Questions and Answers
What is a diagonal?
What does the Polygon Interior Angles Theorem state?
(n-2)*180
The sum of the interior angles of a quadrilateral is 360 degrees.
True
What defines an equilateral polygon?
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What characterizes a regular polygon?
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The sum of the measures of the exterior angles of a convex polygon is 360 degrees.
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What defines a parallelogram?
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What is true about the opposite sides of a parallelogram?
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A rhombus can be defined as:
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A rectangle is a parallelogram with four right angles.
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What is a trapezoid?
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What is the sum of the angles in a rectangle?
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What is a kite?
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Study Notes
Key Definitions in Geometry
- Diagonal: A line segment connecting nonconsecutive vertices of a polygon.
- Polygon Interior Angles Theorem: For a convex polygon with n sides (n-gon), the sum of its interior angles is calculated as (n-2) × 180 degrees.
- Corollary to the Polygon Interior Angles Theorem: The interior angles of a quadrilateral always sum up to 360 degrees.
Types of Polygons
- Equilateral Polygon: All sides are of equal length.
- Equiangular Polygon: All interior angles are equal.
- Regular Polygon: A polygon that is both equilateral and equiangular.
Polygon Exterior Angles Theorem
- The exterior angles of a convex polygon, with one angle per vertex, always sum up to 360 degrees.
Characteristics of Parallelograms
- Parallelogram: A four-sided figure with both pairs of opposite sides parallel.
- Opposite Sides Theorem: In a parallelogram, opposite sides are equal in length (congruent).
- Opposite Angles Theorem: Opposite angles in a parallelogram are congruent.
- Consecutive Angles Theorem: Consecutive angles in a parallelogram are supplementary (sum to 180 degrees).
- Diagonals Theorem: In a parallelogram, the diagonals bisect each other.
Converse Theorems for Parallelograms
- Opposite Sides Converse: If both pairs of opposite sides are equal in a quadrilateral, it is a parallelogram.
- Opposite Angles Converse: If both pairs of opposite angles are equal, it is a parallelogram.
- Opposite Sides Parallel and Congruent Theorem: If one pair of sides is both congruent and parallel, the quadrilateral is a parallelogram.
- Diagonals Converse: If the diagonals bisect each other, the quadrilateral is a parallelogram.
Types of Quadrilaterals
- Quadrilateral: A polygon with four sides.
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Types of Polygons by Sides:
- Pentagon: 5 sides
- Hexagon: 6 sides
- Heptagon: 7 sides
- Octagon: 8 sides
- Nonagon: 9 sides
- Decagon: 10 sides
- Hendecagon: 11 sides
- Dodecagon: 12 sides
Special Parallelograms
- Rhombus: A parallelogram with four congruent sides.
- Rectangle: A parallelogram with four right angles.
- Square: A parallelogram that is both a rhombus and a rectangle.
Corollaries of Special Parallelograms
- Rhombus Corollary: A quadrilateral is a rhombus if it has four congruent sides.
- Rectangle Corollary: A quadrilateral is a rectangle if it has four right angles.
- Square Corollary: A quadrilateral is a square if it meets the conditions of both a rhombus and a rectangle.
Diagonal Theorems
- Rhombus Diagonals Theorem: A parallelogram is a rhombus if its diagonals are perpendicular.
- Rhombus Opposite Angles Theorem: A rhombus is identified when each diagonal bisects a pair of opposite angles.
- Rectangle Diagonals Theorem: A parallelogram is identified as a rectangle if its diagonals are congruent.
Trapezoids
- Trapezoid: A quadrilateral with exactly one pair of parallel sides.
- Isosceles Trapezoid: A trapezoid where the non-parallel (legs) sides are congruent.
- Base Angles Theorem: If a trapezoid is isosceles, then the base angles are equal.
- Base Angles Converse: If a trapezoid has a pair of congruent base angles, it is isosceles.
- Isosceles Trapezoid Diagonals Theorem: An isosceles trapezoid has congruent diagonals.
Midsegments and Kites
- Midsegment of a Trapezoid: Connects the midpoints of the legs and is parallel to the bases, measuring half the sum of the bases' lengths.
- Kite: A quadrilateral with two pairs of adjacent congruent sides and no opposite sides congruent.
- Kite Diagonals Theorem: Diagonals of a kite are perpendicular.
- Kite Opposite Angles Theorem: Exactly one pair of opposite angles in a kite are congruent.
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Description
Test your knowledge of key concepts from Chapter 7 of Big Ideas Math Geometry with these flashcards. This quiz covers essential terms and theorems related to polygons and angles, such as the Diagonal and the Polygon Interior Angles Theorem.
