THE GEOMETRY OF POLYGONS

360°

$$x=360-85-95-110=70°$$

$$90+90+x+x=360 \Rightarrow 2x=180 \Rightarrow x=90°$$

A rectangle (or square, if the sides are also all equal) — any quadrilateral with all four angles equal to 90° qualifies.

2 triangles.

$2 \times 180° = 360°$ — this matches the known angle sum of a quadrilateral exactly.

$$3 \times 180° = 540°$$

The student found the sum of the three KNOWN angles but then incorrectly used that same number as the fourth angle, instead of subtracting it from the total 360° to find what remains. Correct: $360-275=85°$.

$100+85+90+85=360°$ ?

$$x=360-88-92-95=85°$$

The claim is incorrect — a true rectangle requires ALL FOUR angles to be exactly 90°. Here, the angles (88°, 92°, 95°, 85°) are close to but not exactly 90°, so the plot is an irregular quadrilateral, not a true rectangle.

$$x=360-92-88-95=85°$$

$x=360-70-110-70=110°$. Since the angles form two EQUAL pairs (70°,70° and 110°,110°), this could represent a parallelogram (opposite angles equal) — though confirming this fully would require checking the SIDE lengths too, not just angles.

From vertex $A$, only ONE diagonal can be drawn (to the opposite vertex $C$) — diagonals to the two ADJACENT vertices ($B$ and $D$) would just be the existing SIDES of the quadrilateral, not diagonals.

The diagonal from $A$ to $C$ divides quadrilateral $ABCD$ into triangle $ABC$ and triangle $ACD$ — together, these two triangles' angles EXACTLY make up all four angles of the original quadrilateral (with no gaps or overlaps). Since each triangle has an angle sum of 180°, the two triangles combined give a total angle sum of $2\times180°=360°$, which is exactly the quadrilateral's angle sum.

$200+60+50+50=360°$ ? — the sum is correct.

An interior angle greater than 180° corresponds to a REFLEX angle — this means the quadrilateral would need to be CONCAVE (non-convex) at that vertex, with the shape 'caving inward' rather than all vertices pointing outward. While such a concave quadrilateral IS technically a valid quadrilateral (angles summing to 360° still holds even for concave shapes), it would NOT be a simple convex quadrilateral like a typical rectangle or parallelogram — the 200° angle signals the shape must have an unusual, non-convex form.

The student found the SUM of the three known angles but then incorrectly used that number AS the fourth angle, rather than subtracting it from 360° to find the REMAINING angle. Correct: $360-285=75°$.

$88+95+102+75=360°$ ?.

A RECTANGLE (that isn't also a square) has all four angles equal to 90° each, but its SIDES are not all equal length (it has two pairs of different-length sides) — a square specifically requires BOTH all angles equal to 90° AND all sides equal in length; equal angles alone (as in any rectangle) is not sufficient to guarantee a square.

$$x=360-85-95-90=90°$$

Since only ONE corner (90°) matches a true right angle, a rectangular raised bed would fit PERFECTLY only in that specific corner — attempting to place a rectangular bed elsewhere (near the 85° or 95° corners) would leave AWKWARD GAPS or require the bed to be angled/cut to match the plot's actual irregular shape. The design might need custom-shaped beds (trapezoidal or otherwise irregular) for the non-90° corners, or the raised beds could be positioned centrally, away from the irregular edges entirely, to avoid the fitting problem.

$$360-130-70-70=90°$$

Among all quadrilaterals with a fixed perimeter, the SQUARE (with all sides and angles equal) is known to maximize the enclosed area — any deviation from this perfectly regular, symmetric shape (like stretching one angle to 130° while compressing others) typically REDUCES the enclosed area for the same total perimeter, since irregular or elongated shapes tend to 'waste' boundary length relative to the area they enclose, compared to a more compact, symmetric shape like a square.