Common Assesment #3

45-45-90 triangle formula

Leg x square root of 2 = hypotenuse

30-60-90 right triangle formula

short leg x square root of 3 = long leg

short leg x 2 = hypotenuse

Converse, inverse, and contrapositive

converse = flip hypothesis and conclusion (ex: original - if a polygon is a square, then it is a quadrilateral, converse - if a polygon is a quadrilateral, then it is a square)

inverse = negate hypothesis and conclusion (ex: original - if a polygon is a square, then it is a quadrilateral, inverse - if a polygon is not a sqaure, then it is not a quadrilateral)

contrapositive = negate and flip hypothesis and conclusion (ex: original - if a polygon is a square, then it is a quadrilateral, contrapositive - if a polygon is not a quadrilateral, then it is not a square)

Scale factors (perimiter and area)

scale factor = A/B

perimiter = A/B

area = A²/B²

Sine (sin)

angle = opposite leg/hypotenuse (remember opposite leg is NOT touching angle)

Cosine (cos)

angle = adjacent leg/hypotenuse (remember adjacent leg is touching angle)

Tangent (tan)

angle = opposite leg/adjacent leg

Regular polygons are

equilateral and equiangular

Convex polygons have all angles that are

less than 180

Regular polygon sum of interior angles formula

180(x-2)

x = # of sides

Measure of EACH interior angle for regular polygon formula

180(x-2)/x

x = # of sides

Exterior angle is _________ to the interior angle at that vertex

supplementary

The sum of the exterior angles of any polygon add up to

360

To find EACH exterior angle formula

360/x

x = # of sides

Order of shapes

Quadralateral

(2 parallel sides) Parallelogram | (One parallel side) Trapezoid | (Zero parallel sides) Kite ^

Rectangle Rhombus v Square

REMEMBER THAT GOING UP IS TRUE NOT DOWN (EX: all sqaures are rectangles/rhombuses, but not every rectangle/rhombus is a sqaure)

A parrallelogram is a _______ whose opposite sides are _________

quadrilateral, parallel

4 basic properties of a parallelogram

1. Opposite sides are congruent

2. Opposite angles are congruent

3. Consecutive angles are supplementary (consecutive is same side!)

4. The diagonals meet at a midpoint, so the diagonals bisect each other

5 ways to prove a quadrilateral is a parallelogram

1. Both sides of opposite sides are parallel

2. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram

3. If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram

4. If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram

5. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram

Rectangles

a quadrilateral with four right angles

7 properties of rectangles

1. Opposite sides are parallel

2. Opposite sides are congruent

3. Opposite angles are congruent

4. Consecutive angles are supplementary

5. Diagonals bisect each other

6. All angles are right angles

7. Diagonals are congruent

Rhombi

quadrilateral with four congruent sides

9 properties of a rhombus

1. Opposite sides are parallel

2. Opposite sides are congruent

3. Opposite angles are congruent

4. Consecutive angles are supplementary

5. Diagonals bisect each other

6. All four sides are congruent

7. Diagonals are perpendicular

8. Diagonals bisect the opposite angles

9. The small triangles formed by the diagonals are RIGHT and CONGRUENT

Squares

quadrilaterals with four congruent angles and four congruent sides

10 properties of squares

1. Opposite sides are parallel

2. Opposite sides are congruent

3. Opposite angles are congruent

4. Consecutive angles are supplementary

5. Diagonals bisect each other

6. Four right angles

7. Four sdes are congruent

8. Diagonals are congruent

9. Diagonals are perpendicular

10. Diagonals bisect opposite angles

Trapezoid

quadrilateral with exactly one pair of parallel sides

Median of a trapezoid

Segment that joins the midpoints on the legs (sometimes called midsegment), the median of a trapezoid is parallel to the bases and bisects the sides, the lengths of the median is one-half the sum of the lengths of the bases

Median formula

median = ½ (b1 + b2)

Isosceles trapezoid

a trapezoid with congruent legs, both pairs of base angles of an isosceles trapezoid are congruent, the diagonals of an isosceles trapezoid are congruent, consecutive angles are supplementary

Kites

quadrilateral with two pairs of congruent adjacent sides, the diagonals of a kite are perpendicular, the line of symmetry bisects the angles, two sets of congruent right triangles

Distance formula

d = square root (x2 - x1)² + (y2 - y1)² or square root of rise² + run²

Midpoint formula

M = x1 + x2 / 2 , y1 + y2 / 2

Slope formula

m = y2 - y1 / x2 - x1 or rise/run

To determine if a shape is a parallelogram use

slope formula to see if opposite sides are parallel

To determine if a shape is a rhombus

determine if the diagonals are perpendicular using slope formula or determine all four sides congruent using distance formula

To determine if a shape is a rectangle

determine if the diagonals are congruent using distance or determine if all four corners are right angles using slope formula

A shape is a square if

it is both a rhombus and a rectangle

Line segment

Angle

Perpendicular bisector

Angle bisector

Perpendicular Through a Point On the Line

Perpendicular Through a Point NOT On the Line

Parallel lines

Equilateral Triangle Inscribed in a Circle

Regular Hexagon Inscribed in a Circle

Square inscribed in a Circle