Download Key Understandings

Name ____________
Mr. Schlansky
Date _____________
Geometry
Geometry Key Understandings
Logic:
Converse: switch order
Inverse: negate both
Contrapositive: switch order and negate both, Contrapositive is logically equivalent
Negation:
“not,” “it is not the case that”
conjunction:
“and”
disjunction:
“or”
Conditional:
“if, then”
biconditional: “if and only if”
An “and” statement is true if both statements are true
An “or” statement is true if at least one statement is true
Conditional is always true except true implies false
~




Parallel Lines:
If lines are parallel:
-If the angles are the same (both acute or obtuse), set them equal to each other
-If the angles are different (one acute and one obtuse), add them to equal 180º.
If lines are not parallel:
- The same angles are not congruent and different angle are not supplementary
-Supplementary angles add to equal 180º, Complementary angles add to 90º
A
A
O A
O
Triangles: (Look for linear pairs/angles of triangles)
Scalane triangles have 0 congruent sides/angles
Acute triangles have 3 acute angles
Isosceles triangles have 2 congruent sides/angles Obtuse triangles have 1 obtuse angle
Equilateral triangles have 3 congruent
Right triangles have 1 right angle
sides/angles
-An isosceles triangle has congruent angles opposite congruent sides.
-An exterior angle is equal to the sum of the two non-adjacent interior angles.
-The three angle of a triangle add to equal 180º
-The two smallest sides of a triangle must add to be larger than the third side
-The smallest side is opposite the smallest angle in a triangle
-When you see midpoint, think half
-The medians of a triangle are cut in the ratio 2:1
Similar Triangles:
ROCS = Ratio of Perimeters
ROCS 2 = Ratio of Areas
Ratio of Corresponding Angles = 1:1 (The corresponding angles are congruent)
Candy Corn Problems:
If the bases are not involved:
top bottom side


top bottom side
If bases are involved: separate your triangles!
O A
O
When an altitude is drawn to a right triangle:
HLLS and SAAS
H L

