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Ch. 3 – Parallel and
Perpendicular Lines
Section 3.1 – Lines and Angles
1. I CAN identify relationships between figures in space.
2. I CAN identify angles formed by two lines and a transversal.
Key Vocabulary:
Parallel Lines
Skew Lines
Parallel Planes
Transversal
Alternate Interior Angles
Same-Side Interior angles
Corresponding Angles
Alternate Exterior Angles
Section 3.1 – Lines and Angles
Section 3.1 – Lines and Angles
In the solve it, you used relationships
among planes in space to write the
instructions. In Chapter 1, you learned
about intersecting lines and planes. In this
section, we will explore relationships of
nonintersecting lines and planes.
Section 3.1 – Lines and Angles
Not all lines and not all planes intersect.
Segments and rays can also be parallel or skew. They
are parallel if they lie in parallel lines and skew if
they lie in skew lines.
Section 3.1 – Lines and Angles
• Section 3.1 – Lines and Angles
A Transversal is a line that intersects two or more coplanar
lines at distinct points. The diagram below shows the 8 angles
formed by a transversal t and two lines l and m.
NOTICE: Angles 3, 4, 5, and 6 lie between l and m. They are
interior angles. Angles 1, 2, 7, and 8 lie outside of l and m.
They are exterior angles.
Section 3.1 – Lines and Angles
• Pairs of the 8 angles have special names.
Section 3.1 – Lines and Angles
Problem 2
Name the pairs of alternate interior angles.
Name the pairs of corresponding angles.
Section 3.1 – Lines and Angles
Problem 3 – Classifying an Angle Pair
The photo below shows the Royal Ontario Museum in Toronto,
Canada. Are angles 2 and 4 alternate interior angles, same-side
interior angles, corresponding angles, or alternate exterior
angles.
Section 3.1 – Lines and Angles
Section 3.2 – Properties of Parallel Lines
3. I CAN prove theorems about parallel lines.
4. I CAN use properties of parallel lines to find
angle measures.
Key Vocabulary:
Same-Side Interior Angles Postulate
Alternate Interior Angles Theorem
Corresponding Angles Theorem
Alternate Exterior Angles Theorem
Section 3.2 – Properties of Parallel Lines
Section 3.2 – Properties of Parallel Lines
In the solve it, you identified several pairs
of angles that appear congruent. You
already know the relationship between
vertical angles. In this lesson, you will
explore the relationship between the angles
you learned previously when they are
formed by parallel lines and a transversal.
Section 3.2 – Properties of Parallel Lines
The special angle pairs formed by parallel lines
and a transversal are congruent, supplementary,
or both.
Section 3.2 – Properties of Parallel Lines
Problem 1
The measure of <3 is 55. Which angles are
supplementary to <3? How do you know?
Section 3.2 – Properties of Parallel Lines
You can use Same-Side Interior Angles Theorem to prove other
angle relationships.
Section 3.2 – Properties of Parallel Lines
• Proof of Alternate Interior Angles Theorem
Section 3.2 – Properties of Parallel Lines
Problem 2
Given: a||b
Prove <1 and <8 are supplementary
Prove that <1 is congruent to <7.
Section 3.2 – Properties of Parallel Lines
In the previous problem, you proved <1
congruent to <7. These angles are Alternate
Exterior Angles.
Section 3.2 – Properties of Parallel Lines
•
If you know the measure of one of the angles formed by two parallel lines
and a transversal, you can use theorems and postulates to find the
measures of the other angles.
Problem 3
What are the measures of <3 and <4? Justify your answer.
What is the measure of: Justify>
a. <1
b. <2
c. <5
d. <6
e. <7
f. <8
Section 3.2 – Properties of Parallel Lines
Problem 4
What is the value of y?
Section 3.2 – Properties of Parallel Lines
a. In the figure below, what are the values of x
and y?
b. What are the measures of the four angles in
the figure
Section 3.2 – Properties of Parallel Lines
Section 3.3 – Proving Lines Parallel
5. I CAN determine whether two lines are
parallel.
Section 3.3 – Proving Lines Parallel
In the Solve It, you used parallel lines to
find congruent and supplementary
relationships of special angle pairs. In this
section, you will do the inverse. You will
use the congruent and supplement
relationships of the special angle pairs to
prove lines parallel.
Section 3.3 – Proving Lines Parallel
You can use certain angle pairs to decide
whether two lines are parallel.
Section 3.3 – Proving Lines Parallel
Problem 1
Which lines are parallel if <1 is congruent <2?
Justify.
Which lines are parallel if <6 is congruent to
<7?
Section 3.3 – Proving Lines Parallel
We can use the Converse of Corresponding Angles Theorem to
prove converse of the theorems and postulates we learned from
section 3.2.
