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Geo-Trig Midterm
| Question | Answer |
|---|---|
| Point | Has no dimension. Represented by a dot. Named using a capital letter. |
| Line | One dimensional. A set of points that extend infinitely in two opposite directions. Straight. Named using two labeled points or one lowercase letter. |
| Plane | A two dimensional flat surface that extends infinitely in two directions. Represented by a four sided figure. Named using 3-4 points or a capital letter that is not a point. |
| Collinear Points | 3 or more points on the same line |
| Coplanar Points | Points that lie in the same plain |
| Line segment | Part of a line that has two end points. Named by its two end points. |
| Ray | Part of a line. One endpoint that extends infinitely in one direction. Named by its end point and one other point. |
| Opposite Rays | Two rays that share a common end point that extends infinitely in opposite directions. |
| Postulate/Axiom | A statement that is accepted to be true without proof. |
| Ruler Postulate | Given line AB. AB = |B-A| |
| Equality System | System that compares size using measurements. |
| Congruence System | System that compares parts to show that figures are the same size and/or shape. |
| Segment Addition Post | Given segment AC with midpoint B. AB + BC = AC. |
| Distance Formula | States that distance between two points can be found by taking the square root of the difference of the x coordinates squared added to the difference of the y coordinates squared. |
| Angle | A geometric figure that consists of two rays (sides) that meet at a common point (vertex). Usually named by its vertex. |
| Vertex | The common point of an angle. |
| Protractor Postulate | States that when using a protractor, you should put the vertex at 0/180 and then subtract the two numbers that the sides touch to get the measure of the angle. |
| Angle Addition Postulate | States that you can add the parts of an angle to get the whole angle. |
| Acute Angle | An angle that is less than 90 degrees. |
| Right Angle | An angle that is 90 degrees. |
| Obtuse Angle | An angle that is more than 90 degrees but less than 180 degrees. |
| Straight Angle | An angle that is 180 degrees. |
| Adjacent Angles | A pair of angles that share the same vertex and a common side. |
| Bisect | To cut into to equal parts. |
| Midpoint | The point that divides a segment into two equal segments. |
| Segment Bisector | A segment that passes through the midpoint of another segment. |
| Angle Bisector | A ray that divides an angle into two equal angles. |
| The Midpoint Formula | States that you can find the midpoint of a segment in the following way. X coordinate: add the two x coordinates and divide them by 2. Y coordinate: add the two y coordinates and divide them by two. |
| Complementary Angles | A pair of angles that add up to 90 degrees. May or may not be adjacent. |
| Supplementary Angles | A pair of angles that add up to 180 degrees. May or may not be adjacent. |
| Linear Pair | Pair of angles that form a straight angle. Must be adjacent. AKA Adjacent supplementary angles. |
| Vertical Angles | A pair of angles that are across (opposite) from each other that have the same vertex. Never adjacent. Always equal. |
| Perpendicular Lines | Coplanar lines that intersect to form right angles. |
| Line Perpendicular to a Plane | A line that intersects the plane in a point and is perpendicular to every line in the plane that intersects it. |
| Biconditional Statement | A statement that contains the phrase if and only if. (p if and only if q) |
| Conditional Statement | A type of logical statement that has two parts, a hypothesis and a conclusion. (p then q) |
| Converse | The statement formed by switching the hypothesis and conclusion of a conditional statement. (q then p) |
| Negation | The negative of a statement. Represented by ~. |
| Inverse | The statement formed when you negate the hypothesis and conclusion of a conditional statement. (~p then ~q) |
| Contrapositive | The statement formed when you negate the hypothesis and conclusion of the converse of a conditional statement. (~q then ~p) |
| Deductive Reasoning | Uses fact, definitions, and accepted properties in a logical order to write a logical argument. |
| Inductive Reasoning | Uses patterns and observations to form a conjecture. |
| Conjecture | An unproven statement |
| The Law of Detachment | If p --> q is a true conditional statement and p is true, then q is true. |
| The Law of Syllogism | If p --> q and q --> r are true conditional statements, then p --> r is true. |
| Addition Property of Equality | If a=b, then a+c=b+c |
| Subtraction Property of Equality | If a=b, then a-c=b-c |
| Multiplication Property of Equality | If a=b, then ac=bc |
| Division Property of Equality | If a=b, then a/c=b/c |
| Distributive Property of Equality | a(b+c)=ab+ac |
| Reflexive Property of Equality | For any real number a, a=a |