L S
S A

A S
LEG
LEG
ALTITUDE
SEG
SEG
HYPOTENUSE
Regular Polygons:
Exterior angles add to 360º
Interior angles use 180(n – 2)
To find each (one) angle, divide by n angles
Number of Angles Sum
Name
3
180
Triangle
4
360
Quadrilateral
5
540
Pentagon
6
720
Hexagon
Heptagon
7
900
8
1080
Octagon
Nonagon
9
1260
10
1440
Decagon
Equations of Circles and Lines:
Center and radius are key pieces of information for circles
To find center: Negate what is in the parenthesis. If there are no parenthesis, the
coordinate is 0.
Radius is the square root of the right hand side
(x – a) 2 + (y – b) 2 = r 2 where (a,b) is the center and r is the radius
Slope-Intercept form of a line: y = mx + b where m = slope and b = y intercept
When asked for the equation of a line and given a point, substitute into y = mx + b OR Use
Table in Calculator!
y = # is horizontal line, x = # is vertical line
The solution to a system of equations is the point of intersection of the two graphs
Parallel lines have the same slope
Perpendicular lines have negative reciprocal slopes (flip it and negate it)
Transformations:
Reflections: USA A GRAPH OR Keep what you’re reflecting over
ry x ( x, y)  ( y, x) Flip the po int s, rorigin( x, y )  ( x, y ) Negate the po int s
Rotations: USE A GRAPH: Rotate your paper! Positive is counter-clockwise (left)
Translations: Add, Dilations: Multiply
Composition of Transformations: Start at the right!
Isometry: Preserves size: Direct Isometry: Preserves orientation
Opposite Isometry: Does not preserve orientation
Translation and rotation preserves size and orientation
Reflection preserves size but not orientation
Dilation preserves orientation but not size
Geometry Proofs:
If it is not specified, prove triangles are congruent
To prove segments or angles, use CPCTC
To prove triangles are congruent, prove 3 pairs of sides/angles are congruent
*If you get stuck, make something up and keep on going!
1) Do a mini proof with your givens
Altitude creates congruent right angles
Median creates congruent segments
Line bisector creates congruent segments
Midpoint creates congruent segments
Angle bisector creates congruent angles
Perpendicular lines create congruent right angles
When given parallel lines:
Corresponding angles are congruent OR Alternate interior angles are congruent OR
Alternate exterior angles are congruent
2) Use additional tools:
Vertical Angles are congruent (Look for an X)
Reflexive Property (A side/angle is congruent to itself)
Isosceles Triangles (In a triangle, congruent angles are opposite congruent sides)
Addition and Subtraction Property (If you need more or less of a shared side)
*See Circle Proofs Theorems
When given two angles are congruent (alternate interior/exterior or corresponding) conclude that
the lines are parallel
To prove triangles are SIMILAR, prove AA  AA
If asked to prove a proportion/multiplication:
1) Prove triangles are similar
2) Corresponding Sides of Similar Triangle are In Proportion (CSSTIP)
3) In a proportion, the product of the means are equal to the product of the extremes
A parallelogram has:
Two pairs of opposite sides congruent OR
Two pairs of opposite sides parallel OR
One pair of opposite sides congruent and parallel OR
Diagonals that bisect each other OR
Opposite angles congruent
A rectangle is a parallelogram with:
A right angle OR Congruent diagonals
A rhombus is a parallelogram with consecutive sides congruent
A square is a parallelogram with consecutive sides congruent and:
A right angle OR Congruent diagonals
A trapezoid has one pair of opposite sides parallel and one pair of opposite sides not parallel
An isosceles trapezoid is a trapezoid with congruent legs
Coordinate Geometry/Quadrilaterals:
How do you prove…?
1. …two segments are congruent?
Same Distance
Distance (Length) =
x 2  y 2 =
( x 2  x1 ) 2  ( y 2  y1 ) 2
2. …two segments are parallel to each other?
Same Slope
Slope =
y y 2  y1
=
x x2  x1
3. …two segments bisect each other?
Same Midpoint
Midpoint = (avg x, avg y) =  x1  x 2 , y1  y 2 
2 
 2
4. …two segments are perpendicular to each other?
Negative Reciprocal Slopes
Slope =
y y 2  y1
=
x x2  x1
5. …an isosceles triangle? (2 Distances)
Two Congruent Sides
6. … a right triangle? (2 Slopes)
Consecutive Sides Perpendicular
7. … a parallelogram? (4 Distances)
Two Pairs of Opposite Sides Congruent
8. … a rhombus? (4 Distances)
All Sides Congruent
9. … a rectangle? (6 Distances)
1) Opposite Sides Congruent
2) Diagonals Congruent
10. … a square? (6 Distances)
1) All Sides Congruent
2) Diagonals Congruent
11. …a trapezoid? (4 Slopes)
1) 1 pair of opposite sides parallel
2) 1 pair of opposite sides not parallel
12. …an isosceles trapezoid? (4 Slopes, 2 Distances)
1) 1 pair of opposite sides parallel
2) 1 pair of opposite sides not parallel
3) Congruent Legs
13. …a right trapezoid? (4 Slopes)
1) 1 Pair of Opposite Sides Parallel
2) 1 Pair of Opposite Sides Not Parallel
3) Consecutive Sides Perpendicular
Know your parallelogram diamond!
When asked about properties of a shape, draw the shape!
Solids:
Volume = (Area of the base)(height), if it comes to a point, multiply by
1
.
3
Lateral/Surface Area/Sphere: Use formula sheet
Lateral Area does not include the bases, surface area does
When asked about planes and lines, use manipulatives around you (desk and paper for
planes, pens and pencils for lines)
Two planes perpendicular to the same line are parallel!
If a line is perpendicular to two intersecting lines, it is perpendicular to the plane that
contains them!
Locus:
The locus of points from a point is a circle, all other loci are lines.
Draw your loci dashed
Constructions:
Start by placing needle point on the key point(s)
Be able to construct perpendicular bisector, angle bisector, perpendicular lines through a
given point, and equilateral triangle
Circles: (Look for inscribed,central,radii,diameters)
Proofs:
All radii/diameters of a circle are congruent
Congruent arcs have congruent chords have congruent central angles
Angles inscribed to the same/congruent arcs are congruent
An angle inscribed to a radius/diameter is a right angle
Parallel Lines intercept congruent arcs
Tangents drawn from the same point are congruent
60º
The arcs of a circle add to 360º
A diameter cuts a circle into 2 halves of 180º each
120º
180º
Central Angle: Has its vertex at the center of the circle
Central angle is equal to the measure of the intercepted arc
100º
Inscribed Angle: Has its vertex on the circle
Inscribed angle is half of the measure of the intercepted arc
100º
100º
50º
Exterior Angle:
1
Angles: Exterior Angle = (Major Arc – Minor Arc)
2
2(Exterior Angle) = (Major Arc – Minor Arc) 150º
Segments: Exterior  Whole = Exterior  Whole
Intersecting Chords:
Angles: Arc + Arc = Vertical Angle + Vertical Angle
Segments: Part  Part = Part  Part
4
70º
40º
2
6
5
140º
100º 100º 60º
10
5
Two tangents drawn from the same point are congruent
5
Congruent chords intercept congruent arcs
100º
Parallel chords intercept congruent arcs
100º
20º
20º
2
4
Miscellaneous:
USE A GRAPH WHENEVER POSSIBLE!
Mr. x 2 Story:
-Mr. x 2 wants to party to all of his buddies have to come over (every term comes to the side
with x 2 )
-Once everybody comes over, Mr. x 2 parties by playing with bubbles (once every term is
on one side, FACTOR)
*Don’t forget the T-Chart!
When in doubt, set things equal to each other
When given lengths, 95% of the time set them equal to each other
When given angles:
If they are the same (both acute or both obtuse), set them equal to each other
If they are different (one is acute and one is obtuse), add them to equal 180º
To graph a parabola, use your calculator (mirror image)
Look for hidden right triangles (Pythagorean Theorem)
Know how to reduce radicals
Know how to put equation in y = mx + b form
When given a ratio, put an x behind each number
Know area formulas!
Circle: A = r 2
Rectangle: A = LW
1
Triangle: A = LW
2
Points of Concurrency:
Incenter = Intersection of Angle Bisectors: Always inside the triangle
Centroid = Intersection of Medians: Always inside the triangle
Orthocenter = Intersection of Altitudes: Inside acute triangles, on right triangles, outside obtuse
triangles
Circumcenter = Intersection of Perpendicular Bisectors: Inside acute triangles, on right
triangles, outside obtuse triangles
(Non Perpendiculars: always in, Perpendiculars: acute in, right on, obtuse out)
Know your songs, definitions, and how do you proves!!!