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Ways to Prove a Quadrilateral

Right Triangle
(A triangle is a right angle...)

1.. if it contains one right angle (show slopes are negative reciprocals, therefore perpendicular sides, therefore right angles, therefore right angle)
2...if the lengths of its side satisfy the converse of the Pythagorean Theorem (use distance formula to get lengths of the hypotenuse and legs,show a + b =c)

Isosceles Triangle
(A triangle is an isosceles triangle..)

1... if two sides are congruent.
(use distance formula)


1. A quadrilateral is a trapezoid if one pair of opposite sides are parallel and the other pair of opposite side are not parallel (show parallel by slopes being equal, and non-parallel by slopes being non-equal).
2. A trapezoid is an isosceles trapezoid if the pair of non-parallel sides are congruent.(use distance formula).
3. A trapezoid is an isosceles trapezoid if it diagonals are congruent
(use distance formula)

(A quadrilateral is a parallelogram if..)

1...if both pairs of opposite sides are parallel (show slopes equal to show parallel lines)
2...if both pairs of opposite sides are congruent (use distance formula)
3...if one pair of opposite sides is both parallel and congruent (slopes equal and distance formula)
4...if the diagonals bisect each other (use midpoint formula to show that each diagonal has the same midpoint, therefore they bisect each other).

(A quadrilateral is a rectangle if it is...)

1... a parallelogram with one right angle (first show parallelogram, then show slopes of ones belonging to two consecutive sides are negative reciprocals of each other, therefore the two lines are perpendicular, therefore they form a right angle)
2...equiangular (show four pair s of perpendiculars by showing four sets of negative reciprocal slopes, which show four right angles (equiangular))
3...a parallelogram with congruent diagonals (after showing parallelogram, show diagonals congruent using distance formula)

(A quadrilateral is a rhombus if it is...)

1...equilateral (all four sides congruent - distance formula)
2. a parallelogram with two congruent consecutive sides (after showing parallelogram, use distance formula to show two consecutive sides congruent)
3....a parallelogram whose diagonals are perpendicular to each other (show parallelogram, then show slopes of diagonals are negative reciprocals (perpendicular lines), or use converse of Pythagorean Theorem (show perpendicular lines))

(A quadrilateral is a square if it is..)

1.. a rectangle and a rhombus (above)
2...a rectangle with two consecutive sides congruent.(show rectangle, distance formula),(rectangle with perpendicular diagonals, show slopes negative reciprocals or 2)
3....a rhombus with one right angle (show rhombus, then show slopes are negative reciprocals), (or show rhombus with congruent diagonals, distance formula)
4... is both equilateral ( show four congruent sides, distance formula) and equiangular, (show four sets of perpendiculars by showing four sets of negative reciprocal slopes)

Formula: Distance

Distance =
(Rad) (x2-x1)2 + (y2-y1)2

Formula: Slope


Formula: Midpoint

(x1+x2/2 , (y1+y2/2)

Horizontal Line = a
Vertical Line = b

a. = 0
b. = no slope

Area of Rectangle

A = bh or A = lw

Area of a Square

A = s2

Area of a Parallelogram

A = bh

Area of a Triangle

A = 1/2 bh

Area of a Trapezoid

A = 1/2 (b1 + b2) h

Area of a Rhombus

A = 1/2 (d1 X d2)

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