| Symmetric Property of Equality | If a=b, then b=a |
| Transitive Property of Equality | If a=b and b=c, then a=c |
| Substitution Property of Equality | If a=b, then a can be substituted for b in any equation or expression. |
| Reflexive Property of Segment Congruence | For any segment AB, AB is congruent to AB. |
| Symmetric Property of Segment Congruence | If AB is congruent to CD, then CD is congruent to AB. |
| Transitive Property of Segment Congruence | If AB is congruent to CD, and CD is congruent to EF, then AB is congruent to EF. |
| Reflexive Property of Angle Congruence | For any angle A, angle A is congruent to anlge A. |
| Symmetric Property of Angle Congruence | If angle A is congruent to angle B, then angle B is congruent to angle A. |
| Transitive Property of Angle Congruence | If angle A is congruent to angle B, and angle B is congruent to angle C, then angle A is congruent to angle C. |
| Right Angle Congruence Theorem | All right angles are congruent. |
| Congruent Supplements Theorem | If two angles are supplementary to the same angle then they are congruent. |
| Congruent Compliments Theorem | If two angle are complimentary to the same angle then they are congruent. |
| Linear Pair Postulate | If two angles form a linear pair, then they are supplementary. |
| Vertical Angles Theorem | All vertical angles are congruent. |
| Parallel Lines | Coplanar lines that do not intersect. |
| Skew Lines | Lines that do not intersect and are not coplanar. |
| Parallel Planes | Planes that do not intersect. |
| 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 through the point perpendicular to the given line. |
| Transversal | A line that intersects two or more coplanar lines at different points. |
| Interior Angles | Inside Angles |
| Exterior Angles | Outside Angles |
| Corresponding Angles | Two angles that occupy corresponding positions. They are always on the same side of the transversal. |
| Alternate Exterior Angles | Two angles that are exterior and cross over the transversal. |
| Alternate Interior Angles | Two angles that are interior and cross over the transversal. (Z shaped) |
| Consecutive Interior Angles | Two interior angles that are on the same side of the transversal. (Same side interior) |
| Hypothesis | If part of a conditional statement. |
| Conclusion | Then part of a conditional statement. |
| Through any two points there exists exactly one line. | Postulate 5 |
| A line contains at least two points. | Postulate 6 |
| If two lines intersect, then their intersection is exactly one point. | Postulate 7 |
| Through any three noncollinear there exists exactly one plane. | Postulate 8 |
| A plane consists of at least three noncollinear points. | Postulate 9 |
| If two points lie in a plane, then the line containing them lies in the plane. | Postulate 10 |
| If two plans intersect, then their intersection is a line. | Postulate 11 |
| Equivalent Statements | Two statements that are both true or both false. |
| Two-Column Proof | Has numbered statements and reasons that show the logical order of an argument. |
| If two lines intersect to form a linear pair of con ground angles, then the line perpendicular. | Theorem 3.1 |
| If two sides of two adjacent acute angels are perpendicular, then the angles are complimentary. | Theorem 3.2 |
| If two lines are perpendicular, then they intersect to form four right angles. | Theorem 3.3 |
| Corresponding Angles Postulate | If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. |
| Alternate Interior Angles Theorem | If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. |
| Consecutive Interior Angles Theorem | If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. |
| Alternate Exterior Angles Theorem | If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. |
| Perpendicular Transversal Theorem | If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other. |
| Corresponding Angles Converse Postulate | If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel. |
| Alternate Interior Angles Converse Theorem | If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel. |
| Consecutive Interior Angles Converse Theorem | If two lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel. |
| Alternate Exterior Angles Converse Theorem | If two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel. |
| If two lines are parallel to the same line, then they are parallel to each other. | Theorem 3.11 |
| In a plane, if two line are perpendicular to the same line, then they are parallel to each other. | Theorem 3.12 |
| Slopes of Parallel Lines Postulate | In a coordinate, two nonvertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel. |
| Slope Intercept Form | y=mx+b |
| Slope Formula | m=change in y/change in x |
| Rule of Parallel Lines | Two parallel lines must have the same slope but different y-intercepts. |
