Module 2: Transformations and congruence

4 Common types of transformations

1. Rotation 2. Translation 3. Dilation 4. Reflection

kites

quadrilateral with two pairs of congruent, adjacent sides, where diagonals are perpendicular to one another

vector

used to describe objects in motion; represents the magnitude and direction of an object

reflection of y-axis

(x, y) --> (-x, y)

Rotation 90° counterclockwise

(x, y) --> (-y, x)

Rotation 90° clockwise

(x, y) --> (y, -x)

Reflection over x-axis

(x,y) --> (x, -y)

Rotation 180°

(x,y) -> (-x,-y)

Translation Formula

(x,y) -> (x+a, y+b)

Reflection over y=x

(x,y) -> (y,x)

Rectangle rules

1-4. same as parallelogram 5. contains 4 right (90°) angles 6. the diagonals are congruent

square rules

1-4. same as parallelogram 5. contains 4 right angles 6. the diagonals are congruent 7. all 4 sides are congruent 8. the diagonals are perpendicular 9. the diagonals bisect angles

rhombus rules

1-4. same as parallelogram 5. diagonals are perpendicular 6. all sides congruent 7. diagonals bisect angles

parallelogram rules

1. both pairs of opposite sides are congruent and parallel 2. the diagonals bisect each other 3. both pairs of opposite angles are congruent 4. consecutive angles are supplementary

kite rules

1. diagonals bisect their vertex angles 2. vertex diagonal of a kite is a perpendicular bisector of nonvertex angles

AAS postulate

2 angles and non-included side

SAS Postulate

2 sides and included angle are equal

polygon

A closed figure formed by three or more line segments

trapezoids

A quadrilateral with exactly one pair of parallel sides 1. parallel sides are called bases 2. nonparallel sides are called legs 3. two angles that share a base: base angles

Reflection over horizontal/ vertical line

Count how far away it is from line of reflection and mark the vertex same distance over the line

HL Theorem (hypotenuse-leg)

If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.

Hinge Theorem

If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second.

Translation

Moevement of a figure from one location to another without a change in size, shape, or orientation.

Midsegment of a Triangle Theorem

The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long.

Triangle Inequality Theorem

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

Triangle Sum Theorem

The sum of the measures of the interior angles of a triangle is 180 degrees

regular polygon

a polygon with all sides and all angles equal

isosceles trapezoid

a trapezoid with congruent legs • congruent diagonals • base angles in an isosceles trapezoid are congruent

Reflection symmetry

aka line symmetry; figure can be divided into equal halves

corresponding parts

angles, sides, or vertices of two or more figures that are located in the same position when the figures are aligned

Proof of the equidistance of a point on a perpendicualr bisector

any point on a perpendicular bisector is equidistant from the endpoints of the segment it bisects

rigid motion

characteristic of a transformation in which the location of the figure changes, but the shape or size does not

concentric circles

coplanar circles with the same center

Angle of rotation formula

divide 360° by the order of rotation of a figure to find that figure's angle of rotation

Transformation

figure chbages in some way (example: size, orientation, and/ or location)

Rotational Symmetry

figure looks the same after a rotation of less than 360°

SSS Postulate

if 3 sides are the same, they are congruent

iscosceles triangle theorem

if two sides of a triangle are congruent, then the angles opposite those sides are congruent

CPTC (corresponding parts of congruent triangles are congruent)

once you prove two angles are congruent, you have proven their corresponding parts are congruent too

image

resulting figure after pre-image is tranformed • prime sign ' after lettter • example: A --> A'

angle of rotation

the degree measure a figure must rotate in order to look like itself

order of rotation

the number of times an image can be rotated to look like its pre-image

pre-image

the original figure

incenter of a triangle

the point of concurrency of the angle bisectors of a triangle

circumcenter of a triangle

the point of concurrency of the perpendicular bisectors (intersect all sides of triangle at 90° angle at that side's midpoint) of a triangle

orthocenter of a triangle

the point of concurrency of the three altitudes (shortest segment between a vertex of triangle and its opposite side, at 90° angle)of a triangle

centroid of a triangle

the point of concurrency of the three medians (endpoints are vertex of triangle and midpoint of opposite side) of a triangle; always inside

Rotation

transformation in which a figure turns around a fixed point called the center of rotation. Rigid Motion.

Reflection

transformation where the mirror image of a figure is shown directly opposite its line of reflection; regid motion

ASA postulate

two angles and included side are congruent

classifying triangles based on angles

• right- one agle is 90° • equilangular- all there angles measure 60° • obtuse- one angle > 90° • acute- all angles less than 90°

classifying triangles based on sides

• scalene- no congruent sides • isosceles- two congruent sides • equilateral- three congruent sides