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Geometry Midterm
| Question | Answer |
|---|---|
| conjecture | an unproven statement that is based on observations |
| inductive reasoning | looking for patterns and making conjectures |
| counterexample | an example that shows a conjecture false |
| definition | known words that desscribe a new word |
| undefined terms | words that dont have a formal definition |
| point | has no demention, represented by a dot |
| line | extends in one demention and two directions |
| plane | extends in two dementions, doesnt end |
| collinear points | points that are on the same line |
| coplanar | points that are on the same plane |
| line segment | a section of a line with two endpoints |
| ray | consists of an initial point, and is a line only going in one diretion, the direction through the other point across form the initial |
| opposite rays | two rays opposite each other |
| postulate | rules that are accepted without proof |
| axioms | another word for postuate |
| coordinate | the real numbers that coorespond to a point |
| distance | the absolute value of the difference between the two coordinates |
| length | another word for distance |
| between | when three points are on a line the middle one is_______ the other two |
| Distance Formula | a ormula for computing the distance between two points in a coordinate plane |
| congruent segments | segmnts that have the same length |
| angle | two different rays that have the same initial point |
| sides | the rays |
| vertex | the initial point of the angle, where it points |
| congruent angles | angles that have the same measure |
| measure | the absolute value of the difference between the real numbers of the distance of the rays |
| interior | between points that lie on each side of the angle |
| exterior | not in the angle's interior |
| adjacent angles | share a common vertex and side but have no common interior points |
| midpoint | the point that divides or bisects the segment into two congruent segments |
| segment bisector | a segment, ray, line, or plane, that intersects a segment at its midpoint |
| construction | a geometric drwing that uses a limited set of tools, usually a compass and a straightedge |
| angle bisector | aray that divides an angle into two adjacent angles that are congruent |
| vertical angles | their sides form two pair of opposite rays |
| linear pair | two adjacent angles that have their noncommon sides as opposite rays |
| complementary angles | angles thats measures add up to 90 |
| supplementary angles | angles whos measures add up to 180 |
| conditional statement | a statement that contains a hypothesis and a conclusion |
| hypothesis | the if part of a conditional statement |
| conclusion | the then part of a conditional statement |
| converse | switching the hypothesis and conclusion in a conditional statement |
| negation | writning the negative of the statement ex. m<A==30, <A is acute m<A=/=30, <A is not acute |
| inverse | when you negate the hypothesis and conclusion of a conditional statement |
| contrapositive | when you negate the hypothesis and conclusion of the converse of a condito=ional statement |
| equivalent statements | when two statements are both true or both false |
| perpendicular lines | two lines that intersect to form a right angle |
| biconditional statement | a statemnet that contains the phrase "if and only if". it is the equivalent of writing the conditional and its converse |
| Law of detatchment | if p-> q is a true conditional statemnet and p is true, then q is true |
| Law of Syllogism | is p-> q and q-> r are true conditional statementa, then p->r |
| theorem | a true statement that follows as a result of other true statements |
| two column proof | has numbered statements and reasons that show the logical order of an argument |
| transversal | a line that intersects two or more coplanar lines at different points |
| same side interior angles | another name for consecutive interior angles |
| flow proof | uses arrows to show the flow of the logical argument |
| triangle | a figure formed by three segments joining three noncollinear points |
| interior angles | angles on the interior of the triangle |
| exterior angles | angles adjacent to the interior angles |
| corollary | a statement that can be proved easily using the theorem. |
| perpendicular bisector | a segment, ray line, or plane, that is perpendicular to a segment at its midpoint |
| concurrent lines | when three or more lines intersect in the same point |
| point of concurrency | point of intersection of concurrent lines |
| circumcenter | point of concurrency of the perpendicula bisectors of a triangle |
| angle bisector of a triangle | bisector of an angle of a trianle |
| inincener | the point of concurrency of the angle bisectors |
| median of a triangle | segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side |
| centroid | the point of concurrency of the three medians |
| altitude | perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side |
| orthocenter | intersection, or point of concurrency of altitudes |
| midsegment of a triagle | segment that connects the midpoints of two sides of a triangle |
| indirect proof | a proof in which you prove that a statement is true by first assuming that its opposite is true. if this assumption is an impossibility, then you have proved that the origional statement is true |
| midsegment of a trapezoid | the segment that connects the midpoints of its legs |
| proportion | and equation that equates two ratios |
| geometric mean | the positive number x such that a/x=x/b |
| scale factor | the ratio of the lengths of two corresponding sides |
| . | . |
| Ruler Postulate | The points on a line can be matched one to one with the real numbers. The number that corresponds to a point is the coordinate of the point. The distance between points A and B, is the absolute value of the difference between the coordinates of A and B. |
| Segment Addition Postulate | If B is between A and C, then AB+BC=AC. if AB+BC=AC, then B is between A and C. |
