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# Geometry Concepts

### Geometry Concepts Master Deck

Term | Definition |
---|---|

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..." |