U6 - Quadrilaterals

U6: Rectangles are also...

-parallelograms

-quadrilaterals

U6: Trapezoid median

Segment between the midpoint of the non parallel sides of a trapezoid

(Average of the lengths of the 2 bases) —> Also parallel to the bases

Median = (b1 + b2)/2

U6: If a parallelogram has congruent diagonals then its a ———-

If it has perpendicular diagonals it's a ———

If both it's a ——-

Rectangle

Rhombus or kite

Square

U6: Converting units

Multiple fractions till the units cancel out (top and bottom)

Multiply by fractions/units equivalent to 1

U6: Kite/Rhombus area formula

(D1xd2)/2 (splitting into triangles)

U6: Trapezoid area formula

(b1+b2)h/2

—> (two parallelograms)

U6: Perpendicular Bisector

A line that bisects a segment (breaks into two congruent parts), and is perpendicular to the segment (forms right angles)

• Equidistant from endpoints.

• Perpendicular to base.

U6: If a parallelogram has congruent diagonals it is a _________

If a parallelogram has perpendicular diagonals it is a ________.

If it's both that it's a _________.

Rectangle, rhombus, square

U6: Congruent Supplements Theorem

If two angles are supplementary to the same angle (or to congruent angles), then they are congruent.

U6: How are base and height related on a triangle?

Base and height are perpendicular. Height is also called altitude

-base can be any side

U6: Cavaliers principle

2 figures have the same area if they have the same height and width at every point along that height

U6: Rectangle (characteristics/how to prove)

Quadrilateral with 4 right angles

• Diagonals are congruent and bisect eachother

  • They are NOT perpendicular (exception = square)

• 4 right angles

• Parallelogram

U6: Rhombus (characteristics/how to prove)

A quadrilateral with four congruent sides and 4 congruent angles

• 4 congruent sides

• 4 congruent angles

• Paralleogram

• Diagonals are perpendicular bisectors

  • Bisect eachother

  • Form right angles

  • NOT equal

U6: Trapezoid (characteristics/how to prove)

A quadrilateral with exactly one pair of parallel sides

• NOT a paralleogram (only one pair parallel)

  • Non parallel sides = legs, parallel sides = bases

  • the diagonals are generally not congruent and do not bisect each other

U6: Isosceles Trapezoid

An isosceles trapezoid is a trapezoid where the two non-parallel sides (legs) are equal in length. This also means that the base angles are congruent (equal in measure). 

• One pair of parallel and unequal opposite sides (bases

• One pair of congruent non parllel sides (legs

• Diagonals are congruent

  • Do not bisect eachother

• Any lower base angle is supplementary to any upper base angle

U6: Square

A quadrilateral with 4 congruent sides and 4 right angles

• is also (by definition) a rectangle and a rhombus

Characteristics:

• Four equal sides

• Four right angles

• Diagonals are equal in length:

• Diagonals are perpendicular bisectors

  • A square is a rhombus so it shares the same characteristics, diagonals are perpendicular bisectors

  Regular polygon: Because all sides and angles are equal, a square is a type of regular polygon. 

U6: Area of a regular pentagon

(Apothem x perimeter)/2

U6: Equation for the rhombus perimeter (in terms of diagonals)

2√p2+q2

U6: Ways to prove a quadrilateral is a parallelogram

1) Prove both pairs of opposite sides are congruent:

2) Prove both pairs of opposite sides are parallel:

3) Prove one pair of opposite sides is both congruent and parallel:

4) Prove that the diagonals bisect each other:

5) Prove that opposite angles are congruent:

6) Prove consecutive angles are supplementary:

U6: Midsegments of Triangles

Segment between midpoints of 2 sides of a triangle, they are parallel to the opposite side and are half the length of the side they are parallel to

• if all midsegmnets are constructed, the perimeter is ½ the actual permimiter

U6: If a paralleogram has congruent diagonals, its a _______ If it has perpendicular diagonals, its a ____, if both it is a ______.

Congruent = Rectangle

Perpendicular = Rhombus

Both = Square

U6:

U6: Consecutive Angles

Angles in a polygon next to eachother are called consecutive angles, which are supplementary.