Angle Relationships Exploration · 2015-06-17 · 2-16. CORRESPONDING ANGLES FORMED BY PARALLEL LINES The corresponding angles in Julia’s diagram in problem 2-15 have equal measure

Name_____________________________________________ Period____________ Score_________

Angle Relationships Exploration

Materials Needed: Tracing Paper

I. This diagram shows a tiling of parallelograms.

Use tracing paper to trace angle A. Slide it around the figure and mark the angles that are congruent.

a. Name at least two types of angle pairs that you can identify in the figure.

II. An enlarged view of one portion of the parallelogram tiling is shown at the left, with some points and angles labeled.

a. A line that crosses two or more other lines is called a transversal. In the diagram, which line is the transversal?

b. Which lines are parallel?

Trace ∠x on tracing paper and shade its interior. Then translate ∠x by sliding the tracing paper along the transversal until it lies on top of another angle and matches it exactly.

c. Which angle in the diagram corresponds with ∠x?

d. In this diagram, ∠x and ∠b are called corresponding angles because they are in the same position at two different intersections of the transversal. There are 3 other pairs of corresponding angles. Name them all. (Repeat the steps you used above to find each corresponding angle pair if necessary.)

∠a and ______ ∠b and ∠x ∠c and ______ ∠d and ______

e. How would you describe the relationship of each pair of corresponding angles in this figure?

f. Suppose b = 60º. Use what you know about vertical, supplementary, and corresponding angle relationships to find the measures of all the other angles in the diagram.

∠a = ______ ∠b = 60˚ ∠c = ______ ∠d = ______

∠w = ______ ∠x = ______ ∠y = ______ ∠z = ______

Chapter 2: Angles and Measurement 179

2.1.2 What is the relationship? • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Angles Formed by Transversals In Lesson 2.1.1, you examined vertical angles and found that vertical angles are always equal. Today you will look at another special relationship that guarantees angles have equal measure. 2-13. Examine the diagrams below. For each pair of angles marked on the diagram,

quickly decide what relationship their measures have. Your responses should be limited to one of three relationships: same (equal measures), complementary (have a sum of 90°), and supplementary (have a sum of 180°).

a. [ Supplementary ] b. [ Same ] c. [ Complementary ] d. [ Same ]

2-14. Marcos was walking home after school

thinking about special angle relationships when he happened to notice a pattern of parallelogram tiles on the wall of a building. Marcos saw lots of special angle relationships in this pattern, so he decided to copy the pattern into his notebook.

The beginning of Marcos’s diagram is

shown at right and provided on the Lesson 2.1.2 Resource Page. This type of pattern is sometimes called a tiling. In this tiling, a parallelogram is copied and translated to fill an entire page without gaps or overlaps.

a. Since each parallelogram is a translation of another, what can be stated about the angles in the rest of Marcos’ tiling? Use a dynamic geometry tool or tracing paper to determine which angles must have the same measure. Color all angles that must be equal the same color. [ The end result only requires two colors. ]

Problem continues on next page →

a b

d c

f g

h k


In addition, tracing paper can be used to show that that the opposite angles of a parallelogram have the same measure. Place tracing paper on top of the “root” parallelogram and then translate it up and to the right as shown at right. Then rotate the vertical angles 180° about the common vertex to show that the opposite angles in a parallelogram must have equal measure.

Suggested Lesson Activity:

One way to start today’s lesson is to ask for volunteers to share what they learned in Lesson 2.1.1 or you can use Reciprocal Teaching to review the vocabulary from Lesson 2.1.1. Then focus the conversation on the relationships that the students studied: both geometric relationships (such as vertical pairs, straight angle pairs, and right angle pairs) and angle measure relationships (such as same, that is, they have equal measure, supplementary, and complementary). Explain that this lesson will continue to look for relationships between angle pairs. Problem 2-13 is a quick warm-up, which can be done individually or in teams to allow you to assess whether or not students can recognize the relationships they learned of in Lesson 2.1.1. You could use a modified Hot Potato where each team member explains a different part, making sure they justify their answers.

Then move into problem 2-14. If you have access to a computer with a projector, then show the PowerPoint presentation after

students have read the problem statement and part (a). If you do not have access to technology, use a document camera or overhead transparencies to demonstrate the tiling of the parallelogram as described in the Technology Notes section. Much of the angle relationship development will depend on students recognizing that an angle and its translated image must have equal measure. Therefore, emphasize that the diagram is formed by a translated parallelogram and encourage students to trace a parallelogram on tracing paper and physically translate it on the Lesson 2.1.2 Resource Page to verify that corresponding angles must have equal measure. Encourage students to use tracing paper to verify that opposite angles of a parallelogram have equal measures. They can trace a parallelogram in their text and then rotate the tracing paper about the center of the parallelogram to see that the opposite angles must have equal measure. Expect students to make connections between this and what they did as they studied rotation symmetry in Lesson 1.2.6. Move students on to problems 2-15 and 2-16. Note that these problems use transformations to preview the angle relationships students will be studying in Lessons 2.1.2 and 2.1.3: corresponding angles, alternate interior angles, and same-side interior angles. Only corresponding angles will be discussed using their formal name in this lesson. Note that the

