Geometry- All Definitions, Postulates, Theorems and Other properties
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tpsflashcards on November 30, 2011
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142 terms
Terms | Definitions |
|---|---|
 equidistant | equally distant to the same point |
| point | an undefined term; location in space; no size |
| line | an undefined term; extends in 2 directions never ending; no size |
| plane | an undefined term; extends without ending; no thickness; need at least 3 non collinear points to draw a plane |
| space | set of all points (everything) |
| collinear points | points on the same line |
| coplanar points | points on the same plane |
| coplanar | on the same plane |
| collinear | on the same line |
| intersection | set of pints in 2 figures where they meet/cross |
| line segment | 2 points on a line and all the points in between |
| ray | starting at 1 point and going infinitely in one direction |
| parallel lines | 2 lines that are coplanar and never touch |
| skew lines | not coplanar; not parallel; but never touch |
| length | distant between endpoints |
| congruent | same size and shape |
| congruent segments | segments that have the same length |
| midpoint | a point that divides the segment into two ≈ segments |
| bisector | line that crosses at the midpoint |
| opposite rays | (forms a line) must have connecting end point |
| postulate (axiom) | a statement accepted without proof |
| angle | a figure formed by two rays that have a common end point |
| acute ∠ | less than 90º |
| obtuse ∠ | greater than 90º |
| right ∠ | equals 90º |
| strait ∠ | equals 180º |
| congruent ∠s | have equal measures |
| adjacent ∠s | 2 coplanar ∠s with a common vertex and a common side but no common interior points (don't have to be congruent) |
| Bisector of an ∠ | a ray that divides ∠ into 2 ≈ adjacent ∠s |
| protractor postulate | can use a protractor to measure |
| ∠ addition postulate | 1st part of ∠ + 2nd part of ∠ = a whole ∠ ( ∠AOC+ ∠COD = ∠AOD) |
| unnamed postulate/theorem #1 | a line contains at least 2 points |
| unnamed postulate/theorem #2 | a plane contains at least 3 points that are not collinear |
| unnamed postulate/theorem #3 | space contains at least 4 points that are non coplanar |
| unnamed postulate/theorem #4 | through any 3 points there is at least one plane |
| unnamed postulate/theorem #5 | if 2 points are on a plane then the line containing those points is also on the plane |
| unnamed postulate/theorem #6 | if 2 planes intersect then their intersection forms a line |
| theorem | a statement that has to be proven |
| unnamed postulate/theorem #7 | if 2 lines intersect they intersect at exactly 1 point |
| unnamed postulate/theorem #8 | through a line and a point not in the line there is exactly 1 plane |
| unnamed postulate/theorem #9 | if 2 lines intersect then exactly 1 plane contains both lines |
| conditional statement/ if then statement | if p then q |
| converse | if q then p |
| counter example | an example used to prove a statement is false (the hypothesis is true but the statement is false) (only takes 1 counter example to prove a statement false) |
| hypothesis of "if ellen studies then she will get an A" | ellen studies |
| conclusion of "if ellen studies then she will get an A" | she will get an A |
| Biconditional | if and only if (ex. a triangle is acute if and only if it has 3 acute∠s) |
| 6 properties of Equality (algebra) | addition, subtraction, multiplication, division, substitution, distributive |
| reflexive property | a=a; a≈a |
| symmetric property | if a=b then b=a |
| transitive property | if a=b and b=c then a=c |
| midpoint theorem | parts = half the whole ( A_______M_______B) )if m is midpoint of AB then AM=1/2 AB and MB=1/2 AB) |
| Angle bisector theorem | part= half the whole |
| rule with vertical ∠s | vertical ∠s are ≈ |
| perpendicular lines | 2 lines that intersect to form right ∠s |
| unnamed postulate/theorem #10 (related to ⊥ lines) | 2 lines are ⊥ if they form ≈ adjacent ∠s |
| converse of unnamed postulate/theorem #10 (related to ⊥ lines) | if 2 lines form ≈ adjacent ∠s then they are ⊥ |
| unnamed postulate/theorem #11 | if exterior sides of 2 adjacent acute ∠s are ⊥ then the ∠s are complementary |
| unnamed postulate/theorem #12 | if 2 ∠s are ≈ then their supplements are also ≈ |
| unnamed postulate/theorem #13 | if 2 acute ∠s are ≈ then their complements are also ≈ |
