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# Geometry Concepts

### Geometry Concepts Master Deck

TermDefinition
Addition Property of Equalities You can add the same thing to both sides of an equation.
Subtraction Property of Equalities You can subtract the same thing to both sides of an equation.
Multiplication Property of Equalities You can divide by the same thing in both sides of an equation.
Division Property of Equalities You can divide by the same thing in both sides of an equation.
Substitution Property Simplifies. y+x=3 2+y+x --->2+3
Reflexive Property Anything is equal to itself.
Symmetric Property If x=y, then y=x.
Transitive Property If x=y and y=z, then x=z. Connects things. You can compare more than 3 things. For example, if x=y, y=z, z=a, a=b, b=c, then c=all of them. All Transitive Properties are Substitutions, but not all Substitutions are Transitive Properties.
Distributive Property The product of the sums is equal to the sums of the products (multiplying then adding = adding then multiplying). Addition and multiplication are both commutative and associative.
Postulate: Segment Addition Postulate If B is between A and C, then AB+BC=AC. "...B is between A and C..." means all 3 points are colinear.
Postulate: Ruler Postulate 1. The points on a line can be paired with the real numbers in such a way that any 2 points can have coordinates 0 and 1 2. Once a coordinate system has been chosen in this way, distance between any 2 points = absolute value of distance of their coords
Postulate: Angle Addition Postulate If point B lies on the interior of angle AOC, then m∠AOB+m∠BOC=m∠AOC.
Postulate: The Linear Pair Postulate If angle AOC is a straight angle and B is any point not on line AC, then m∠AOB+m∠BOC=180 degrees. NOT m∠AOB+m∠BOC=m∠AOC.
Theorem: The Midpoint Theorem If a point M is the midpoint of a line segment AB, then AM=1/2AB and BM=1/2AB. Not to be confused with def. of midpoint. Theorem creates proportion, says 1 piece is half the length of larger piece, def. says midpoint splits segment into 2 congruent pieces
Theorem: Angle Bisector Theorem If ray BX is the bisector of angle ABC, then m∠ABX=1/2m∠ABC and m∠XBC=1/2m∠ABC. Definition does not equal theorem.
Theorem: Vertical Angles are ______ Congruent Definition does not say they are congruent, only theorem does. With theorems that do not have names, make sure to write the entire thing out in a proof.
Theorem: If 2 lines are perpendicular, then they form _____ angles. Congruent Adjacent
Theorem: If 2 lines form congruent adjacent angles, then the lines are _____ Perpendicular
Theorem: If the exterior sides of 2 adjacent angles are perpendicular, then the acute angles are _____ Complementary
Theorem: Supplements Theorem If 2 angles are supplementary to congruent angles, then they are congruent. In other words, if ∠1 is supplementary to ∠2 and ∠3 is supplementary to ∠2, then ∠1 ≅ ∠3.
Theorem: Complements Theorem If 2 angles are complementary to congruent angles, then they are congruent. In other words, if ∠1 is complementary to ∠2 and ∠3 is complementary to ∠2, then ∠1 ≅ ∠3.
Fill in the Blank: Vertical Angles ___ have a common vertex. Always.
Fill in the Blank: Always, Sometimes, or Never: Two right angles are ___ complementary. Never.
Fill in the Blank: Always, Sometimes, or Never: Right angles are ___ vertical angles. Sometimes.
Fill in the Blank: Always, Sometimes, or Never: Angles A, B, and C are ___ complementary. Never. Note: The term "complementary angles" only refers to a pair, or 2, angles.
Fill in the Blank: Always, Sometimes, or Never: Vertical Angles ___ have a common supplement. Always.
Fill in the Blank: Always, Sometimes, or Never: Perpendicular Lines ___ lie in the same plane. Always.
Fill in the Blank: Always, Sometimes, or Never: 2 lines are perpendicular if and only if they ___ form congruent Adjacent Angles. Always.
Fill in the Blank: Always, Sometimes, or Never: Perpendicular lines ___ form 60° angles. Never.
Fill in the Blank: Always, Sometimes, or Never: If the exterior sides of 2 Adjacent Angles are perpendicular, then the angles are ___ supplementary. Never.
Fill in the Blank: Always, Sometimes, or Never: If a pair of Vertical Angles are supplementary, the lines forming the angles are ___ perpendicular. Always.
Fill in the Blank: Always, Sometimes, or Never: Lines in 2 parallel planes are ___ parallel. Sometimes.
Fill in the Blank: Always, Sometimes, or Never: Lines in 2 parallel planes are ___ skew. Always.
