Coordinate Proofs Properties and Formulas (Ch. 6)

Proving Quadrilaterals Properties/Formulas flashcards

Parallelogram

Properties (THE BIG 5):

• Opposite sides are congruent

• Opposite sides are parallel

• Opposite angles are congruent

• Consecutive angles are supplementary

• Diagonals bisect each other

To Prove: (ONE of the following)

• Diagonals bisect each other

• Both pairs of opposite sides are parallel

• Both pairs of opposite sides are congruent

• One pair of opposite sides is parallel and congruent

Rectangle

Properties:

• The big 5

• 4 right angles (equiangular)

• Diagonals are congruent

To Prove: (Prove it is a parallelogram and ONE of the following)

• One right angle (Opposite recipricol slopes)

• Diagonals are conrgeunt

Rhombus

Properties:

• The big 5

• 2 congruent consecutive sides (equilateral)

• Diagonals are perpendicular

• Diagonals bisect opposite angles

To Prove: (Prove it is a parallelogram and ONE of the following)

• Diagonals are perpendicular

• 2 adjacent sides are congruent

Square

Properties:

• The big 5

• Rectangle with 2 congruent consecutive sides

• equilateral

• 4 right angles

• Diagonals bisect the angles

• Diagonals are congruent and perpendicular

To Prove:

• Prove it is a rectangle and that 2 adjacent sides are congruent.

• Prove it’s a rhombus and has one right angle.

Trapezoid

Properties:

• Only 2 parallel sides

• Parallel sides: bases

• Non-parallel sides: legs

To Prove:

• Prove that 1 pair of opposite sides are parallel and the other pair of sides are not parallel.

Isosceles trapezoid

Properties:

• Legs are congruent

• Base angles are congruent

• The mid-segment is parallel to the bases

• The length of the mid-segment is equal to ½ the sum of the lengths of the bases

To Prove: (Prove the figure is a trapezoid and ONE of the following)

• Legs are congruent

• Diagonals are congruent

Right Trapezoid

Properties:

• One leg is perpendicular to the base

To Prove:

• Prove figure is a trapezoid and one leg is perpendicular to a base.

Kite

Properties:

• Quadrilateral with 2 pairs of consecutive sides congruent and no opposite sides congruent.

• Diagonals are perpendicular

To Prove:

• Prove 2 consecutive sides are congruent and the opposite sides are not congruent.

• The longer diagonal is the perpendicular bisector of the shorter diagonal

Line segments are congruent → lengths are equal

Distance Formula

d = √(x₂ - X₁)² + (Y₂ - Y₁)²

Lines are parallel → slopes are equal

Slope Formula

m = (Y₂ - Y₁)/(X₂ - X₁)

Line segments bisect each other → midpoints are the same

Midpoint Formula

M_x = X₁ + X₂/2

M_y = Y₁ + Y₂/2

Lines are perpendicular → slopes are negative reciprocals

Slope Formula

m = (Y₂ - Y₁)/(X₂ - X₁)