| unnamed postulate/theorem #14 | if 2 ll planes are cut by a third plane then the lines of intersection are ll |
| transversal | a line that intersects 2 or more coplanar lines in different points |
| when dealing with Transversals and lines these 4 kinds of ∠s are formed | alt. int. ∠s alt. ext. ∠s same side int. ∠s corresponding ∠s |
| what types of ∠s are ≈ when 2 ll lines are cut by a trans. | corresponding ∠s are ≈ and alt. int. ∠s are ≈ |
| what type of ∠s are supp. when 2 ll lines are cut by a trans. | same side int. ∠s |
| unnamed postulate/theorem #15 | if a trans. is ⊥ to 1 of 2 ll lines then it is ⊥ to the other line also |
| 5 ways to prove lines are ll | if 2 lines are cut by a trans. and ... alt. int. ∠s are ≈ corresponding∠s are ≈ same side int. ∠s are supplementary; if in a plane 2 lines are ⊥ to the same line if 2 lines are ll to the same line |
| unnamed postulate/theorem #16 | though a point outside a line there is exactly 1 line ll to the given line |
| unnamed postulate/theorem #17 | through a point outside a line there is exactly 1 line ⊥ to the given line |
| triangle | a figure formed by 3 segments joining 3 non collinear points |
| scalene ∆ | no ≈ segments |
| isosceles ∆ | at least 2 ≈ sides |
| acute ∆ | all ∠s of the ∆ are ≈ |
| equilateral | all sides are ≈ |
| equiangular | all ∠s are ≈ |
| auxiliary line | a line ray or segment added to a diagram to help in a proof |
| corollary | a statement that can be proved easily by applying the theorem |
| obtuse ∆ | a ∆ with 1 obtuse ∠ |
| right ∆ | a ∆ with 1 right ∠ |
| exterior ∠ theorem | the measure of the ext. ∠s of a ∆ equals the sum of the measures of the 2 remote interior ∠s |
| 4 corollaries of ext. ∠ theorem | 1. if 2 ∠s of 1 ∆ are ≈ of to 2 ∠s of another ∆ then the 3rd ∠s of both ∆s are ≈ 2. Each ∠ of an equiangular has a measure of 60º 3. in a ∆ there can be at most 1 right or 1 obtuse ∠ 4. the acute ∠s of a right ∆ are complementary |
| polygon | a figure formed by coplanar segments such that 1. each segment intersects exactly with 2 other segments one at each endpoint and 2. no 2 segments with a common endpoint are collinear |
| diagonal | a segment joining 2 non consecutive vertices |
| convex polygon | a polygon such that NO line containing a side of the polygon contains a point in the interior of the polygon OR a polygon is convex if NO diagonal contains points outside the polygon |
| concave polygon | if a diagonal contains points outside the polygon |
| regular polygon | a polygon that is equiangular and equilateral |
| equation to find the sum of the measures of of a convex polygon | (n-2)180 (n= # of sides in the polygon) |
| equation to find an ∠ of a regular polygon | [(n-2)180]÷n |
| unnamed postulate/theorem #18 | the sum of the measure of the ext. ∠s of any convex polygon is 360 |
| deductive reasoning | proving statements by reasoning from accepted postulate, definitions, theorems, and given info (must be true) |
| inductive reasoning | a kind of reasoning in which the conclusion is based on several past observations (conclusion is probably true but not necessarily true) |
| 5 ways to prove ∆s are ≈ | SSS postulate, SAS postulate, ASA postulate, HL theorem, and AAS theorem |
| SSS postulate (side, side, side) | if 3 sides of one ∆ are ≈ to 3 sides of another ∆ then the ∆s are ≈ |
| SAS postulate (side, angle, side) | if 2 sides and the included ∠ of 1 ∆ are congruent to 2 sides and the included ∠ of the other ∆ then the ∆s are ≈ |
| ASA postulate (angle side angle) | if 2 ∠s and the included side of 1 ∆ are ≈ to 2 sides and included ∠s of another ∆ then the ∆s are ≈ |
| CPCTC (corresponding parts of ≈ ∆s are ≈) | by finding 2 ∆s ≈ then you can prove their corresponding parts are ≈ |
| Isosceles ∆ | ∆ with at least 2 sides ≈ |
| the isosceles ∆ theorem | if 2 sides are ≈ then the ∠s opposite those sides are ≈ |
| converse of the isosceles ∆ theorem | if 2 ∠s of a ∆ are ≈ then the sides opposite those ∠s are ≈ |
| unnamed corollary A (related to equilateral ∆s) | an equilateral ∆ is also equiangular |
| unnamed corollary B (related to isosceles ∆s) | the bisector of the vertex ∠ of an isosceles ∆ is ⊥ the base of the midpoint |
| AAS theorem (angle, angle side) | if 2 ∠s and a non included side of 1 ∆ are ≈ to the corresponding parts of the other ∆ then the ∆s are ≈ |
| HL theorem | if the hypotenuse and a leg of 1 right ∆ are ≈ to the corresponding parts of another right ∆ then the ∆s are ≈ |
| quadratic formula | x= [-b ± √b²-4ac] ÷ 2a |
