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Level 324

Geometry Essential Theorems


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midpoint theorem
if M is the midpoint of AB, then AM is congruent to MB
What defines a line?
Through any two points, there is exactly one line. (p. 105), A line contains at least two points. (p. 106)
What defines a plane?
Through any three non-collinear points, there is exactly one plane. (p. 105) A plane contains at least three non-collinear points. (p. 106)
How can you tell if an entire line lies in a plane?
If two points lie in a plane, then the entire line containing those points lies in that plane. (p. 106)
How do two lines intersect?
If two lines intersect, then their intersection is exactly one point. (p. 106)
How do two planes intersect?
If two planes intersect, then their intersection is a line. (p. 106)
Supplement Theorem
If two angles form a linear pair, then they are supplementary angles.
Complement Theorem
If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary angles.
Vertical Angle Theorem
If two angles are vertical angles, then they are congruent. (p. 127)
Perpendicular Transversal Theorem
in a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other
Parallel and Perpendicular Lines
Two nonvertical lines have the same slope if and only if they are parallel. Two nonvertical lines are perpendicular if and only if the product of their slopes is -1. (p. 158)
Shortest Distance
The perpendicular segment from a point to a line is the shortest segment from the point to the line. (p. 298)
Triangle Angle Sum Theorem
The sum of the measures of the angles of a triangle is 180.(p. 210)
Third Angle Theorem
If two angles of a triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent.
Exterior angle theorem
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles.
CPCTC
Corresponding parts of congruent triangles are congruent.
Side-Side-Side Congruence (SSS)
If three sides of a triangle are congruent to three sides of another triangle, then the triangles are congruent.
Side-Angle-Side Congruence (SAS)
If two sides and the included angle of a triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
Angle-Side-Angle Congruence (ASA)
If two angles and the included side of a triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.
Angle-Angle-Side Congruence (AAS)
If two angles and a non-included side of a triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent.
Isosceles Triangle Theorem
If 2 sides of a triangle are congruent, then the angles opposite those sides are congruent
Equilateral Triangle Corollaries
A triangle is equilateral if and only if it is equiangular. Each angle of an equilateral triangle measures 60°. (p. 247)
Perpendicular Bisector
A segment or Ray the perpendicularly bisects a side of a 🔺
Median
A segment or Ray that joins a vertex to the midpoint of the opposite side
altitude
What is the apothem called In a triangle?
Centroid Theorem
The centroid of a triangle is located 2/3 of the distance from each vertex to the midpoint of the opposite side.
Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
Interior Angle Sum Theorem
If a convex polygon has n sides and S is the sum of the measures of its interior angles, then S = 180(n - 2). (p. 318)
Exterior Angle Sum Theorem
If a polygon is convex, then the sum of the measures of the exterior angles, one at each vertex, is 360. (p. 320)
Parallelogram Theorem - Sides
Opposite sides of a parallelogram are congruent and parallel. (p. 326)
Parallelogram Theorem - Angles
Opposite angles of a parallelogram are congruent and consecutive angles in a parallelogram are supplementary. If a parallelogram has one right angle, then it has 4 right angles. (p. 326)
Parallelogram Theorem - Diagonals
The diagonals of a parallelogram bisect each other and each diagonal of a parallelogram separates the parallelogram into two congruent triangles. (p. 328)
Rectangle Theorem
If a parallelogram is a rectangle, then the diagonals are congruent. (p. 340) If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. (p. 342)
Rectangle Definition
A Quadrilateral with all right angles, but is not a square
Square Properties
-Four right angles
Trapezoid Definition
A Quadrilateral with exactly one set of parallel sides
Isosceles Trapezoid Theorems
Each pair of base angles are congruent, the diagonals are congruent, and the legs are congruent. (p. 356)
Trapezoid Median Theorem
The median of a trapezoid connects the midpoints of the legs, is parallel to the bases, and its measure is one-half of the measures of the bases. (p. 359)
Area Postulate
Congruent figures have equal areas. (p. 642)
Area of a Regular Polygon
The area of a regular n-gon with sides length s is half the product of the apothem a and the peroimeter P, so A=1/2aP, or A=1/2a*ns.
Area of a Circle
pi r ^ 2
Area of a Region
The area of a region is the sum of the areas of all of its non-overlapping parts. (p. 658)
Congruence Transformations
The congruence transformations are those transformations that result in a congruent image: translation, reflection, and rotation.
Reflection as Rotation
Reflecting an image successively in two perpendicular lines results in a 180° rotation. (p. 512)
Transitive Property
if two segments or two angles are congruent to the same segment of angle, they are congruent to each other
Substitution property
a(b) = (ab)
translation
It is a transformation that moves points the distance and in the same direction.
reflection
It is a transformation that flips a figure over a line of reflection.
rotation
It is a transformation that turns a figure about a fixed point.
line of reflection
The line over which a figure is reflected. The vertices of the original and new figure are the same distance from this. The line of reflection can be the x-axis, the y-axis, or any line on the coordinate plane.
Reflexive Property
Anything equals itself; a shared piece.
Deductive Argument
if the premises are true, the conclusion must be true (it is impossible for the conclusion to be false)
inductive argument
if the premises are true, then the conclusion is probably true (it is improbable for the conclusion to be false)
Slopes of Parallel Lines
If lines are parallel, then their slopes are exactly the same.
Slopes of Perpendicular Lines
If two lines are perpendicular, then the product of their slopes is -1. One slope is the negative reciprocal of the other slope.
Got coordinates?
Graph them!
Distance "formula"
Graph the points and connect.
Congruent right triangles
If the legs are congruent, the hypotenuse of each triangle must be congruent.
Got a geometric shape?
Label it with all the givens.
Sketch the shape!
Got info about a shape but no shape?
Triangle inequality - longest side
In a triangle, the longest side is across from the largest angle.
Triangle inequality - largest angle
In a triangle, the largest angle is across from the longest side.
Kite theorem
A quadrilateral is a kite if it has two pairs of adjacent sides congruent and no opposite sides congruent
Parallel Postulate
if there is a line and a point not on the line then there is exactly one line through the point parallel to the given line