Polygons, Triangles, Quadrilaterals, and Applications

A polygon is a closed plane figure made up of three or more segments (sides) that intersect only at their endpoints (vertices).

If a figure is not closed, sides intersect between endpoints or are collinear, does not have segments forming all sides, or is not a plane figure, then it is not a polygon.

Examples of polygons include triangles, quadrilaterals, pentagons, and hexagons, each defined by the number of sides they possess.

Polygons are named by their vertices, listing the letters consecutively (by going around a polygon’s vertices).

Classifying Polygons — Number of Sides

< 3 sides = Not possible

3 sides = triangle

4 sides = quadrilateral

5 sides = pentagon

6 sides = hexagon

7 sides = heptagon

8 sides = octagon

9 sides = nonagon

10 sides = decagon

11 sides = hendecagon

12 sides = dodecagon

n sides = n-gon (For any number of sides > 12)

Classifying Polygons — Convex vs. Concave

Diagonal — A segment that joins two nonconsecutive vertices of a polygon.

Convex polygon — Has no diagonals with points outside the polygon.

Concave polygon — Has at least one diagonal with points outside the polygon.

Classifying Polygons — Irregular vs. Regular

Regular polygon — Equilateral and equianglar

Irregular polygon — Neither equilateral nor equiangular

Note: If a polygon is equilateral then it is equiangular. There are some cases where the opposite is true.

Formula — Sum of Interior Angles of a Polygon: (n - 2) * 180 degrees

Theorems:

  • The sum of the interior angles of a convex polygon is: (n-2) * 180 where n is the number of sides.

  • The sum of the exterior angles of a convex polygon is: 360 degrees

  • The sum of the interior angles of a triangle is 180 degrees.

  • The sum of the interior angles of a square is 360 degrees.

  • Etc.

Triangles:

  • Equilateral — All 3 sides are congruent

  • Isoceles — 2 sides are congruent

  • Scalene — No sides are congruent

  • Acute — All 3 angles acute

  • Equiangular — All 3 angles are congruent

  • Right — Contains 1 right angle

  • Obtuse — Contains 1 obtuse angle

Angle/Side Relationships in Triangles:

  • The smallest angle is opposite the smallest side.

  • The largest angle is opposite the largest side.

  • For isosceles triangles:

    • If a triangle has 2 congruent sides then the angles opposite them are congruent.

      • (The converse is also true)

  • For equilateral triangles:

    • If a triangle is equilateral then it is equiangular.

      • (The converse is also true)

Triangle Inequality Theorem: The sum of two side lengths of a triangle is greater than the third side length. (Ex: Side lengths 1, 2, and 3 do not form a triangle, but side lengths 1, 3, and 4 works)

Pythagorean Theorem Converse

  • If a² + b² = c² then a triangle is a right triangle, where a and b are the smallest side lengths.

  • If a² + b² > c² then a triangle is acute, where a and b are the smallest side lengths.

  • If If a² + b² < c² then a triangle is obtuse, where a and b are the smallest side lengths.

Special Segments of Triangles

  • Median — A segment from a vertex of the triangle to the midpoint of the opposite side. The there medians of a triangle meet at a point called the centroid.

  • Altitude — A segment from a vertex of the triangle perpendicular to the opposite side. The three altitudes of a triangle meet at a point called the orthocenter.

  • Midsegment — A segment from the midpoint of one side of the triangle to the midpoint of another side.

    • The Midsegment Theorem: The midsegment is parallel to the base segment, and the base segment is equivalent to the length of two midsegments.

Classifying Quadilaterals

If a quadrilateral is a parallelogram then…

  • Its opposite sides are congruent.

  • Its opposite sides are parallel.

  • Its opposite angles are congruent.

  • Its consecutive angles are supplementary.

  • Its diagonals bisect each other.

How to prove if a Quadrilateral is a Parallelogram

  • If BOTH pairs of opposite sides of a quadrilateral are congruent.

  • If BOTH pairs of opposite sides of a quadrilateral are parallel.

  • If BOTH sides of opposite angles of a quadrilateral are congruent.

  • If an angle of a quadrilateral is supplementary to BOTH of its consecutive angles.

  • If BOTH diagonals of a quadrilateral bisect each other.

  • If the SAME pair of opposite sides of a quadrilateral is congruent and parallel.

Rhombuses

  • If a quadrilateral is a rhombus, then all sides are congruent.

  • If a quadrilateral is a rhombus, then diagonals are perpendicular, forming 90-degree angles.

  • If a quadrilateral is a rhombus, then diagonals bisect the angle.

  • Plus all properties of a parallelogram.

Rectangles

  • If a quadrilateral is a rectangle then all angles are right angles.

  • If a quadrilateral is a rectangle then diagonals are congruent.

  • Plus all properties of a parallelogram.

Squares: Both Rhombuses and Rectangles

  • If a quadrilateral is a square then all sides are congruent and right angles.

Not a Parallelogram: Trapezoid

  • If a quadrilateral is a trapezoid then the bases are parallel.

  • Parts of a Trapezoid:

    • Bases - Parallel sides.

    • Legs - Nonparallel sides.

    • Pair of base angles — Two angles that share a base.

  • Midsegment of a Trapezoid

    • Definition: A segment connecting the midpoints of the legs.

    • Theorems:

      • The midsegment is parallel to the bases.

      • The midsegment’s length is 0.5(b1+b2) the sum of the bases’ lengths.

        • Where b1 = base 1 and b2 = base 2

  • Isosceles Trapezoid

    • If a quadrilateral is an isosceles trapezoid then legs are congruent.

    • If a quadrilateral is an isosceles trapezoid then base angles are congruent.

    • If a quadrilateral is an isosceles trapezoid then diagonals are congruent.

Not a Parallelogram: Kite

  • If a quadrilateral is a kite, then 2 pairs of consecutive sides are congruent, but no opposite sides are congruent.

  • If a quadrilateral is a kite then diagonals are perpendicular.

  • The diagonals bisect the angles in a Kite.