What is the degree of rotation that maps the Hexagon and Octagon onto themselves, but leaves the Pentagon out?
Show thinking:
Rotation Angle
Shape Behavior: 90° vs. 180° Turns
9. The Non-Square Rectangle Opposite sides congruent
Length > Width
Does a 90° turn map this rectangle onto itself?
Yes No (it turns vertical)
Does a 180° turn map this rectangle onto itself?
Yes No
10. Equilateral Triangle at 180° Base angles: 120°
Original (Point UP)
→
Sketch here
Does a 180° turn map the equilateral triangle onto itself? Explain why:
11. Rigid Motion Notation: Which transformation is NOT equivalent to a 180° rotation about the origin?
Recall rule: R180°(x, y) = (-x, -y). Opposite signs for both coordinates.
A) Reflection over x-axis then y-axis (ry ∁ rx)
B) Reflection over y-axis then x-axis (rx ∁ ry)
C) Rotation of 180° about origin (R180°, (0,0))
D) Translation 4 units left and 4 down (T-4, -4)
Self-Check: How confident do you feel finding rotation angles?
👍 Got it! ✋ Need practice 🤔 Need help
• Triangle: 240° ÷ 120° = 2 (Whole number ✓)
• Hexagon: 240° ÷ 60° = 4 (Whole number ✓)
• Pentagon: 240° ÷ 72° = 3.33 (✗)
• Octagon: 240° ÷ 45° = 5.33 (✗)
Q7: Maps at 315°? D only
Answer: D) Regular Octagon.
• Octagon base angle = 45°
• 315° ÷ 45° = 7 (Exact match! ✓)
• 45, 90, 135, 180, 225, 270, 315°, 360°
• Square (90), Pentagon (72), Hexagon (60) do not divide 315 evenly.
Q8: Shared Mystery Angle (Hexagon & Octagon YES, Pentagon NO) θ = 180°
Step-by-step logic: We need a common multiple of Hexagon's base (60°) and Octagon's base (45°) that is NOT a multiple of Pentagon's base (72°).
Hexagon multiples: 60, 120, 180, 240...
Octagon multiples: 45, 90, 135, 180...
Pentagon multiples: 72, 144, 216... (180 is not here!)
*Note: 360° also maps all onto themselves, but 180° is the intended rotation strictly between 0° and 360°.
Reasoning & Coordinate Transformation Solutions
Q9: Rectangle
90°: NO. Rectangle becomes vertical; dimensions do not align with original horizontal orientation.
180°: YES. Opposite sides are parallel and congruent; horizontal orientation is restored.
Q10: Triangle at 180°
Answer: NO.
Explanation: The apex rotates from pointing UP to pointing DOWN. A regular triangle only maps onto itself at multiples of 120° (120°, 240°, 360°).
Q11: Rigid Motion
Answer: D (T-4,-4)
R180° changes (x,y) to (-x,-y). Both composite reflections (ry ∁ rx and rx ∁ ry) also yield (-x,-y). A translation shifts points without inverting coordinates.
Special Education Co-Teaching & Accommodation Strategies
1. Tracing Paper (Patty Paper)
Have students trace the polygon, put a pencil point in the center, and rotate until the outline snaps back into alignment.
2. Calculator Routine
Teach the "Test by Division" routine: target angle ÷ base angle = whole number? Yes = maps onto itself. Decimal = fails!
3. Odd vs. Even Side Anchor
Highlight that even-sided regular polygons always hit 180°, while odd-sided ones never do. Keep this on the board as a reference!
Geometry 3.08 • Rotational Symmetry Page 2 Answer Key
