Rigid Motions Preserve Angle Measure

You will be able to:

• Verify that translating, reflecting, or rotating an angle keeps its measure
• Explain why angle preservation defines a rigid motion
• Find angle measures in figures moved by a sequence of rigid motions

Learning Objectives for This Lesson

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

1. Verify that a translated angle has the same measure as the original
2. Verify that a reflected angle has the same measure as the original
3. Verify that a rotated angle has the same measure as the original
4. Explain why angle preservation is a defining property of rigid motions
5. Find unknown angle measures after a sequence of rigid motions

Same Angle, Two Spots: How Do We Know?

Two angles on the grid look identical but sit in different places.

• Could you be sure their measures match just by looking?
• What tool settles it without guessing?

A protractor — measure both and compare.

Measure, Translate, Then Measure Once More

• A angle, slid across the grid, still measures

Why Translation Keeps the Measure

• Each ray shifts the same way — so it stays parallel to its original
• Same directions in, same angle between them

Translating an Obtuse Angle Keeps Its Measure

Angle measures about .

Translate by : vertex .

• Image angle measures about — unchanged

Quick Check: Why No Change?

After translating an angle, its measure is unchanged.

Which reason is correct?

• The rays shifted the same way, so their directions — and the angle — held

Translation never rotates the rays, so the opening is fixed.

Flipping and Turning Change Direction

Translation was the easy case — nothing rotated.

But a reflection flips a figure, and a rotation turns it.

• Both clearly change direction. Do they change the measure?

Reflecting an Angle: Opens Left, Same Size

• Reflect a angle across the -axis
• The image opens to the left — but still measures

Orientation Reverses, but the Measure Holds

A reflection is a mirror. Mirrors flip left and right.

• A right angle in a mirror is still a right angle
• A angle in a mirror is still

Orientation is a direction. Measure is a number. Only one flips.

Rotation Turns Both Rays Equally

• Both rays rotate by the same amount — the gap between them is unchanged

Rotating an Acute Angle About the Origin

Angle measures about , vertex at the origin.

Rotate counterclockwise: .

• Image still measures about — not

Edge Case: A Straight Angle

A straight angle measures — two rays forming a line.

Rotate it about its vertex.

• The image is a line tilted — still a straight angle, still

Every Rigid Motion: Same Measure

• Three motions, three angle types — every image measure equals its original

Your Turn: Reflect and Check

Angle measures .

Reflect it across the -axis. Before measuring, predict the image measure.

• Then place the protractor on the image and confirm

Is That True of EVERY Transformation?

Every rigid motion we tested preserved the measure.

But "rigid" is a specific category. Let's test one that is not:

• A horizontal stretch — does it keep angle measures?

A Stretch Changes the Angles

• Right angle at origin stays ; the other two angles change
• Not rigid — and not congruent to the original

Length Plus Angle Equals Congruence

On Your Own: Angles After a Sequence

Triangle has angles , , and .

It is reflected, then translated, then rotated.

• Without measuring, state the three angles of the final image

Three Common Traps to Avoid Here

• Reflection changed it. Flipping reverses which way the angle faces, never how wide it opens.
• Rotation added degrees. The rotation amount is how far the figure turned — not the angle's measure.
• Only nice angles work. Preservation holds for , , any measure at all.

What Stays Fixed: Angle Measure

A rigid motion can flip an angle to open the other way, or turn it to point anywhere — but it never changes how many degrees wide it is.

• Translation, reflection, rotation: all preserve angle measure
• True for every angle type: acute, right, obtuse, straight

What's Next: Do Parallel Lines Survive?

You've shown rigid motions preserve length and angle measure.

A figure can also have parallel sides.

• Next lesson: does a rigid motion keep parallel lines parallel?