Theorem: If 2 parallel planes are cut by a third, then the lines of intersection are ___. Parallel.
Parallel Line Postulate (This postulate is biconditional - "If 2 lines are intersected by a transversal and Corresponding Angles are congruent, then they are parallel." Parallel lines if and only if Corresponding Angles.) If 2 lines are parallel and intersected by a transversal, then Corresponding Angles are congruent. They are congruent because 1 line is pre-image of other after translation across part of transversal. Translations=isometric, congruence is preserved.
Theorem: If 2 lines are parallel and intersected by a transversal, then ___ are congruent. Alternate Interior/Exterior Angles. Biconditional - "If 2 lines are intersected by a transversal and Alternate Interior/Exterior Angles are congruent, then they are parallel." Parallel lines if and only if Alternate Interior/Exterior Angles.
Fill in the Blank: Always, Sometimes, or Never: When there is a transversal of 2 lines, the 3 lines are ___ coplanar. Always.
Fill in the Blank: Always, Sometimes, or Never: 2 lines that are not coplanar ___ intersect. Never.
Fill in the Blank: Always, Sometimes, or Never: 2 lines skew to a third line are ___ skew to each other. Sometimes.
Fill in the Blank: Always, Sometimes, or Never: 2 planes parallel to the same line are ___ parallel to each other. Sometimes.
Theorem: If 2 lines are parallel and intersected by a transversal, then ___ are supplementary. Same-Side Interior/Exterior Angles. Note: This theorem is biconditional - "If 2 lines are intersected by a transversal and Same-Side Interior/Exterior Angles are supplementary, then they are parallel."
Theorem: 2 parallel lines are perpendicular to the ___ transversal. Same. Note: The word "...transversal." implies that the lines must be coplanar.
Theorem: In a plane, if 2 lines are perpendicular to the same transversal, then they are ___. Parallel. In other words, if line AB is perpendicular to line YZ and line CD is also perpendicular to line YZ, then AB∥CD. Note: The word "...transversal..." implies that the lines must be coplanar.
Theorem: Through a point not on a line, there is exactly 1 line ___ to the given line. Parallel/Perpendicular. In other words, if point P is not on line AB, then there is only 1 line parallel/perpendicular to AB that also goes through P.
Theorem: 2 lines parallel to a third line are ___ to each other. Parallel. In other words, if line AB is parallel to line YZ and line CD is also parallel to line YZ, then AB∥CD. NOT to be confused with Transitive Property. Note: Says nothing about all 3 lines being parallel, just they are parallel to one another.
Theorem: The sum of the angles of a triangle is ___°. 180.
Theorem: The Exterior Angles of a triangle is equal to the sum of 2 ___. Remote Interior Angles. Note: An Exterior Angle is formed using a side of the triangle and extending another side. Note: The Remote Interior Angles are the Interior Angles that are not adjacent to the Exterior Angle.
Corollary: In an equiangular triangle, each angle is ___°. 60.
Corollary: In an obtuse triangle, there is at most ___ obtuse angle. 1. Note: This is because if there were 2 or more obtuse angles present, the sum of the angles would be more than 180°.
Corollary: In a right triangle, there is at most ___ right angle. 1. Note: This is because if there were 2 or more right angles present, the sum of the angles would be more than 180°. If there were 2 right angles, the third angle would have to be 0°, which is a ray.
Corollary: The acute angles of a right triangle are ___. Complementary.
Corollary: If 2 angles of 1 triangle are congruent to 2 angles of another triangle, then the third angles are ___. Congruent.
Corollary: Equilateral triangles are ___. Equiangular. Note: This corollary is biconditional. The statement "Equiangular triangles are equilateral." is also true. An easy way to convey this is "Equilateral triangles if and only if equiangular triangles".
Corollary: The bisector of the vertex angle of an isosceles triangle is ___ to the base at the base's midpoint. Perpendicular.
Theorem: The Isosceles Triangle Theorem If 2 sides of a triangle are ≅, then the ∠s opposite those sides are ≅. Biconditional. Statement "If ∠s opposite 2 sides are ≅, then those sides are ≅." is also true. Easy way to convey this is "2 ≅ angles of a triangle if and only if ≅ opposite sides".
Theorem: Angle Angle Side Theorem If 2 ∠s and a NON-included side of 1 △ are ≅ to corr. parts (2 ∠s and NON-included side) of another △, then △s are ≅. Theorem derived from corollary that states that if 2 ∠s of 1 △ are ≅ to 2 ∠s of another △, then the third ∠s are ≅. "...NON-included..."
Created by: tianm27
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