lines that do no intersect and are non-coplanar (means they are not parallel or intersecting). Parallel Planes – planes that do not intersect or touch.

1. Identify each of the following using the box below:

(a) all segments parallel to BC (b) a segment skew to EH (c) a plane parallel to plane ABG

Transversal – a line that cuts or intersects 2 or more coplanar lines at 2 different points (ex: line t)

Homework – Page 174 – 176 (13-19ODD, 21-43, 50, 51)

Section 2 – Angles and Parallel Lines I can prove and apply theorems about the angles formed by parallel lines and a transversal

(corresponding, alternate interior, alternate exterior, and consecutive interior).

Corresponding angles postulate – If 2 parallel lines are cut by a transversal, then corresponding angles are congruent. (ex: ∠1 ≅ ∠3, ∠8 ≅ ∠6; ∠2 ≅ ∠4, ∠7 ≅ ∠5)

Alternate Interior angles theorem – If 2 parallel lines are cut by a transversal, then alternate interior angles are congruent. (ex: ∠2 ≅ ∠6, ∠3 ≅ ∠7)

Alternate Exterior angles theorem – If 2 parallel lines are cut by a transversal, then alternate exterior angles are congruent. (ex: ∠1 ≅ ∠5, ∠4 ≅ ∠8)

Consecutive Interior angles theorem – If 2 parallel lines are cut by a transversal, then consecutive interior angles are supplementary (2 angles that add to 180.)(ex: m∠2 + m∠3 = 180, m∠7 + m∠6 = 180)

1. In the figure, m∠11 = 51. Find the measure of each angle. Tell which postulates or theorems you used.

(a) m∠15 (b) m ∠16

2. The following diagram represents floor tiles in Michelle’s house. If m∠2 = 125, find m∠3.

3. Use the figure below to find the indicated variable. Explain your reasoning.

(a) If m ∠5 = 2x – 10 and m ∠7 = x + 15, find x. (b) Find y, if m∠6 = 4(y – 25) and m ∠8 = 4y.

Perpendicular Transversal Theorem – in a plane, if a line is perpendicular to 1 of 2 parallel lines, then it is perpendicular to the other.

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Homework – Page 181 – 183 (11 – 30, 38, 39,42, 43, 45)

Section 5 – Proving Lines Parallel I can prove and apply theorems about the angles formed by parallel lines and a transversal

(corresponding, alternate interior, alternate exterior, and consecutive interior).

Converse of Corresponding angles postulate – if 2 lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel.(ex: If ∠1 ≅ ∠3, then a∥b; If ∠8≅ ∠6, then a∥b; If ∠ 2≅ ∠4, then a∥b; If ∠7 ≅ ∠ 5, then a∥b)

Alternate Exterior angles converse – if 2 lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel(ex: If∠ 1 ≅ ∠5, then a∥ b OR ∠8 ≅ ∠4, then a∥b)

Alternate Interior angles converse – if 2 lines are cut by a transversal so that alternate interior angles are congruent, then the liens are parallel (ex: If∠ 2 ≅∠6, then a∥b OR ∠7 ≅ ∠3, then a∥b)

Consecutive Interior angles converse - if 2 lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel.(ex: m∠2 + m∠3 = 180, then a∥b OR m∠ 7 + m∠6 = 180, then a∥b)

Perpendicular Transversal converse – if 2 lines are perpendicular to the same line, then the 2 lines are parallel.

1. Given the following information, is it possible to prove that any of the lines shown are parallel? If so, state the postulate or theorem that justifies your answer.

(a) ∠1 ≅ ∠3 (b) m∠1 = 103 and m∠4 = 100

#1 Guided practice: Given the following information, is it possible to prove that any of the lines shown are parallel? If so, state the postulate or theorem that justifies your answer:

(a) ∠2 ≅ ∠8

(b)

∠

3

≅

∠

11

(c) ∠12 ≅ ∠14

(d) ∠1 ≅ ∠15

(e) m∠8 + m∠13 = 180

(f) ∠8 ≅ ∠ 6

2. (a) Find m ∠ZYN so that PQ∥ ´MN . Show your work.

(b) Find y so that e∥f. Show your work.

3. In order to move in a straight line with maximum efficiency, rower’s oars should be parallel. Refer to the photo at the right. Is it possible to prove that any of the oars are parallel? If so, explain how. If not, explain why not.

FIREWORKS A fireworks display is being readied for a celebration. The designers want to have four fireworks shoot out along parallel trajectories. They decide to place two launchers on a dock and the other two on the roof of a building. To pull off this display, what should the measure of angle 1 be?

Homework – Page 209 - 211 (8 – 21, 23, 33 – 35, 37,42) ALLSection 3 – Slopes of Lines

I can find slopes of lines. I can use slope to prove lines are parallel or perpendicular.

1. Find the slope of each:(a) (6, -2) and (-3, -5) (b) (4,2) and (4, -3)

(c) (8, -3) and (-6, -2) (d) (-3, 3) and (4, 3)

Find the slope of each line.

Slope: also known as a rate of change, the change in y over the change in x.

