The sum of 2-sides of a triangles must be larger than the 3rd side.
Properties of a Parallelogram
Parallelogram
• Opposite sides are congruent.
• Opposite sides are parallel.
• Opposite angles are congruent.
• Diagonals bisect each other.
• Consecutive (adjacent) angles are supplementary (+ 180 degrees).
• Sum of the interior angles is 360 degrees.
Properties of a Rectangle
Rectangle
• All properties of a parallelogram.
• All angles are 90 degrees.
• Diagonals are congruent.
Properties of a Rhombus
Rhombus
• All properties of a parallelogram.
• Diagonals are perpendicular (form right angles).
• Diagonals bisect the angles.
Properties of a Square
Square
• All properties of a parallelogram.
• All properties of a rectangle.
• All properties of a rhombus.
Properties of an Isosceles Trapezoid
Isosceles Trapezoid
• Diagonals are congruent.
• Opposite angles are supplementary + 180 degrees.
• Legs are congruent
Median of a Trapezoid
DISTANCE FORMULA
MIDPOINT FORMULA
SLOPE FORMULA
PROVE PARALLEL LINES
EQUAL SLOPES
PROVE PERPENDICULAR LINES
OPPOSITE RECIPROCAL SLOPES (FLIP/CHANGE)
PROVE A PARALLELOGRAM
Prove a Parallelogram
• Distance formula 4 times to show opposite sides congruent.
• Slope 4 times to show opposite sides parallel (equal slopes)
• Midpoint 2 times of the diagonals to show that they share the same midpoint which means that the diagonals bisect each other.
How to prove a Rectangle
Prove a Rectangle
• Prove the rectangle a parallelogram.
• Slope 4 times, showing opposite sides are parallel and consecutive (adjacent) sides have opposite reciprocal slopes thus, are perpendicular to each other forming right angles.
How to prove a Square
Prove a Square
• Prove the square a parallelogram.
• Slope formula 4 times and distance formula 2 times of consecutive sides.
Prove a Trapezoid
Prove a Trapezoid
• Slope 4 times showing bases are parallel (same slope) and legs are not parallel.
Prove an Isosceles Trapezoid
Prove an Isosceles Trapezoid
• Slope 4 times showing bases are parallel (same slopes) and legs are not parallel.
• Distance 2 times showing legs have the same length.
Prove Isosceles Right Triangle
Prove Isosceles Right Triangle
• Slope 2 times showing opposite reciprocal slopes (perpendicular lines that form right angles) and Distance 2 times showing legs are congruent.
• Or Distance 3 times and plugging them into the Pythagorean Theorem
Prove an Isosceles Triangle
Prove an Isosceles Triangle
• Distance 2 times to show legs are congruent.
Prove a Right Triangle
Prove a Right Triangle
• Slope 2 times to show opposite reciprocal slopes (perpendicular lines form right angles).
Sum of the Interior Angles
180(n-2)
Measure of one Interior Angle
Measure of one interior angle
180( 2)n
n
Sum of an Exterior Angle
360 Degrees
Measure of one Exterior Angle
360/n
Number of Diagonals
2
)3( nn
1-Interior < + 1-Exterior < =
180 Degrees
Number of Sides of a Polygon
Ext1
360
Converse of PQ
Change OrderQP
Inverse of PQ
Negate
~P~Q
Contrapositive of PQ
Change Order and Negate
~Q~PLogically Equivalent: Same
Truth Value as PQ
Negation of P
Changes the truth value
~P
Conjunction
And (^)
P^QBoth are true to be true
Disjunction
Or (V)
P V Qtrue when at least one is true
Conditional
If P then QPQ
Only false when P is true and Q is false
Biconditional
(iff: if and only if)TT =TrueF F = True
Locus from 2 points
The locus of points equidistant from two points, P and Q, is the perpendicular bisector of the line segment
determined by the two points.
Locus of a Line
Set of Parallel Lines equidistant on each side of the line
Locus of 2 Parallel Lines
3rd Parallel Line Midway in between
Locus from 1-Point
Circle
Locus of the Sides of an Angle
Angle Bisector
Locus from 2 Intersecting Lines
2-intersecting lines that bisect the angles that are formed by the intersecting lines
Reflection through the x-axis
(x, y) (x, -y)
Reflection in the y-axis
(x, y) (-x, y)
Reflection in line y=x
(x, y) (y, -x)
Reflection in the origin
(x, y) (-x, -y)
Rotation of 90 degrees
(x, y) (-y, x)
Rotation of 180 degrees
(x, y) (-x, -y)Same as a reflection in the
origin
Rotation of 270 degrees
(x, y) (y, -x)
Translation of (x, y)
Ta,b(x, y) (a+x, b+y)
Dilation of (x, y)
Dk (x, y) (kx, ky)
Isometry
Isometry: Transformation that Preserves Distance
• Dilation is NOT an Isometry
• Direct Isometries
• Indirect Isometries
Direct Isometry
Direct Isometry
• Preserves Distance and Orientation (the way the vertices are read stays the same)
• Translation
• Rotation
Opposite Isometry
Opposite Isometry
• Distance is preserved
• Orientation changes (the way the vertices are read changes)