| Protractor Postulate | Consider a point A on one side of line OB. The rays of the form ray OA can be matched one to one with the numbers from 0 to 180. The measure of<AOB is equal to the absolute value of the distance between the numbers for ray OA and ray OB. |
| Angle Addition Postulate | If P is in the interior of<RST, then m<RSP+m<PST=m<RST |
| any two points | Through any two points there exists exactly one line |
| what a line contains | A line contains at least two points |
| two line inersection | If two lines intersect, then their intersection is exactly one point |
| three noncollinear points | Through any three noncollinear there exists exactly one plane |
| what a plane contains | A plane contains at least three noncollinear points |
| line in a plane | If two points lie on a plane, then the line containing them lies in the plane |
| Plane intersection | If two planes intersect, then their intersection is a line |
| Linear Pair Postulate | If two angles form a linear pair, then they are supplementary |
| 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. |
| Perpendicular Postulate | If there is a line and a point not on the line, then there is exactly one line throught the point perpendicular to the given line. |
| Corresponding Angles Postulate | If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. |
| Corresponding Angles Converse | If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are paralel |
| Slopes of Parallel Lines | In a coordinate plane, two nonvertical lines are parallel if and only if they have th same slope. Any two vertical lines are parallel. |
| Slopes of Perpendicular lines | In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is -1. Vertical and horizontal lines ar perpendicular. |
| SSS Congruence Postulate | f three sides of one triangle are congruent to three sides of a second triangle, then the two triangles aare congruent. |
| SAS Congruence Postulate | If two sides and the included angle of one triangle are congruent to two side and the included angle of a second triangle, then the two triangles are congruent |
| ASA Congruence Postulate | If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent |
| Area of a Square Postulate | The area of a square is the square of the length of its side, or A=s (squared) |
| Area Congruence Postulate | If two polygons are congruent, then they have the same area |
| Area Addititon Postulate | The area of a region is the sum of the areas of its nonoverlapping parts. |
| Angle-Angle (AA) Similarity Postulate | If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. |
| . | . |
| Properties of Segment Congruence | segment congruence is reflexive, symetric, and transitive |
| Reflexive property of congruence | for any segment AB, AB is congruent to AB |
| Symetric Poroperty of congruence | if AB is congruent to CD, then CD is congruent to AB |
| Transitive property of congruence | if AB is cong. to CD and CD is cong. to EF, then AB is cong. to EF |
| Property of angle congruence | angle congruence is reflexive, symetric, and transitive |
| Right Angle Congruence Theorem | All right angles are congruent |
| Congruent Supplements Thoeroem | If two angles are suplementary to the same angle (or to cong. angles) then the two angles are congurent |
| Congruent Complements Theorem | If two angles are complementary to the same angle (or to cong. angles) then the two angles are congurent |
| Vertical Angles Theorem | Vertical Angles are congruent |
| perpendicular lines | if two lines intersect to form a linear pair of congruent angles, then th lines are perpendicular |
| complementary angles | if two sides of two adjacent acute angles are perpendicular, then the angles are complementary |
| 4 right angles | if two lines are perpendicular, they intersect to form four right angles |
| Alternate Interior Angles | If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent |
| Consecutive Interior Angles | If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary |
| Alternate Exterior Angles | If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent |
| Perpendicular Transversal | If a transversal id perpendicular to one of two parallel lines, then it is perpendicular to the other |
| Alternate Interior Angles Converse | If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel |
| Consecutive Interior Angles Converse | If two lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel. |
| Alternate Exterior Angles Converse | If two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel |
| parallel to the same line | if two lines aree parallel to the same line, then they are parallel to each other |
| perpendicular to the same line | in a plane, if two lines are perpendicular to the same line, then they are parallel to each other |
| Triangle Sum theorem | Te sum of the measures of the inteior angles of a trinagle is 180 |
| Corollary to the triangle sum theorem | The acute angles of a right triangle are complementary |
| Exterior Angels Theorem | The measure of an exterior angle of a triangle is equal to the sum of the measure of the two nonadjacent interior angles |
| Thrid Angle Theorem | If two angles of one trainagle are congruent to two angles of another triangle, then the third angles are also congruent |
| Reflexive Property of Congruent Triangles | Every triangle is congruent to itsellf |
| AAS Congruence Theorem | If two angles and a nonincluded side of one triangle are congruent to two andgles and the corresponding nonincluded side of a second triangle, then the two triangles are congruent |
| Base Angle Theorem | If two sides of a triangle are congruent, then the angles opposite them are congruent |
| Corollary to the base angles Theorem | If a triangle is equlatteral, then it is equiangluar |
| Converse of the Base angles Theorem | If two angles of a triangle are congruent, then the sides opposite them are congruent |
| Conerse of the base angles therorem corollary | Is a triangle is equiangulal, then it is equlateral |
| HL Congruence Theorem | If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right trianlge, then the two triangles are congruent. |
