math class 678 4-in-1 skill sheets
8.G.A.2 8th Grade Geometry

Reflections & Congruence

Understand that a two-dimensional figure is congruent to another if one can be obtained from the other by a sequence of rigid motions.

How to explain it

The anchor students hold onto: A reflection flips a figure across an axis. Each point and its image lie the same distance from the axis on opposite sides. Image vertices take primed names. Apply the first op to every preimage vertex, then apply the second op to that result. The image is named with double primes after a 2-op chain.

Students extend reflection thinking into rotations (#91), dilations (#92), and compose multiple rigid motions in #93 Congruence through Transformations.

Worked examples

Example 1 Reflect a point — x-axis
Reflect (3, 4) over the x-axis.
Step 1Keep x; negate y.
Step 2(3, 4) maps to (3, -4).
AnswerImage: (3, -4).
Example 2 Reflect a triangle — y-axis
Reflect △PQR over the y-axis.
Step 1Negate x; keep y for each vertex.
Step 2P(3,2)→P'(-3,2); Q(6,2)→Q'(-6,2); R(4,5)→R'(-4,5).
AnswerP'(-3,2); Q'(-6,2); R'(-4,5).
Example 3 Reflect a triangle — y = x
Reflect △ABC over y = x.
Step 1Swap x and y for each vertex.
Step 2A(1,5)→A'(5,1); B(4,5)→B'(5,4); C(2,8)→C'(8,2).
AnswerA'(5,1); B'(5,4); C'(8,2).

Common mistakes

What students write Negating the wrong coordinate — using (−x, y) for an x-axis reflection instead of (x, −y).
The fix x-axis: negate y only. y-axis: negate x only. The axis name tells you which coordinate stays.
Try this A student reflects point A(4, -3) over the line y = x. The student uses the rule (x, y) → (−x, −y) and writes: A′(−4, 3). Find and correct the error.
What students write Using the 180° rotation rule (−x, −y) for a y = x reflection instead of the swap (y, x).
The fix Reflection over y = x swaps coordinates only: (x, y) → (y, x). No negation.
Try this A student reflects J(3, −5) over the line y = −x. The student uses the rule (x, y) → (−y, x) and writes: J′(5, 3). Find and correct the error.
What students write Applying the second op to the original preimage instead of the intermediate image.
The fix Each step uses the previous result as its new preimage. Chain the operations in order.
Try this A student composed two rigid motions on P(3, 4): Step 1: Translate by (1, 2) → P′ = (4, 6). ✓ Step 2: Reflect over x-axis → P″ = (4, 6). ✗ Describe the student’s error. State the correct final image P″.
What students write Assuming the image is NOT congruent after two rigid motions.
The fix Compositions of rigid motions preserve distance and angle — the final image is ALWAYS congruent.
Try this A student translated △ABC (A(1,2), B(3,2), C(2,4)) by (−1, 3), then reflected over y = x. Translate: A(0,5), B(2,5), C(1,7). ✓ Reflect over y = x: A″(0,5), B″(5,2), C″(7,1). ✗ Identify the error. State the correct A″B″C″.

Teacher tip

Head off the two predictable errors before they happen. First: x-axis: negate y only. y-axis: negate x only. The axis name tells you which coordinate stays. Second: Reflection over y = x swaps coordinates only: (x, y) → (y, x). No negation.