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# Geo-Trig Midterm

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).
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.
Created by: monopoly10