2. In 2000, the annual sales for one manufacturer of camping equipment was $48.9 million. In 2005, the total sales were $85.9 million. If sales increase at the same rate, what will be the total sales in 2015?

3. After Take Two began renting DVDs at their video store, business soared. Between 2005 and 2010, profits increased at an average rate of $9000 per year. Total profits in 2010 were $45,000. If profits continue to increase at the same rate, what will the total profit be in 2014?

Parallel Lines – If lines are parallel, they have the same or equal slopes. (ex: if a∥b, slope a = slope b) If ma = ½ then mb = ½

Perpendicular Lines – if lines are perpendicular, then they have slopes with a product of -1. (ex: if a⊥b, slope a *slope b = -1) If ma = ½ , the mb = -2

4.

5. Graph the line that satisfies each condition

Homework – Page 191 – 193 (15-33ODD, 34-39ALL, 41, 48, 50, 52, 53, 55)Section 4 – Equations of Lines

I can find the equation of a line parallel or perpendicular to a given line that passes through a given point.

Slope – Intercept Form : y = mx + b, where ‘m’ is the slope and ‘b’ is the y-intercept (where the line crosses the y-axis)

(ex: m = 3 and b = -2 so y = 3x – 2)

Point- Slope Form : y – y1 = m(x – x1), where ‘m’ is the slope and (x1, y1) is a point on the line.(ex: m = ½ and (3, 5) is a point on the line so y – 5 = ½ (x – 3))

1. Write an equation in slope-intercept form given a slope and a y-intercept. Then graph the line.(a) m = 6 and y-intercept = -3 (b) m = ½ and b = 8

2. Write an equation in point-slope form given a slope and a point on the line. Then graph the line.(a) m = -3/5 and contains (-10, 8) (b) m = 4 and contains (-3, -6)

3. Write an equation of a line through each pair of points in slope-intercept form:(a) (4, 9) and (-2, 0)

(b) (-3, -7) and ((-1, 3)

Horizontal & Vertical Line Equations

The equation of a horizontal line is y = b where b is the y-intercept. All horizontal lines have points that use the same y-coordinate Horizontal lines have a slope of zero.

The equation of a vertical line is x = a where a is the x-intercept. All vertical lines have points that use the same x-coordinate. All vertical lines have a slope that is undefined.

What is the equation of a line that has a slope of 0 and passes through (-3,8)? Then graph the line.

What is the equation of a line that has an undefined slope and passes through (-1,4)? Then graph the line.

Key Things to Remember:*m = 0 is perpendicular to m = undefined

*m = 0 is a horizontal line and the equation is y = b (it never crosses the x-axis)*m = undefined is a vertical line and the equation is x = a (it never crosses the y-axis)

Example:-Write an equation in slope-intercept form of (3,2) and (1,2)

-Write and equation in slope-intercept form of (-2, 6) and (-2, -1)

4. Write an equation of the line through (5, -2) and (0, -2) in slope-intercept form.

5. Write an equation in slope-intercept form for a line perpendicular to the line y = 1/5 x + 2 through (2,0)

6. Write an equation in slope-intercept form for a line parallel to the line y = -3/4 x + 3 and containing (-3, 6)

7. An apartment complex charges $525 per month plus a $750 annual maintenance fee.(a) Write an equation to represent the total first year’s cost ‘A’ for ‘r’ months of rent.

(b) Compare this rental cost to a complex with no annual maintenance fee but $600 per month for rent. If a person expects to stay in an apartment for one year, which complex offers the better rate?

Homework – Page 200-202 (13-29ODD, 32, 38, 40, 41, 46, 49, 51, 52, 56, 58)Section 6 – Perpendiculars and Distance

I can find the distance between a point and a line and between parallel lines.

Distance between a point and a line – the distance between a line and a point not on the line is the length of the segment perpendicular to the line from the point.

Perpendicular Postulate – if given a line and a point not on the line, then there exists exactly 1 line through the point that is perpendicular to the line.

Example:

1. Construct and name the segment that represents the distance from Q to PR(a) distance from Q to PR (b) distance from Y to TS

(c) distance from C to AB

2. (a) Line s contains points at (0,0) and (-5,5). Find the distance between line s and point V(1,5).

2. (b) Line l contains points at (1,2) and (5,4). Construct a line perpendicular to l through P(1,7). Then find the distance from P to l.

Distance Between Parallel Lines The distance between parallel

lines is the length of a segment that has an endpoint on each line and is perpendicular to them. Parallel lines are everywhere equidistant, which means that all such perpendicular segments have the same length.

Find the distance between each pair of parallel lines with the given equations.

Find the distance between each pair of parallel lines with the given equation.

Hint: If you have two lines that are parallel and the perpendicular segment does not land on a point that you can read, you will have to write the equation of the line for the perpendicular segment and solve to find the point that they have in common.

3. (a) Find the distance between the parallel lines with equations y = 2x + 3 and y = 2x – 1, respectively.

3. (b) Find the distance between parallel lines with equations x + 3y = 6 and x + 3y = -15, respectively.

Homework – Page 218 – 221 (9 – 12 ALL, 15 – 25 ODD)