| Perpendicular Bisector Theorem | If a point is on a perpendicular bisector of a segment, then it is equideistant fom the endpoints of the segment |
| Converse of the Perpendicular Bisector Theorem | If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment |
| Angle Bisctor Theorem | If a point is on the bisector of an angle, then it is equidistant from the two sides of an angle |
| Converse of the Angle Bisector Theorem | If a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the bisector of the angle |
| Concurrency of Perpendicular Bisectors of a Triangle | The perpendicular bisectors of a triangle intersect at a pont that i equidistant from the verticies of the triangle |
| Concurrency of Angle Bisectors of a Triangle | The angle bisetors of a triangle intersect at a point that is equidistant from the sides of the triangle |
| Concurrency of Medians of a Triangle | The medians of a triangle intersect at a point that is twothrids of the distnace from eachvertex to the midpoint of the opposite sid |
| Concurrency of Altitudes of a Triangle | The lines containing the altitudes of a triangle are concurrent |
| Midsegment Theorem | The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long |
| side longer=larger angle | If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side |
| larger angle=longer side | If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle |
| Exterior Angle Inequality | The measure of an exterior angle of a triangle is greater than the measure of either of the two nonadjacent interior angles |
| Triangle Inequality | The sum of the lengths of any two sides of a triangle is greater than the length of the third side |
| Hinge Theorem | If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the thrid side of the first is longer than the thir side of the second |
| Converse of the Hinge | If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the second, then the included angle of the first is larger than the included angle of the second |
| Interior Angles of a Quadrilateral | The sum of the measures of the interior angles of a quadrilateral is 360 |
| parallelogram-opposite sides | If a quadriateral is a parallelogram, then its opposite sides are congruent |
| parallelogram-opposite angles | If a quadrilateral is a parallelogram, then its opposite angles are congruent |
| parallelogram-consecutive angles | If a quadrilateral is a parallelogram, then its consecutive angles are supplementary |
| parallelogram-diagonals | If a quadrilateral is a parallelogram, then its diagonals bisect each other |
| parallelogram-opposite sides converse | if both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram |
| parallelogram-opposite angles converse | if both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram |
| parallelogram-consecutive angles | if an angle of a quadrilateral is supplementary toboth of its consecutive angles, then the quadrilateral is a parallelogram |
| parallelogram-diagonals | if the diagonals of a quadrilateral bisect each othe, then the quadrilateral is a parallelogram |
| parallelogram-sides are congurent and parallel | if ne pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram |
| Rhombus Corollary | A quadrilateral is a rhombus if and only if it has four congruent sides |
| Rectangle Corollary | A quadrilateral is a rectangle i and only if it has four right angles |
| Square Corollary | A quadrilateral is a square is and only if ut us a rhobus and a rectangle |
| rhombus-diagnolas | a parallelogram is a rhombus if and only if its diagonals are perpendicular |
| rhombus-diagonals bisect angles | a parallelogram is a rhombus if and only if each diagonals bisect a pair of opposite angles |
| rectangle-diagonals | a parallelogram is a rectngle if and only if its diagonals are congruent |
| trapezoid -base angles | if a trapezoid is isoceles, then eachpair of base angles is congruent |
| trapezoid-base angles converse | if a trapezoid has a pair of congruent base angles, then it is an isisceles trapezoid |
| trapezoid-diagonals | a trapezoid is isosceles if and only if its diagonals are congruent |
| Midsegment Theorem for Trapezoids | the midsegement of a trapezoid is parallel to each base and its length is one hlf he sum of the lengths of the bases. |
| kite-opposite angles | If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent |
| kite-diagonals | if a quadrilateral is a kite, then its diaginals are perpendicular |
| Area of a Rectangle | The area of a rectangle is the product of its base and height A=bh |
| Area of a Parallelogram | The area of a parallelogram is the product of a base and its corresponding height A=bh |
| Area of a Triangle | The area of a triangle is one half the product of a base and its corresponding height A=1/2bh |
| Area of a Trapezoid | The area of a trapezoid is one half the product of the height and the sum of the bases A=1/2h(b1+b2) |
| Area of a Kite | The area of a kiet is one half the product of the length of its diagonals A=1/2 d1d2 |
| Area of a Rhombus | The area of a rhombus is equal to one half the product of the lengths of the diagonalls A=1/2d1d2 |
| similar polygons-perimeter ratios | if two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths |
| SSS Similarity Theorem | If the lengths of the corresponding sides of two triangles are proportional, then the triangles are similar |
| SAS Similarity Theorem | If an angle of one triangle is congruent to an angle of a second triangle, and the lengths of the sides including these angls are proportional, then the triangles are similar |
| Triangle Proportionality Theorem | If a line parallel to one side of a triangle intersects the other two sides, then it devides the two sides proportionally |
| Converse of the Triangle Proprotionality Theorem | If a line divides two sides of a triangle proportionally, then it is parallel to the thrid side |
| three parallel lines-two transversals | If three parallel lines intersect two transversals, then they divide the transversals proportionally |
| ray-bisecting angle | If a ray bisects an angle of a triangle, then it divides the opposite side into segmenta whose lengths are proportional to the lengths of the other two sides. |
Created by:
rtgiggle