Geometry Chapter 7 Flashcards

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.
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.

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.