The least angle measure is 180° for the specific question you’re referencing.

Quick Scoop: What this question is really asking

The phrase “least angle measure by which this figure can be rotated so that it maps onto itself” is about rotational symmetry.

  • A figure “maps onto itself” if, after you rotate it, it looks exactly the same as it did before.
  • The least angle is the smallest positive rotation (greater than 0° but less than 360°) that does this.

In the common multiple‑choice version of this problem with options 45°, 90°, 180°, and 360°, the worked solution shows that:

  • 45° does not map the figure onto itself.
  • 90° makes it perpendicular to the original position.
  • 180° does map it onto itself.
  • 360° also works, but it is larger than 180°.

So the correct (least) angle is 180°.

Mini concept: How these problems usually work

When you do not see the actual picture, you can still understand the pattern:

  • Many textbook problems use shapes that have half‑turn symmetry (like certain “double‑arrow” figures, some rectangles with special markings, or shapes that look the same upside‑down). In those, a 180° rotation sends the figure onto itself.
  • The least angle of rotation is always less than 360° unless the shape has no rotational symmetry at all.

For a regular polygon with nnn sides, the least angle that maps it onto itself is 360∘/n360^\circ /n360∘/n.

  • Example: a regular octagon has least angle 360∘/8=45∘360^\circ /8=45^\circ 360∘/8=45∘.

In your particular question, the solved example explicitly concludes that the given figure’s smallest such angle is 180° , not 45° or 90°.

Rotational symmetry: a quick story version

Imagine you place the figure on a transparent sheet and pin it at the center.

  • You slowly rotate the sheet.
  • For most angles, the figure looks “different.”
  • At 180° , suddenly it lines up perfectly with where it started; you can’t tell it was turned, just like flipping something upside‑down that looks the same that way.

Any further angles that also work (like 360°) are just multiples of that fundamental 180°.

Key term you might see in class

  • Angle of rotation : The smallest angle through which a shape is rotated so that it lies on itself again.

For the figure in the problem you’re referencing, that angle of rotation is 180°.

TL;DR:
For the usual textbook figure behind the question “what is the least angle measure by which this figure can be rotated so that it maps onto itself?” (with choices 45°, 90°, 180°, 360°), the answer is 180°.

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