| Slopes of Perpendicular Lines Postulate | In a coordinate plane, two nonvertical lines are perpendicular only if the product of their slopes are -1 or negative reciprocals. |
| Triangle | A figure formed by three segments joining three noncollinear points. |
| Equilateral Triangle | A triangle with 3 congruent sides |
| Isosceles Triangle | A triangle with at least 2 congruent sides |
| Scalene Triangle | A triangle with no congruent sides |
| Acute Triangle | A triangle with 3 acute angles |
| Equiangular Triangle | A triangle with 3 congruent angles |
| Right Triangle | A triangle with 1 right angle |
| Obtuse Triangle | A triangle with 1 obtuse angle |
| Triangle Sum Theorem | The sum of the measures of the interior angles of a triangle is 180 degrees. |
| Exterior Angle of a Triangle Theorem | The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles. |
| Corollary to a Theorem | A statement that can be proved easily using a theorem. |
| Congruence Statement | A statement that shows two triangles are congruent to each other. |
| Third Angle Theorem | If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent. |
| SSS Congruence Postulate | If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. |
| SAS Congruence Postulate | If two sides and the included angle of one triangle are congruent to two sides 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. |
| AAS Congruence Theorem | If two angles and a nonincluded side of one triangle are congruent to two angles and a non included sife of a second triangle, then the two triangles are congruent. |
| Corresponding Parts of Congruent Triangles are Congruent | CPCTC |
| Base Angles | The two angles adjacent to the base of an isosceles triangle. |
| Vertex Angles | The angle opposite the base in an isosceles triangle. |
| Base Angles Theorem (Isosceles Triangle Theorem) | If two sides of a triangle are congruent, then the opposite them are congruent. |
| Base Angles Converse Theorem | If two angles of a triangle are congruent then the sides opposite them are congruent. |
| Corollary to the Base Angles Theorem | If a triangle is equilateral, then it is equiangular. |
| Corollary to the Base Angles Converse Theorem | If a triangle is equiangular, then it is equilateral. |
| Hypotenuse Leg Congruence Theorem | If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the triangles are congruent. |
| Perpendicular Bisector | A segment, ray, line, or plane that is perpendicular to a segment at its midpoint. |
| Equidistant | Same distance |
| Distance from a Point to a Line | The length of the perpendicular segment form the point to the line. |
| Perpendicular Bisector Theorem | If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of a segment. |
| Perpendicular Bisector Converse Theorem | If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. |
| Angle Bisector Theorem | If a point is on the angle bisector of an angle, then it is equidistant from the sides of the angle. |
| Angle Bisector Converse Theorem | If a point in the interior of an angle and is equidistant from the sides of the angle, then it lies on the bisector of the angle. |
| Perpendicular Bisector of a Triangle | A line that is perpendicular to a side of a triangle at the midpoint of the side. |
| Concurrent Lines | Three or more lines that meet at the same point. |
| Point of Concurrency | The point of intersection of concurrent lines. |
| Circumcenter | The point of concurrency of the perpendicular bisectors of a triangle. |
| Angle Bisector of a Triangle | A bisector of an angle of a triangle. |
| Incenter | The point of concurrency of the angle bisectors of a triangle. |
| Concurrency of Perpendicular Bisectors of a Triangle Theorem | The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle. |
| Concurrency of Angle Bisectors of a Triangle Theorem | The angle bisectors of a triangle intersect at a point that is equidistant from the sides of a triangle. |
| Median of a Triangle | A segment whose endpoints are a vertex of a triangle and the midpoint of the opposite side. |
| Centroid | The point of concurrency of the three medians of a triangle. |
| Altitude of a Triangle | The perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side of a triangle. |
| Orthocenter | The point of concurrency of the three altitudes of a triangle. |
| Concurrency of Medians of a Triangle Theorem | The medians of are concurrent at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side. |
| Concurrency of Altitudes of a Triangle | The lines containing the altitudes of a triangle are concurrent. |
| Midsegment of a Triangle | A segment that connects the midpoints of two sides of a triangle. |
| Midsegment Theorem | The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long. |
| If one side of a triangle is longer than another side, the angle opposite the longer side is larger than the angle opposite the shortest side. | Theorem 5.10 |
| If one angle of a triangle is lager than than another angle, then the side opposite the larger angle is longer than the side opposite the smallest angle. | Theorem 5.11 |
