Geometry Chapter 7 Flashcards

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.
• 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.