Questions and Answers
If one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a:
If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.
True
If a parallelogram is both a ______ and a ______, then the parallelogram is a square.
rectangle, rhombus
If the diagonals of a parallelogram are congruent, then the parallelogram is a square.
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Quadrilaterals with four right angles are:
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Every square is a rhombus.
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The opposite sides of any rhombus are parallel and congruent.
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The diagonals of a rhombus must bisect each other.
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The diagonals of a rectangle must be perpendicular.
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A regular polygon is always convex.
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What is the sum of the interior angles of a quadrilateral?
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A quadrilateral with two pairs of parallel sides is a ______.
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What is the definition of a polygon?
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What is the sum of the exterior angle measures of a polygon?
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Study Notes
Properties of Parallelograms
- A parallelogram with one pair of consecutive sides congruent is classified as a rhombus.
- If the diagonals of a parallelogram are perpendicular, it is identified as a rhombus.
- A parallelogram becomes a rectangle if its diagonals are congruent.
- A parallelogram is defined as a rectangle if it contains at least one right angle.
- If the diagonals of a parallelogram bisect opposite angles, it qualifies as a rhombus.
- A quadrilateral is categorized as a parallelogram when both pairs of opposite sides are congruent.
- A quadrilateral becomes a parallelogram if opposite angles are congruent or if angles are supplementary to consecutive angles.
Types of Quadrilaterals
- Quadrilaterals featuring four right angles include squares and rectangles.
- A rhombus and a square have diagonals that are perpendicular to each other.
- A square, rhombus, rectangle, and general parallelogram have diagonals that bisect each other.
- Quadrilaterals that show two pairs of opposite sides also fall under parallelograms.
- Shapes where the diagonals bisect angles are either rhombuses or squares.
- Shapes categorized with congruent diagonals include rectangles and squares.
Definitions and Characteristics
- A polygon is defined as a closed plane figure formed by three or more line segments.
- Each segment of a polygon is termed a side, while the intersection of two sides is referred to as a vertex.
- A diagonal connects two non-consecutive vertices in a polygon.
- A regular polygon is both equilateral (all sides congruent) and equiangular (all angles congruent).
- A concave polygon possesses at least one diagonal that lies outside of the polygon, while a convex polygon has all diagonals inside.
Angles in Polygons
- The sum of interior angles for triangles is 180 degrees, while for quadrilaterals, it is 360 degrees.
- A pentagon's interior angles sum to 540 degrees, and a hexagon sums to 720 degrees.
- The sum of exterior angles of any polygon always equals 360 degrees.
Trapezoids
- A trapezoid is defined as a quadrilateral with exactly one pair of parallel sides.
- The parallel sides of a trapezoid are known as bases, while the non-parallel sides are called legs.
- Base angles of a trapezoid are two consecutive angles sharing a base.
- In an isosceles trapezoid, the legs are congruent, resulting in congruent base angles and congruent diagonals.
- The midsegment of a trapezoid connects the midpoints of the legs and is parallel to the bases, measuring half the sum of the bases' lengths.
Theorems and Properties
- The properties of parallelograms include having two pairs of opposite congruent sides and angles and bisecting diagonals.
- A kite's diagonals are always perpendicular, with only one pair of opposite angles being congruent.
- A parallelogram will have opposite sides that are both congruent and parallel.
True or False Statements
- Every square is a rhombus and every rhombus is a parallelogram.
- The opposite sides and angles of any rhombus are congruent.
- The diagonals of both rectangles and rhombuses bisect each other; however, rectangles do not have perpendicular diagonals.
- Congruent diagonals do not guarantee a square but do confirm a rectangle.
This organized structure outlines critical geometry concepts related to quadrilaterals, parallelograms, trapezoids, and their properties, aiding in understanding and recall.
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Test your knowledge of geometric properties related to parallelograms with these flashcards. Each card presents a key concept or definition that will help reinforce your understanding of this chapter. Perfect for quick review before exams!
