Parallel Lines Cut by a Transversal

4/22/2020 National Go Math Middle School, Grade-8

L E S S O N 8.G.5 Parallel Lines Cut by Use informal arguments to establish facts about… the angles 11.1 a Transversal created when parallel lines are cut by a transversal… . ESSENTIAL QUESTION What can you conclude about the angles formed by parallel lines that are cut by a transversal?

8.G.5 Parallel Lines and Transversals A transversal is a line that intersects two lines in the same plane at two different points. Transversal t and lines a and b form eight angles.

Angle Pairs Formed by a Transversal

Term Example Corresponding angles lie on the same side of the ∠1 and ∠5 transversal t, on the same side of lines a and b. Alternate interior angles are nonadjacent angles that lie on opposite sides of the transversal t, ∠3 and ∠6 between lines a and b.

y Alternate exterior angles lie on opposite sides of n

a ∠1 and ∠8

p the transversal t, outside lines a and b. m o C

n Same-side interior angles lie on the same side of i

h ∠3 and ∠5 s i

l the transversal t, between lines a and b. b u P

t r u o c r a H

n i l f

f Use geometry software to explore the angles formed i M

n when a transversal intersects parallel lines. o t h g u o H Construct a line and label two points on the line © A and B. Create point C not on Then construct a line parallel to through point C. Create another point on this line and label it D.

Lesson 11.1 347

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Create two points outside the two parallel lines and label them E and F. Construct transversal Label the points of intersection G and H.

Measure the angles formed by the parallel lines and the transversal. Write the angle measures in the table below.

Drag point E or point F to a different position. Record the new angle measures in the table.

Angle ∠CGE ∠DGE ∠CGH ∠DGH ∠AHG ∠BHG ∠AHF ∠BHF

Measure

Reflect Make a Conjecture Identify the pairs of angles in the diagram. Then make a conjecture about their angle measures. Drag a point in the diagram to confirm your conjecture. 1. corresponding angles

2. alternate interior angles y n a

p m o C

g n i h s i l b u P

3. alternate exterior angles r u o c r a H

n i l f f i M

n o t h g u o H

4. same-side interior angles ©

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8.G.5 Justifying Angle Relationships You can use tracing paper to informally justify your conclusions from the first Explore Activity.

Recall that vertical Lines a and b are parallel. (The black angles are the opposite arrows on the diagram indicate angles formed by two parallel lines.) intersecting lines. ∠1 and ∠4 are vertical Trace the diagram onto tracing paper. angles. Position the tracing paper over the original diagram so that ∠1 on the tracing is over ∠5 on the original diagram. Compare the two angles. Do they appear to be congruent? What do you notice about the special angle pairs formed by the transversal?

Use the tracing paper to compare all eight angles in the diagram to each other. List all of the congruent angle pairs.

Finding Unknown Angle Measures

y You can find any unknown angle measure when two parallel lines are cut by n a p a transversal if you are given at least one other angle measure. m o C

g n i h s i l

b 8.G.5 u EXAMPLE 1 P

t r u o c r a H Find m∠2 when m∠7 = 125°. n i l f f i M

n ∠2 is congruent to ∠7 because they o t h

g are alternate exterior angles. u o H

Find m∠VWZ. ∠VWZ is supplementary to ∠YVW because they are same-side interior angles. m∠VWZ + m∠YVW = 180° Lesson 11.1 349

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From the previous page, m∠VWZ + m∠YVW = 180°, m∠VWZ = 3x°, and m∠YVW = 6x°.

Replace m∠VWZ with 3x° and m∠YVW with 6x°. Combine like terms.

Divide both sides by 9. Simplify.

Find each angle measure. 5. m∠GDE =

6. m∠BEF =

7. m∠CDG =

Guided Practice

Use the figure for Exercises 1–4. (Explore Activity 1 and Example 1)

1. ∠UVY and are a pair of corresponding angles.

2. ∠WVY and ∠VWT are angles.

3. Find m∠SVW. y n a

4. Find m∠VWT. m o C

g n i h

5. Vocabulary When two parallel lines are cut by a transversal, s i l b u P

angles are supplementary. (Explore Activity 1) u o c r a H

n i l f f i M

ESSENTIAL QUESTION CHECK-IN n o t h g u o H 6. What can you conclude about the interior angles formed when two parallel lines are cut by a transversal? ©

350 Unit 5

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