Question

which of these shapes is an example of a regular polygon?

select one:
a. a right triangle.
b. a pentagon where all sides and angles are equal.
c. a parallelogram with opposite sides equal.
d. a rhombus with two \\(60^\circ\\) angles and two \\(120^\circ\\) angles.

which of the following is not a characteristic of a concave polygon?

select one:
a. at least one interior angle is greater than \\(180^\circ\\).
b. all interior angles are less than \\(180^\circ\\).
c. at least one diagonal lies outside the polygon.
d. it may have an indentation.

a convex polygon must have which of the following properties?

select one:
a. all diagonals are inside the polygon.
b. its exterior angles sum to more than \\(360^\circ\\).
c. it has at least one reflex angle.
d. at least one diagonal is outside the polygon.

Explanation:

Identify the regular polygon

$$ \text{Regular Polygon} \iff \text{Equilateral (all sides equal)} \land \text{Equiangular (all angles equal)} $$

$$ \text{Option b: A pentagon where all sides and angles are equal} \implies \text{Regular Polygon} $$

Identify the non-characteristic of a concave polygon

$$ \text{Concave Polygon} \implies \exists \text{ interior angle } > 180^\circ \land \exists \text{ diagonal outside the polygon} $$

$$ \text{Option b: All interior angles are less than } 180^\circ \implies \text{NOT characteristic of concave (it is characteristic of convex)} $$

Identify the property of a convex polygon

$$ \text{Convex Polygon} \implies \forall \text{ interior angles } < 180^\circ \land \forall \text{ diagonals lie entirely inside the polygon} $$

$$ \text{Option a: All diagonals are inside the polygon} \implies \text{True property of convex polygon} $$

Answer:

Question 1

• a. A right triangle.
• b. A pentagon where all sides and angles are equal. (Correct answer)
• c. A parallelogram with opposite sides equal.
• d. A rhombus with two 60° angles and two 120° angles.

Question 2

• a. At least one interior angle is greater than 180°.
• b. All interior angles are less than 180°. (Correct answer)
• c. At least one diagonal lies outside the polygon.
• d. It may have an indentation.

Question 3

• a. All diagonals are inside the polygon. (Correct answer)
• b. Its exterior angles sum to more than 360°.
• c. It has at least one reflex angle.
• d. At least one diagonal is outside the polygon.