12 multiple-choice questions, progressively harder.
Which classification is justified by the five-sided boundary shown?
Solution
Correct answer: C
The boundary has five straight sides and no crossing, so it is a simple pentagon.
The inward corner at C is reflex, and segment BD leaves the polygonal region. Thus the pentagon is concave.
A convex parallelogram has four congruent sides and two different interior angle measures, 70 degrees and 110 degrees. Which description is justified?
Correct answer: A
The four congruent sides make it equilateral.
The two stated angle measures differ, so the polygon is not equiangular and not regular.
A rectangle has consecutive side lengths 6 cm and 11 cm. Which name is not applicable?
Correct answer: D
The rectangle is a parallelogram and hence an inclusive trapezoid.
Its unequal adjacent side lengths prevent all four sides from being congruent, so it is not a rhombus.
Which claim is true under the inclusive hierarchy?
Correct answer: B
A rhombus is convex and has four congruent sides.
They can be grouped into two disjoint adjacent pairs, which meets the kite definition. The reverse or stronger claims require conditions that are not guaranteed.
A simple nonagon has vertices A through I in boundary order. How many diagonals have endpoint A?
From A, do not join A to itself, and exclude its two adjacent vertices B and I.
The six remaining vertices give six diagonals.
A turned convex quadrilateral has two parallel side pairs and four right-angle marks. No side lengths are supplied. What most specific name is guaranteed?
The parallel pairs and four right angles meet the rectangle definition.
There is no evidence that adjacent side lengths are equal, so the more restrictive square condition is not guaranteed.
Which extra condition must accompany congruent sides to make a convex polygon regular?
Regularity requires the equilateral and equiangular conditions together.
Diagonal, parallel-side, or orientation conditions do not replace congruent interior angles.
An unmarked convex regular 15-gon has what least positive matching rotation?
Fifteen equal rotation steps complete one full turn.
Which describes the reflection axes of an unmarked convex regular heptagon?
An odd side count has no pair of opposite vertices.
Each of the seven vertices supplies one mirror that meets the midpoint of the opposite side.
An unmarked convex regular dodecagon has 12 sides. Which statement about a half-turn is correct?
The least rotation step is found from the full turn.
A half-turn therefore maps every vertex and joining side onto the polygon.
Two equal-side pairs in a proposed kite use AB with BC and BC with CD. Why is this insufficient for the definition?
The definition requires two disjoint adjacent pairs.
Here the pairs overlap at side BC, and side DA belongs to neither. Four equal sides are permitted; reused sides are not.
For a trapezoid with a chosen pair of parallel bases, what is the defining isosceles condition used in this course?
Our inclusive isosceles-trapezoid definition uses congruent angles at one chosen base.
A rectangle meets this condition, since its base angles are right angles. Later proofs establish other properties and tests.
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