Download Geometry Chapter 3: Parallel and Perpendicular Lines Term Example

Geometry Chapter 3: Parallel and Perpendicular Lines
Lesson 1: Lines and Angles
Learning Targets
Success Criteria
LT-1: Identify parallel lines, perpendicular lines,
skew lines, and the angles formed by two
lines and a transversal.
•
•
•
Identify types of lines in the plane.
Classify angle pairs.
Identify angle pairs and transversals.
Parallel lines: Coplanar lines that do not intersect.
Perpendicular lines: Lines that intersect at 90º angles.
Skew lines: Lines that are not coplanar and do not intersect.
Parallel Planes: Planes that do not intersect.
Term
Example
Transversal - A line that intersects two coplanar
lines at two different points. The transversal t and
the other two lines r and s form eight angles.
Corresponding Angles - Angles that lie on the
same side of the transversal t, on the same sides of
lines r and s.
Alternate Interior Angles - Nonadjacent angles
that lie on opposite sides of the transversal t,
between lines r and s.
Alternate Exterior Angles – Angles that lie on
opposite sides of the transversal t, outside lines r
and s.
Same-Side-Interior Angles – Angles that lie on
the same side of the transversal t, between r and s.
Same-Side-Exterior Angles – Angles that lie on
the same side of the transversal t, outside lines r
and s.
Page 1
Ex #1: Identify Types Of Lines In The Plane.
Use the diagram to identify the following:
1a. a pair of parallel segments
1b. a pair of skew segments
1c. a pair of perpendicular segments
1d. a pair of parallel planes
1e. a pair of perpendicular planes
Ex #2: Classify Pairs of Angles.
Give an example of each angle pair.
2a. corresponding angles
2b. alternate interior angles
2c. alternate exterior angles
2d. same-side interior angles
2e. same-side exterior angles
2f. vertical angles
Ex #3: Identify Angle Pairs and Transversals.
Use the diagram to identify the transversal and classify each angle pair.
3a. ∠1 and ∠3
3b. ∠2 and∠ 6
3c. ∠4 and ∠6
Page 2
Lesson 2 – Angles Formed by Parallel Lines and Transversals
Learning Targets
Success Criteria
LT-2: Prove and apply the theorems about the
angles formed by parallel lines and a
transversal.
•
•
Use the Corresponding Angles Postulate.
Find angle measures given parallel lines
and a transversal.
Corresponding Angle Postulate
If two parallel lines are cut by a transversal, then
corresponding angles are congruent.
Alternate Interior Angle Theorem
If two parallel lines are cut by a transversal, then alternate interior
angles are congruent.
Alternate Exterior Angle Theorem
If two parallel lines are cut by a transversal, then alternate exterior
angles are congruent.
Same-Side Interior Angle Theorem
If two parallel lines are cut by a transversal, then same-side interior
angles are supplementary.
Same-Side Exterior Angle Theorem
If two parallel lines are cut by a transversal, then same-side exterior
angles are supplementary.
Page 3
Ex #1: Use The Corresponding Angles Postulate.
A. Find m∠QRS.
B.
If CD // EF ,
find m∠DEF.
Ex #2 Find Angle Measures Given Parallel Lines And A Transversal.
A. Find x.
m∠ABD = (2x + 10)º
m∠BDE = (3x - 15)º
B. Find m∠CAY.
Ex #3: Find Angle Measures Given Parallel Lines And A Transversal.
Find x and y in the diagram below.
m∠1 = (2x+27)º
m∠2 = (2x + y)º
Page 4
Lesson 3 – Proving Lines Parallel
Learning Targets
Success Criteria
LT-3: Use the angles formed by a transversal to
prove/disprove that two lines are parallel.
•
•
•
Prove lines parallel using the converse of:
◦ Corresponding Angles Postulate.
◦ Alternate Interior/Exterior Angles Thm
◦ Same-Side-Interior/Exterior Angles Thm
Write proofs to prove lines are parallel.
Construct a line parallel to a line through a
point not on it.
Recall that the converse of a theorem is found by exchanging the hypothesis and conclusion. The
converse of a theorem is not automatically true. If it is true, it must be stated as a postulate or proved
as a separate theorem.
Original conditional: If p, then q.
converse:
If
q,
then
Ex #1: Prove Lines Are Parallel (Using the Converse)
Use the diagram at the right, each set of given information, and
the theorems that you have learned to show that l // m
A. ∠4 ≅ ∠8
B. ∠1 ≅ ∠8
C. m∠3 = (4x – 80), m∠7 = (3x – 50), x = 30
D. m∠3 = (10x + 8), m∠5 (25x – 3), x = 5
Ex #2: Prove Lines are Parallel (Using the Converse)
A carpenter is creating a woodwork pattern and wants
two long pieces to be parallel. m∠1= (8x + 20)° and
m∠2 = (2x + 10)°. If x = 15, show that
pieces A and B are parallel.
Page 5
p.
3.3 Construct a line parallel to a given line through a point not on it.
Ex #3: Write A Two-Column Proof To Prove Lines Parallel.
Given: l // m, ∠1 ≅ ∠3
Prove: p // r
Statements
Reasons
1.
2.
3.
4.
Ex #4: Write A Two-Column Proof To Prove Lines Parallel.
Given: ∠1≅∠2, ∠3 ≅∠1
Prove: XY // WV
Statements
Reasons
1.
2.
3.
Page 6
*Ex #5: Write A Two-Column Proof To Prove Lines Parallel.
