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Ch. 3 – Parallel and Perpendicular Lines
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 AnglesNot 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
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Section 3.1 – Lines and AnglesA 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 AnglesProblem 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 LinesThe 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 ParallelWe 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 TrianglesProof 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 TrianglesProblem 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
