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