Rigid Motions Preserve Parallel Lines

You will be able to:

• Verify that translation, reflection, and rotation keep parallel lines parallel
• Compare image slopes to confirm parallelism
• Explain why parallelism is preserved, using angle preservation

Learning Objectives for This Lesson

By the end of this lesson, you will be able to:

1. State that rigid motions map parallel lines to parallel lines
2. Verify parallelism is preserved under a translation
3. Verify parallelism is preserved under a reflection
4. Verify parallelism is preserved under a rotation
5. Explain why rigid motions preserve parallelism, using angle preservation

Could a Rotation Make Them Meet?

Two parallel lines never intersect.

• A translation just slides them — easy to believe they stay parallel
• But a rotation swings both lines to new directions

Could it swing them toward each other until they cross?

Translate Two Parallel Lines Together

• Slide both lines by the same vector — the images are still parallel

Equal Slopes Mean the Lines Are Parallel

Two lines are parallel exactly when they have equal slopes (and never meet).

• Translation changes each line's direction by zero
• Same slope before same slope after still parallel

Translation Worked Example With Oblique Lines

Parallel lines and (both slope ).

Translate by .

• Both images keep slope — still parallel

Quick Check: Slopes After Sliding

Two parallel lines both have slope .

After any translation, the image slopes are:

• Both still — translation never changes a line's slope

Reflection and Rotation Change Direction

Translation never tilted the lines. But:

• Reflection flips the lines like a mirror
• Rotation swings them to new directions

Both change the slope. Can the lines still be parallel?

Reflection: Slopes Flip, Stay Equal

• and reflected across the -axis: both images slope

Rotation Turns Horizontal Lines Into Vertical

• and rotated become and — both vertical, parallel

Any Rotation Keeps Them Parallel

• A rotation turns both lines by the same angle — the gap in direction is unchanged

Three Motions, and Still Parallel Throughout

• Every rigid motion kept the pair parallel

Your Turn: Reflect and Compare

Parallel lines and .

Reflect both across the -axis: .

• Find both image slopes. Are the images parallel?

From Checking to Reasoning: Does It Always Work?

We verified parallelism in many specific cases.

• But we cannot check every pair and every rigid motion

So: why must parallelism always be preserved?

Parallel Lines and a Transversal

• Parallel lines make equal corresponding angles with any transversal
• And the reverse: equal corresponding angles force the lines to be parallel

Angles Are Preserved, So Parallelism Is Too

• Equal angles map to equal angles — so the image lines stay parallel

The Argument Laid Out in Four Steps

• The conclusion follows for any rigid motion — no case checking needed

Why a Rectangle Stays a Rectangle

A rectangle has two pairs of parallel sides.

Apply any rigid motion:

• Parallel sides stay parallel, lengths and right angles stay fixed
• So the image is still a rectangle — a congruent copy

On Your Own: Verify and Decide

Parallel lines and .

Apply a rotation about the origin: .

• Find both image slopes and state whether the images are parallel

Three Common Traps to Avoid Here

• Rotation makes them meet. Both lines turn the same angle, so they cannot converge.
• It looks parallel, so it is. Slopes and look parallel but cross — check slopes.
• Distance is the reason. The clean argument runs through angles, not spacing.

Parallelism Survives Every Rigid Motion

Parallel stays parallel — not as a fact to memorize, but because rigid motions preserve angles.

• Holds for any pair of lines, any slope, any rigid motion
• A rectangle stays a rectangle; a parallelogram stays a parallelogram

What's Next: Transversal Angles Directly

You've shown a transversal's angle relationships travel with parallel lines under any rigid motion.

• Next lesson: studying those angle relationships head-on (8.G.A.5)
• Corresponding, alternate interior, and co-interior angles