8.G.1: Properties of Rigid Motions
Also written as 8.G.A.1 and CCSS.MATH.CONTENT.8.G.A.1 · Cluster: Understand congruence and similarity using physical models, transparencies, or geometry software.
The Official Wording
Verify experimentally the properties of rotations, reflections, and translations:
Common Core State Standards, Grade 8, Geometry · Read it at the source · Checked October 14, 2026
What 8.G.1 Means in Plain English
Eighth graders trace a figure, then slide, turn or flip the tracing and measure what changed. A segment 5 units long is still 5 units, a 40° angle is still 40°, and the parallel sides of a trapezoid stay parallel. Only position and direction move.
Measuring rather than being told is the point: students check each property with tracing paper, grids or geometry software. These facts underpin congruence in 8.G.2, and in high school the three motions get precise definitions (HSG-CO.4).
8.G.1 I Can Statements
- I can show that a slide, turn or flip keeps segment lengths.
- I can show that rigid motions keep every angle the same size.
- I can show that parallel lines stay parallel after a rigid motion.
8.G.1 Worked Example
One trapezoid, one quarter turn on tracing paper, and three things to measure, one for each lettered part.
Trapezoid ABCD has A(1, 1), B(5, 1), C(3, 3) and D(1, 3). Rotate it 90° counterclockwise about the origin with tracing paper. Check that its lengths, angles and parallel sides survive the turn.
- Trace the trapezoid and the axes, pin the origin with a pencil point, and turn the paper a quarter turn counterclockwise. Read off the new corners.A′(−1, 1), B′(−1, 5), C′(−3, 3), D′(−3, 1)
- Part a, lengths: the horizontal and vertical sides can be counted square by square on the grid.AB = A′B′ = 4, DC = D′C′ = 2, AD = A′D′ = 2
- Part b, angles: measure with a protractor. The right angle at A and the 45° angle at B come through unchanged.∠A = ∠A′ = 90°, ∠B = ∠B′ = 45°
- Part c, parallels: AB and DC were both horizontal, so parallel. After the turn both are vertical, so still parallel.AB ∥ DC, and A′B′ ∥ D′C′
Check it: Use the rule for a 90° counterclockwise turn, (x, y) to (−y, x): B(5, 1) becomes (−1, 5), the same corner the tracing gave. Then lay the tracing over the image: it fits exactly.
Common Mistakes with 8.G.1
Turning about the wrong point
A′ left at (1, 1) because the tracing was pinned at A, not at the origin
Try this: Mark the center of rotation on the tracing paper before turning, and keep the pencil point on it the whole time.
Mixing up clockwise and counterclockwise
B(5, 1) turned 90° counterclockwise and placed at (1, −5)
Try this: Draw the turning arrow on the tracing first. Counterclockwise from the positive x-axis heads up toward the positive y-axis.
Thinking a flip changes the angles
“The reflected triangle points the other way, so its angles must be different.”
Try this: Measure the angles before and after with a protractor. A reflection reverses the orientation, but every angle measure stays the same.
Where 8.G.1 Fits
Practice 8.G.1
Quick Check
- Segment PQ runs from (2, 1) to (2, 7). It is reflected across the y-axis. How long is P′Q′?
- A 72° angle is translated 5 units right and 3 units down. What does the image angle measure?
- Lines ℓ and m are parallel. Both are rotated 90° about the same point. Are their images parallel?