SUPPLEMENTARY ANGLES

2-angles that add up to 180 degrees.

COMPLEMENTARY ANGLES

2-angles that add up to 90 degrees

Vertical Angles are congruent to each other

PARALLEL LINES CUT BY A TRANSVERSAL

SUM OF THE INTERIOR ANGLES OF A TRIANGLE

180 DEGREES

LARGEST ANGLE OF A TRIANGLE

ACROSS FROM THE LONGEST SIDE

SMALLEST ANGLE OF A TRIANGLE

ACROSS FROM THE LONGEST SIDE

LONGEST SIDE OF A TRIANGLE

ACROSS FROM THE LARGEST ANGLE

SMALLEST SIDE OF A TRIANGLE

ACROSS FROM THE SMALLEST ANGLE

TRIANGLE INEQUALITY THEOREM

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)

• Reflection

• Glide Reflection

What Transformation is NOT an Isometry?

Dilation

Area of a Triangle

bh2

1 Triangle a of Area

Area of a Parallelogram

Area of a Rectangle

Area of a Trapezoid

)(2

1 Area 21 bbhTrapezoid

Area of a Circle

Circumference of a Circle

Surface Area of a Rectangular Prism

Surface Area of a Triangular Prism

)()()()2

1( 332211 hbhbhbbhSA

Surface Area of a Trapezoidal Prism

)()()()()](2

1[ 4433221121 hbhbhbhbbbhSA

H

Surface Area of a Cylinder

Surface Area of a Cube

)(6 2SSA

Volume of a Rectangular Prism

Volume of a Triangular Prism

HbhV )2

1(

Volume of a Trapezoidal Prism

prism theofHeight H

trapezoid theofheight h

)](2

1[ 21

HbbhV

H

Volume of a Cylinder

Volume of a Triangular Pyramid

pyramid theofheight H

triangle theofheight h

]2

1[

3

1

HbhV

Volume of a Square Pyramid

pyramid theofheight H

square a of sideS

][3

1 2

HSV

Volume of a Cube

cube a of sideS

3

SV