Quadrilaterals and Polygons — Free Geometry Review Games.
This unit covers parallelogram properties, special quadrilaterals, polygon angle sums and regular polygons — essential concepts for Geometry. Use our interactive study games to test your understanding, or review questions in traditional format below.
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This unit covers parallelogram properties, special quadrilaterals, polygon angle sums and regular polygons — essential concepts for Geometry. Use our interactive study games to test your understanding, or review questions in traditional format below.
Key Concepts Breakdown
1 Parallelogram Properties
A parallelogram has two pairs of parallel sides, and this single fact produces four provable properties that exams test directly. Students must be able to set up and solve equations using opposite sides equal, opposite angles equal, consecutive angles supplementary, and diagonals bisecting each other. Any of these four properties may appear as the given or the unknown in an exam question.
Key Points
- Opposite sides are congruent: AB = CD and BC = DA
- Opposite angles are congruent: ∠A = ∠C and ∠B = ∠D
- Consecutive angles are supplementary: ∠A + ∠B = 180°
- Diagonals bisect each other (each diagonal cuts the other into two equal halves)
In parallelogram ABCD, ∠A = (3x + 10)° and ∠B = (5x − 6)°. Find x and both angle measures.
Since consecutive angles in a parallelogram are supplementary, set (3x + 10) + (5x − 6) = 180. Simplifying gives 8x + 4 = 180, so 8x = 176 and x = 22. Substituting back: ∠A = 76° and ∠B = 104°, and you can verify 76 + 104 = 180.
2 Special Quadrilaterals
Rectangles, rhombuses, and squares are all parallelograms, so they inherit all four parallelogram properties plus have their own additional ones. Exams frequently require students to identify which extra property distinguishes each shape, especially regarding diagonals. Trapezoids and isosceles trapezoids are not parallelograms and follow different rules.
Key Points
- Rectangle: all angles = 90°; diagonals are congruent (equal in length)
- Rhombus: all sides congruent; diagonals are perpendicular and bisect the vertex angles
- Square: has ALL properties of both rectangle and rhombus
- Isosceles trapezoid: one pair of parallel sides; base angles congruent; diagonals congruent
The diagonals of rhombus PQRS intersect at T. If PT = 6 and QT = 8, find the length of side PQ.
Rhombus diagonals are perpendicular bisectors of each other, so triangle PTQ is a right triangle with legs 6 and 8. Using the Pythagorean theorem: PQ² = 6² + 8² = 36 + 64 = 100, so PQ = 10. This is the length of every side of the rhombus.
3 Polygon Angle Sums
The interior angle sum of any polygon depends only on the number of sides, given by the formula (n − 2) × 180°. The exterior angle sum of any convex polygon is always 360°, regardless of the number of sides. Students must apply both formulas to find missing angles and must distinguish between interior and exterior angle questions.
Key Points
- Interior angle sum = (n − 2) × 180°, where n = number of sides
- Exterior angle sum of any convex polygon = 360° (always)
- One interior angle + its exterior angle = 180° (they form a linear pair)
- To find n given the angle sum, solve (n − 2) × 180 = given sum
The sum of the interior angles of a polygon is 1260°. How many sides does the polygon have?
Set up the equation (n − 2) × 180 = 1260. Divide both sides by 180 to get n − 2 = 7, then add 2 to find n = 9. The polygon is a nonagon (9 sides), and you can verify: (9 − 2) × 180 = 7 × 180 = 1260°.
4 Regular Polygons
A regular polygon has all sides congruent AND all angles congruent, so each interior angle equals the total interior sum divided by n. Exams test finding one interior angle, one exterior angle, and the number of sides given one of those angles. The relationship between interior and exterior angles (summing to 180°) is frequently used.
Key Points
- Each interior angle of a regular n-gon = (n − 2) × 180° ÷ n
- Each exterior angle of a regular n-gon = 360° ÷ n
- Interior angle + exterior angle = 180° for any single vertex
- If given one exterior angle, find n by dividing 360 by that angle measure
Each exterior angle of a regular polygon measures 24°. How many sides does it have, and what is each interior angle?
Since all exterior angles of a regular polygon sum to 360°, divide 360 ÷ 24 = 15 sides. Each interior angle is the supplement of the exterior angle: 180° − 24° = 156°. You can verify using the interior formula: (15 − 2) × 180 ÷ 15 = 13 × 180 ÷ 15 = 2340 ÷ 15 = 156°.
Questions, answered.
What is Quadrilaterals and Polygons?
Quadrilaterals and Polygons is Unit 6 of Geometry, covering parallelogram properties, special quadrilaterals, polygon angle sums and regular polygons.
How to study for Geometry Unit 6?
Start with the Quick Summary above, review the Key Concepts, then test yourself with our interactive study games. Aim for 80%+ accuracy before moving on.
How many questions are in this unit?
This unit has 28+ review questions across 5 different game modes.