A

Core Connections Geometry 180

2-14. Problem continued from previous page.

b. Consider the angles inside a single parallelogram. Which angles must have equal measure? How can you justify your claim? [ Opposite angles must be equal because translation preserves angle and vertical angles have equal measure. ]

c. What about relationships between lines? Can you identify any lines that must be parallel? Mark all the lines on your diagram with the same number of arrows to show which lines are parallel. [ Opposite sides of a parallelogram are parallel because of the definition of parallelogram (from Lesson 1.3.2). ]

2-15. Julia wants to learn more about the angles

in Marcos’s diagram and has decided to focus on just a part of his tiling. An enlarged view of that section is shown in the image below right, with some points and angles labeled.

a. A line that crosses two or more other lines is called a transversal. In Julia’s diagram, which line is the transversal? Which lines are parallel? [ JK! "#

is the transversal; LM! "##

and NP! "##

are parallel. ]

b. Trace ∠x on tracing paper and shade its interior. Then translate ∠x by sliding the tracing paper along the transversal until it lies on top of another angle and matches it exactly. Which angle in the diagram corresponds with ∠x? [ ∠b ]

c. In this diagram, ∠x and ∠b are called corresponding angles because they are in the same position at two different intersections of the transversal. What is the relationship between the measures of angles x and b? Must one be greater than the other, or must they be equal? Explain how you know. [ They are equal, because they were copied from the same angle of the original parallelogram. As learned in Chapter 1, translations are rigid transformations, so they preserve both length and angle. ]

w N

x

b J

a

y

d M c

z

J

K

L

P

Exploration: Do corresponding angles always have equal measure? III. For parts (a) through (d) below, use tracing paper to decide if corresponding angles have the same measure. Then determine if you have enough information to find the measures of x and y. If you do, find the angle measures and state the relationship.

a. b.


2-16. CORRESPONDING ANGLES FORMED BY PARALLEL LINES The corresponding angles in Julia’s diagram in problem 2-15 have equal measure

because they were formed by translating a parallelogram.

a. Name all the other pairs of corresponding angles you can find in Julia’s diagram from problem 2-15. [ d and z, a and w, c and y ]

b. Suppose b = 60º. Use what you know about vertical, supplementary, and corresponding angle relationships to find the measures of all the other angles in Julia’s diagram. [ x, y, and c all measure 60°; a, d, w, and z measure 120°. ]

2-17. Frank wonders whether corresponding angles always have equal measure. For parts

(a) through (d) below, use tracing paper to decide if corresponding angles have the same measure. Then determine if you have enough information to find the measures of x and y. If you do, find the angle measures and state the relationship.

a. b.

[ x = 135° (corresponding angles), [ y = 45° (straight angle pair), not y = 135° (vertical angles) ] enough information for x ]

c. d.

[ x = 135° (corresponding angles), [ x = 45° (straight angle pair), y = 45° (straight angle pair) ] not enough information for y ]

e. Answer Frank’s question: Do corresponding angles always have equal measure? If not, when are their measures equal? [ No, corresponding angles have equal measure only when the lines intersected by the transversal are parallel. ]

f. Conjectures are often written in the form, “If…, then…”. A statement in if-then form is called a conditional statement. Make a conjecture about corresponding angles by completing this conditional statement: “If …, then corresponding angles have equal measure.” [ If lines are parallel, then corresponding angles have equal measure. ]

g. Prove that your conjecture in part (f) is always true. That is, explain why this conjecture is a theorem. [ If lines are parallel, then corresponding angles are congruent because there is a rigid transformation that carries one angle onto the other (translation). ]

x y

x

y

y

x

x y








a. b.


c. d.





x y

x

y

y

x

x y

c. d.








a. b.


c. d.





x y

x

y

y

x

x y








a. b.


c. d.





x y

x

y

y

x

x y

e. Do corresponding angles always have equal measure? If not, when are their measures equal?

f. Conjectures are often written in the form, “If…, then…”. A statement in if-then form is called a conditional statement. Make a conjecture about corresponding angles by completing this statement:

“If _______________________________________________________________________, then corresponding angles have equal measure.”

IV. For each diagram below, find the value of x , if possible. If it is not possible, explain how you know. State the relationships you use. Be prepared to justify every measurement you find.

a. b. c.


2-18. For each diagram below, find the value of x , if possible. If it is not possible, explain how you know. State the relationships you use. Be prepared to justify every measurement you find to other members of your team. [ There are multiple methods for each, so make sure students justify their responses. ]

a. b. c.

[ x = 65° ] [ Not enough information ] [ 4x ! 25° = 3x + 10° because the lines are not so x = 35° ] parallel. ]

53°

x

115°

x



a. b. c.


53°

x

115°

x



a. b. c.


53°

x

115°

x

Notice that the transversal is horizontal in examples a and b.In both, the 135˚ angle and ∠x are a pair of corresponding angles.

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Angle Relationships Exploration · 2015-06-17 · 2-16. CORRESPONDING ANGLES FORMED BY PARALLEL LINES The corresponding angles in Julia’s diagram in problem 2-15 have equal measure