Given: ∠1 ≅ ∠4, ∠3 and ∠4 are supplementary
Prove: l // m
Statements
Reasons
1.
2.
3.
4.
5.
6.
7.
Lesson 4 – Perpendicular Lines
Learning Targets
Success Criteria
LT-4: Construct, prove, and apply theorems
about perpendicular lines.
•
•
•
Find the shortest distance from a point to a
line.
Prove properties of lines.
Construct a line perpendicular to a given line
through a point not on it.
Perpendicular Bisector (of a segment): A line perpendicular to a segment at the segment's midpoint.
Distance from a point to a line: The length of the perpendicular segment from the point to the line.
3.3 Construct a line perpendicular to a given line through a point not on it.
Page 7
Ex #1: Find The Shortest Distance From A Line To A Point Not On It.
A. Name the shortest segment from A to
BC .
B. Write and solve the inequality for x.
C. Name the shortest segment from A to
BC .
D. Write and solve an equation for x.
Theorem
Hypothesis
3.4.1 If two intersecting lines
form a linear pair of congruent
angles, then the lines are
perpendicular.
3.4.2 Perpendicular
Transversal Theorem
In a plane, if a transversal is
perpendicular to one of two
parallel lines, then it is
perpendicular to the other line.
3.4.3 If two coplanar lines are
perpendicular to the same line,
then the two linew are parallel to
each other.
Page 8
Conclusion
Ex #2: Prove Lines Perpendicular.
Given: r // s, ∠1 ≅ ∠2
Prove: r ⊥ t
Statements
Reasons
1.
2.
3.
4.
Ex #3: Prove Lines Perpendicular.
Statements
Reasons
1.
2.
3.
Lesson 5 – Slopes of Lines
Learning Targets
Success Criteria
LT-5: Prove that two lines are parallel or
perpendicular by calculating their slopes.
•
•
•
Use the slope formula to calculate the slope
of a line.
Calculate and interpert rate of change.
Determine whether lines are parallel,
perpendicular, or neither.
The slope of a line in a coordinate plane is a number that describes the steepness of the line. Any two
points on a line can be used to determine the slope.
Page 9
Rise: The difference in the y-values of two points on a line.
Run: The difference in the x-values of two points on a line.
Slope: The ratio of rise to run.
Ex #1: Find The Slope Of A Line.
Use the slope formula to determine the slope of each line.
a.
b b.
c.
d.
Ex #2: Calculate And Interpret Rate Of Change.
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.
Page 10
Parallel Line Theorem: In a coordinate plane, two nonvertical lines are parallel if and only if they
have the same slope. Any two vertical lines are parallel.
Ex:
Perpendicular Line Theorem: In a coordinate plane, two nonvertical lines are perpendicular if and
only if the product of their slopes is -1. Vertical and horizontal lines are perpendicular.
Ex:
Ex #3: Determine Whether Lines Are Parallel, Perpendicular, Or Neither.
Graph each pair of lines. Use their slopes to determine
whether they are parallel, perpendicular, or neither.
A.
B.
C.
Lesson 6 – Lines in the Coordinate Plane
Learning Targets
Success Criteria
LT-6: Graph lines and write their equations in
slope-intercept and point-slope form.
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•
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Page 11
Write equations for lines in slope-intercept
form:
◦ Given a point and the slope of a line.
◦ Given two points on a line.
◦ Given a point and the y-intercept
Write equations for lines in point-intercept
form.
◦ Given a point and the slope of a line.
◦ Given two points on a line.
◦ Given a point and the y-intercept
Graph lines in the coordinate plane.
Classify pairs of lines.
The equation of a line can be written in many different forms. The point-slope and slope-intercept
forms of a line are equivalent. Because the slope of a vertical line is undefined, these forms cannot be
used to write the equation of a vertical line.
Name
Form
Point-Slope Form
Slope-Intercept Form
The equation of a vertical line is:
The equation of a horizontal line is:
Ex #1: Write The Equation Of Each Line
In The Given Form.
Ex #2: Graph Each Line In the Coordinate
Plane.
A. the line with slope 6 through (3, –4) in point-slope form 2. Graph each line.
a. y = ½ x + 1
B. the line through (–1, 0) and (1, 2) in slope-intercept
form
b. y – 3 = -2(x + 4)
c. y = -3
C. the line with the x-intercept 3 and y-intercept –5 in point
slope form
d. y – 1 = -
2
(x + 2)
3
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.
Page 12
Ex #3: Determine Whether Lines Are Parallel, Intersecting, Or Coinciding.
A.
y = 3x + 7, y = –3x – 4
C. 2y – 4x = 16, y – 10 = 2(x - 1)
B.
D.
3x + 5y = 2, 3x + 6 = -5y
Ex #4: Write Equations For Lines Given A Point And The y-intercept.
Erica is trying to decide between two car rental plans. For how many miles will the plans cost the
same?
Page 13
Lesson
Problems
3.1 p. 149
#14-32, 34-43, 58, 60-62
3.2 p. 158
#6-25, 27, 28, 30-32, 34, 35, 41, 42
3.3 p. 166
#12-14, 16-35, 38, 40, 41-45, 58, 60, 64
3.4 p. 175
#6-14, 15 challenge, 16-22, 24 challenge, 26, 27, 31-34, 40.
44
3.5 p. 186
#10-22, 26-28, 29, 30, 34, 38-40
3.6 p. 194
#13-25, 27-34, 37, 39, 45-47, 51, 58-60
Page 14