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