| Exterior Angle Inequality Theorem | The measure of an exterior angle of a triangle is greater than the measure of either of the two nonadjacent interior angles. |
| Triangle Inequality Theorem | The sum of the length of any two sides of a triangle is greater than the length of the third side. |
| Polygon | A plane figure that meets the following conditions: 1. Formed by three or more segments called sides, such that no two sides with a common endpoint are collinear. 2. Each side intersects exactly two other sides, one at each endpoint(vertex). |
| Quadrilateral | A 4 sided polygon |
| Pentagon | A 5 sided polygon |
| Hexagon | A 6 sided polygon |
| Heptagon | A 7 sided polygon |
| Octagon | An 8 sided polygon |
| Nonagon | A 9 sided plygon |
| Decagon | A 10 sided polygon |
| Dodecagon | A 12 sided polygon |
| n-gon | A n sided figure. |
| Convex Polygon | A polygon in which no line that contains a side of the polygon contains a point in the interior of the polygon. |
| Concave/Nonconvex Polygon | A polygon that is not convex. |
| Regular Polygon | A polygon that is both equilateral and equiangular |
| Diagonal of a Polygon | A segment that joins two nonconsecutive vertices of a polygon. |
| Interior Angles of a Quadrilateral Theorem | The sum of the measures of the interior angles of a quadrilateral is 360 degrees. |
| Polygon Interior Angles Theorem | The sum of the measures of the interior angels of a convex n-gon is (n-2)*180 |
| Corollary to the Polygon Interior Angles Theorem | The measure of each interior angle of a regular n-gon is 1/n*(n-2)*180. |
| Polygon Exterior Angles Theorem | The sum of the measures of the exterior angles of a convex, on angle at each vertex, is 360 degrees. |
| Corollary to the Polygon Exterior Angles Theorem | The measure of each exterior angle of a regular n-gon is 360/n. |
| Parallelogram | A quadrilateral with both pairs of opposite sides parallel. |
| Opposite sides of a parallelogram are congruent | Theorem 6.2 |
| Opposite angles of a parallelogram are congruent | Theorem 6.3 |
| Consecutive angles of a parallelogram are supplementary | Theorem 6.4 |
| The diagonals of a parallelogram bisect each other | Theorem 6.5 |
| Proving Quadrilaterals are Parallelograms by Definition | If you can show that both pairs of opposite sides of a quadrilateral are parallel then it is a parallelogram |
| If you can show that both pairs of opposite sides of a quadrilateral are congruent then it is a parallelogram | Theorem 6.6 |
| If you can show that both pairs of opposite angles of a quadrilateral are congruent then it is a parallelogram | Theorem 6.7 |
| If you can show that both pairs of consecutive angles in a quadrilateral are supplementary then it is a parallelogram | Theorem 6.8 |
| If you can show that the diagonals of a quadrilateral are bisect each other then it is a parallelogram | Theorem 6.9 |
| If you can show that one pair of opposite sides of a quadrilateral are both congruent and parallel then it is a parallelogram | Theorem 6.10 |
| Rhombus | A parallelogram with 4 congruent sides. |
| Rectangle | A parallelogram with 4 right angles. |
| Square | A parallelogram with 4 right angles and 4 congruent sides. |
| Special Parallelograms Memory Tip | A square is a rectangle and a rhombus, but a rectangle and a rhombus are never a square. Also when you combine a rectangle and a rhombus you get a square. |
| A parallelogram is a rhombus if and only if its diagonals are perpendicular. | Theorem 6.11 |
| A parallelogram is a rhombus if only if diagonal bisects a pair of opposite sides. | Theorem 6.12 |
| A parallelogram is a rectangle if and only if its diagonals are congruent. | Theorem 6.13 |
| Trapezoid | A quadrilateral with exactly one pair of parallel sides. |
| Trapezoid Information | The parallel sides of a trapezoid are called the bases. The nonparallel sides are called legs. The angles that border the bases are called base angles. A trapezoid has two pairs of base angles. |
| Isosceles Trapezoid | A trapezoid where the two legs are congruent. |
| If a trapezoid is isosceles, then each pair of base angles are congruent. | Theorem 6.14 |
| If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. | Theorem 6.15 |
| A trapezoid is isosceles if and only if its diagonals are congruent. | Theorem 6.16 |
| Midsegment of a Trapezoid Theorem | The midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases. |
| Kite | A quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent. |
| If a quadrilateral is a kite, then its bisectors are perpendicular. | Theorem 6.18 |
| If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent. | Theorem 6.19 |
| Circumscribe Circle | A circle with an inscribed polygon. |
| Inscribed Polygon | A polygon whose verticies all lie on a circle. |
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monopoly10
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