Geometry Honors - Chapter 3 – Parallel and Perpendicular Lines
Section 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).
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Parallel Lines – coplanar lines that do not intersect or touch.
Skew Lines – are lines that do no intersect and are not coplanar (means they are not parallel to 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
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(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)
Interior angles – angles that lie on the transversal between 2 lines (ex: ∠4, ∠3, ∠5, ∠ 6)
Exterior angles – angles that lie on the transversal outside 2 lines (ex: ∠1, ∠2, ∠7, ∠8)
Special Angle Pairs:
o
o
o
o
Consecutive Interior angles - interior angles on the same side of the transversal
(ex: ∠4 and ∠5; ∠ 3 and ∠6)
Alternate Interior angles – interior angles on the opposite side of the transversal
(ex: ∠4 and ∠6; and ∠3 and∠5)
Alternate Exterior angles – exterior angles on the opposite side of the transversal
(ex: ∠1 and ∠7; ∠2 and ∠8)
Corresponding angles – angles that lie on the same side of the transversal with one angle is interior and
one angle is exterior in the pair (ex: ∠ 1 and ∠5; ∠2 and ∠6; ∠4 and ∠8; ∠3 and∠7)
2 – 3:Name the transversal connecting each pair of angles. Then classify as alternate interior, alternate
exterior, corresponding, or consecutive interior angles.
2.
3.
(a) ∠2 and ∠6
(a) ∠3 and ∠5
(b) ∠1 and ∠7
(b) ∠2 and ∠8
(c) ∠3 and ∠8
(c) ∠5 and ∠7
(d) ∠3 and ∠5
(d) ∠2 and ∠ 9
Homework – Page 174 – 176 (13-19ODD, 21-43, 50, 51)
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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 line 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∠1 = 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.
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Perpendicular Transversal Theorem – in a plane, if a line is perpendicular to 1 of 2 parallel lines,
then it is perpendicular to the other.
4.
5.
6.
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Review
If 2 parallel lines are cut by a transversal, name 3 angle pairs that are congruent to each other:
o _____________________________
o _____________________________
o _____________________________
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If 2 parallel lines are cut by a transversal, name an angle pair that is supplementary:
o ____________________________
Homework – Page 181 – 183 (11 – 29, 38, 39, 43, 45)
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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 Interior angles converse – if 2 lines are cut by a transversal so that alternate interior angles
are congruent, then the lines are parallel
(ex: If∠ 1 ≅ ∠5, then a∥ b OR
∠8 ≅ ∠4, then a∥b)
Alternate Exterior angles converse – if 2 lines are cut by a transversal so that alternate exterior
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 ∠YN 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.
Homework – Page 209 - 211 (8 – 21, 23, 33 – 35, 42) ALL
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Section 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)
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?
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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
3. Determine whether FG and HJ are parallel, perpendicular or neither.
(a) F(1, -3), G(-2, -1), H(5, 0) and J(6, 3)
(b) F(3, 6), G(-9, 2), H(5, 4) and J(2, 3)
(c) F(5, 1), G(5, 6), H(-2, 3) and J(-6, 3)
4. Graph the line that contains P(0,1) and is perpendicular to QR with Q(-6, -2) and R(0, -6)
5. Graph the line that contains Q(5, 1) and parallel to MN with M(-2, 4) and N(2, 1).
Homework – Page 191 – 193 (15-33ODD, 34-39ALL, 41, 48, 52, 53, 55)
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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.
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Slope – Intercept Form: y = mx + b, where ‘m’ stands for the slope and ‘b’ stands for the y-intercept
(where the line crosses the y-axis)
(ex: m = 3 and b = -2
so
y = 3x – 2)
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Point- Slope Form: y – y1 = m(x – x1), where ‘m’ stands for 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)
(c) (-1, 3) and (7, 3)
(d) (-3, 1) and (-3, 6)
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 + 3 and containing (-3,
6)
7. An apartment complex charges $525 per month plus a $750 annual maintence 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 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, 56, 58)
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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’.
3. (a) Find the distance between the parallel lines ‘a’ and ‘b’ with equations y = 2x + 3 and y = 2x – 1,
respectively.
3. (b) Find the distance between parallel lines ‘a’ and ‘b’ with equations x + 3y = 6 and x + 3y = -15,
respectively.
Homework – Page 218 – 221 (9 – 12 ALL, 15 – 25 ODD)