Inequalities and Parallelograms 😒

Trichotomy property

X > Y

X < Y

X = Y

Addition and subtraction property of inequality

FOR REAL NUMBERS a,b,c,d:

if a > b then a + c > b + c

if a > b then a - c > b - c

if a > b and c (is greater than or equal to) d then a + c > b + d

Multiplication and division property of inequality

FOR REAL NUMBERS x,y,z:

if x > y then: x times z > y times z (if z < 0, x times z < y times z)

if z = 0: x times z = y times z

(JUST LOOK AT SHEET BC I CANT MAKE THIS MAKE SENSE)

The transitive property of inequality

if x > y and y > z then x > z

The whole is greater than the some of its parts

If a = b+c and c > 0 then a > b

Theorem (Longer Side Larger Angle)

In a triangle, the longer sides are opposite the larger angles, and the shorter sides are opposite the smaller angles

Converse (Larger Angle Longer Side)

In a triangle, the larger angles are opposite the longer sides, and the smaller angles are opposite the shorter sides.

The Exterior Angle Inquality Theorem

The exterior angle of a triangle is greater than either remote interior angle

The Triangle Inequality Theorem

The Sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Extension:

the difference of the lengths of any two sides of a triangle is less than the length of the third side

Corollary 1

The perpendicular segment from a point to a line is the shortest segment from the point to the line

Corollary 2

The perpendicular segment from a point to a plane is the shortest segment from the point to the plane.

If a quad is a parallelogram, the opposite sides are parallel

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opposite sides of a parallelogram are congruent

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opposite angles of a parallelogram are congruent

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consecutive angles of a parallelogram are supplement

yayayayay

a diagonal of a parallelogram creates 2 congruent triangles

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The diagnals of a parallelogram bisect eachother

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