| median of a ∆ | a segment from a vertex to the midpoint of the opposite side |
| altitude of a ∆ | is the ⊥ segment from a vertex to the opposite side |
| perpendicular bisector of a segment | a ray or segment that is ⊥ to the segment at its midpoint |
| unnamed postulate/theorem #19 | if a point lies on the ⊥ bisector or a segment then the point is equidistant from the endpoints of a segment |
| unnamed postulate/theorem #20 | if a point is equidistant from the endpoints of a segment then the point is equidistant from the sides of the ∠ |
| unnamed postulate/theorem #21 | if a point lies on the bisector of an ∠ then the point is equidistant from the sides of the ∠ |
| unnamed postulate/theorem #22 | if a point is equidistant from the sides of the ∠ then the point lies on the bisector of the ∠ |
| parallelogram | a quadrilateral with both pairs of opposite sides are ≈ |
| 3 theorems relating to parallelograms | opp. sides of a parallelogram are ≈; opp. ∠s of a parallelogram are ≈; diagonals of a parallelogram bisect each other |
| 4 ways to prove the quad is a parallelogram | 1. if both pairs of opp. sides of a quad. are ≈ 2. if 1 pair of opp. sides of a quad are both ≈ and ll 3. if both pairs of opp. ∠s of a quad are ≈ 4. if the diagonals of a quad bisect each other |
| 2 unnamed postulates/theorems related to ll lines | if 2 lines are ll then all the points on 1 lines are equidistant to all the points on the other line; if 3 ll lines cut 1 transversal into ≈ segments then they cut every transversal into ≈ segments |
| 2 unnamed postulates/theorems that relate to medians of a ∆ | 1. a line that contains the midpoint of 1 side of a ∆ and is ll to another side passes through the midpoint of the 3rd side; the segment that joins the midpoints of 2 sides of a ∆ is ll to the 3rd side and is 1/2 as long as the 3rd side |
| rectangle | a quad w/ 4 right ∠s |
| rhombus | a quad w/ 4 ≈ sides |
| square | a quad w/ 4 right ∠s and 4 ≈ sides |
| theorems/rules about rectangles (1) | the diagonals of a rectangle are ≈ |
| theorems/rules about rhombuses (2) | the diagonals of a rhombus are ⊥; each diagonal of a rhombus bisects 2 ∠s if the rhombus |
| unnamed postulate/theorem #23 | the midpoint of the hypotenuse of a right ∆ is equidistant from the 3 vertices |
| unnamed postulate/theorem #24 (related to proving a parallelogram is a rectangle) | if an ∠of a parallelogram is a right ∠ then the parallelogram is a rectangle |
| unnamed postulate/theorem #25 (related to proving a parallelogram is a rhombus) | if 2 consecutive sides of a parallelogram are ≈ then the parallelogram is a rhombus |
| trapezoid | a quad with exactly 1 pair of ll sides |
| isosceles trapezoid | a trapezoid with 1 pair of ≈ sides |
| median of a trap. | 1. is ll to the bases 2. has a length equal to the avg of the base length |
| unnamed postulate/theorem #26 (related to isosceles trap. and their base ∠s) | base ∠s of and isosceles trap are ≈ |
| inequality | a mathematical sentance that contains < > ≤ ≥ |
| properties of inequalities #1 | if a>b and c≥d then a+c>b+d |
| properties of inequalities #2 | if a>b and c>0 then ac>bc and a/c>b/c |
| properties of inequalities #3 | if a>b and c<0 then ac<bc and a/c<b/c |
| properties of inequalities #4 (like transitive) | if a>b and b>c then a>c |
| properties of inequalities #5 | if a=b+c and c>0 then a>b |
| ext. ∠ inequality theorem | the measure of an exterior ∠ of a ∆ is greater than the measure of either remote ∠s |
| inverse | if not p then not q |
| contrapositive | if not q then not p |
| venn diagram | a circle diagram that may be used to represent a conditional statement |
| indirect proof | a proof in which you assume temporarily that the conclusion is not true and then deduce a contradiction |
| 3 theorems/postulates regarding inequalities of 1 ∆ | 1. the sum of 2 sides of a ∆ is greater than the 3rd side 2. if 1 side of a ∆ is longer than the 2nd side then ∠ opp. the 1st side is larger than the ∠ opp. the 2nd side 3. if 1 ∠ of a ∆ is larger than a 2nd ∠ then the side opp. the 1st ∠ is longer than the side opp. the 2nd ∠ |
| SAS inequality | if 2 sides of a ∆ are ≈ to 2 sids of another ∆ but included 1st ∠ of the 1st ∆ is larger than the included ∠ of the 2nd ∆ then the 3rd side of the 1st ∆ is longer than the 3rd side of the 2nd ∆ |
| SSS inequality | if 2 sides of 1 ∆ are ≈ to 2 sides of another ∆ but the 3rd side of the 1st ∆ is larger than the included ∠of the 2nd ∆ |
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