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
Problem 1
In the figure, assume that lines and planes that appear to be parallel are
parallel.
A) Which segments are parallel to 𝐴𝐵?
B) Which segments are skew to 𝐶𝐷?
C) What are two pairs of parallel lines?
D) What are two segments parallel to plane BCGF?
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