Section 3.3 – Proving Lines Parallel
Proving Converse of the Alternate Interior Angles
Theorem
Given: <4 is congruent <6
Prove: L||m
Section 3.3 – Proving Lines Parallel
Prove Converse of the Same-Side Interior Angles
Postulate
Given: m<3 + m<6 = 180
Prove: L||m
Section 3.3 – Proving Lines Parallel
Proving Converse of the Alternate Exterior Angle
Theorem
Given: <4 is congruent <6
Prove: L||m
Section 3.3 – Proving Lines
Parallel
Problem 3
The fence gate at the right is made up of pieces of wood
arranged in various directions. Suppose <1 is congruent to <2.
Are lines r and s parallel? Explain.
What is another way to explain why r||s. Justify.
Section 3.3 – Proving Lines Parallel
Problem 4
What is the value of x for which a||b?
Section 3.3 – Proving Lines Parallel
What is the value of w for which c||d?
Section 3.3 – Proving Lines Parallel
Section 3.3 – Proving Lines Parallel
Section 3.3 – Proving Lines Parallel
Section 3.4 – Parallel and Perpendicular
Lines
5. I CAN relate parallel and perpendicular lines.
Section 3.4 – Parallel and Perpendicular
Lines
In the Solve It, you likely made your
conjecture about Oak Street and Court Road
based on their relationships to Schoolhouse
Road. In this section, you will use similar
reasoning to prove that lines are parallel
and perpendicular.
Section 3.4 – Parallel and Perpendicular
Lines
You can use the relationships of two lines to
a third line to decide whether the two lines
are parallel or perpendicular to each other.
Section 3.4 – Parallel and Perpendicular
Lines
There is also a relationship with
perpendicular lines.
Section 3.4 – Parallel and Perpendicular
Lines
Problem 1
A carpenter plans to install molding on the sides and the top of
the doorway. The carpenter cuts the ends of the top piece and
one end of each of the sides pieces 45 degrees as shown. Will
the side pieces of molding be parallel? Explain.
Can you assemble the pieces at the right to form a picture frame
with opposite sides parallel?
Section 3.4 – Parallel and Perpendicular
Lines
The previous theorems proved lines parallel. The
perpendicular transversal theorem allows us to
conclude that lines are perpendicular.
Section 3.4 – Parallel and Perpendicular
Lines
Proving a relationship between two lines.
Given: In a plane, c | b, b | d, and d |a.
Prove: c | a
Section 3.4 – Parallel and Perpendicular
Lines
Section 3.4 – Parallel and
Perpendicular Lines
Section 3.4 – Parallel and Perpendicular
Lines
Section 3.5 – Parallel Lines and Triangles
6. I CAN use parallel lines to prove a theorem
about triangles.
7. I CAN find measures of triangles.
Key Vocabulary
Auxiliary Line
Exterior Angle of a Polygon
Remote Interior Angles
Section 3.5 – Parallel Lines and Triangles
Section 3.5 – Parallel Lines and Triangles
In the Solve It, you may have discovered
that you can rearrange the corners of the
triangles to form a straight angle. You can
do this for any triangle.
Section 3.5 – Parallel Lines and Triangles
The sum of the angle measures of a triangle
is always the same. We will use parallel
lines to prove this theorem.
Section 3.5 – Parallel Lines and Triangles
To prove the Triangle Angle-Sum Theorem,
we must use an auxiliary line. An auxiliary
line is a line that you add to the diagram to
help explain relationships in proofs. The
red line is the diagram is an auxiliary line.
Section 3.5 – Parallel Lines and Triangles
Proof of Triangle Angle-Sum Theorem
Section 3.5 – Parallel Lines and Triangles
Problem 1
What are the values of x and y in the diagram.
Section 3.5 – Parallel Lines and Triangles
An exterior angle of a polygon is an angle formed by a
side and an extension of an adjacent side. For each
exterior angle of a triangle, the two nonadjacent
angles are its remote interior angles. In the triangle
below <1 is an exterior angle and <2 and <3 are its
remote interior angles.
Section 3.5 – Parallel Lines and Triangles
The theorem below states the relationship
between an exterior angle and its two
remote interior angles.
Section 3.5 – Parallel Lines and Triangles
Problem 2
What is the measure of <1?
What is the measure of <2?
Section 3.5 – Parallel Lines and Triangles
Problem 3
When radar tracks an object,
the reflection of signals off
the ground can result in
clutter. Clutter causes the
receiver to confuse the real
object with its reflection,
called a ghost. At the right,
there is a radar receiver at A,
an airplane at B, and the
airplane’s ghost at D. What
is the value of x?
Section 3.5 – Parallel Lines and Triangles
Section 3.5 – Parallel Lines and Triangles
Section 3.5 – Parallel Lines and Triangles
Section 3.5 – Parallel Lines and Triangles