Question

1. which regular polygon can completely tessellate a plane?

an equilateral triangle
a obtuse scalene triangle
a regular decagon
a regular nonagon

Explanation:

Step1: Recall the formula for the interior angle of a regular polygon

The formula for the interior angle of a regular polygon is \(\theta=\frac{(n - 2)\times180^{\circ}}{n}\), where \(n\) is the number of sides of the polygon. For a polygon to tessellate a plane, the interior angle \(\theta\) must divide \(360^{\circ}\) evenly (\(360\div\theta\) is an integer).

Step2: Calculate the interior angle of an equilateral triangle

For an equilateral triangle \(n = 3\). Using the formula \(\theta=\frac{(3 - 2)\times180^{\circ}}{3}=60^{\circ}\). And \(360\div60 = 6\) (an integer).

Step3: Analyze the obtuse scalene triangle

A scalene triangle is not a regular polygon (sides and angles are not equal), so it does not meet the requirement of a "regular polygon" in the question.

Step4: Calculate the interior angle of a regular decagon

For a regular decagon \(n = 10\). \(\theta=\frac{(10 - 2)\times180^{\circ}}{10}=144^{\circ}\). And \(360\div144 = 2.5\) (not an integer).

Step5: Calculate the interior angle of a regular nonagon

For a regular nonagon \(n=9\). \(\theta=\frac{(9 - 2)\times180^{\circ}}{9}=140^{\circ}\). And \(360\div140=\frac{18}{7}\approx2.57\) (not an integer).

Answer:

an equilateral triangle