geo


Section 1.6

  • Angle Pairs - Adjacent angles with a common vertex and side but no index points in common. 

  • Vertical Angles - Angles whose sides form opposite rays

  • Linear Pair - Angles whose noncommon sides are opposite rays, and form a line. 

  • Complementary Angles - Angles whose measures add up to 90 degrees.

  • Supplementary Angles - Angles whose measures add up to 180 degrees.

  • Angle Bisector - Array that divides an angle into 2 adjacent angles that are congruent.


Section 1.7 

  • Midpoint Formula 

  •  Distance Formula 

  • Simplifying Radicals

Find the largest perfect square that's also a multiple of the radical. Factor. Simplify the perfect square


Section 1.8

  • Perimeter - Distance around a geometric figure. 

    • Circumference - Distance around a circle

  • Area - Number of square units that a geometric figure encloses.



Section 2.2 

  • Inductive Reasoning - The process of arriving at a general conclusion based on observing patterns or specific examples. 

    • Conjecture - educated guess based on inductive reasoning


Section 2.3 

Our conditional statement is p -> q

  • Converse - Formed by switching the hypothesis and conclusion (q -> p)

  • Inverse - Formed by negating the hypothesis and conclusion of the conditional statement. (~p -> ~q)

  • Contrapositive - Formed by negating the hypothesis and conclusion of the converse statement (~q -> ~p)

  • Logically Equivalent Statements 

    • Conditional = Contrapositive (p -> q = ~q -> ~p)

    • Converse = Inverse (q -> p = ~p -> ~q)


2.5 Lesson

  • Deductive Reasoning - The process of proving a specific conclusion from one or more general statements.

  • Law of Detachment - If p -> q is a true statement and p is true, then q is true.

    • Ex: If there's lightning, it's not safe. Marla saw lightning.

    • Its not safe!

  • Law of Syllogism - If p -> q is true, and q -> r is true, then p -> r is a true statement.

    • Ex: If today is Wednesday, then I’m going to basketball practice. If I'm going to practice, then I need my shoes.

    • Therefore… If today is Wednesday, then I need my shoes.


Lesson 2.6

Algebraic Properties of Equality

  • Addition: If a=b, then a+c = b+c

  • Subtraction: If a=b, then a-c = b-c

  • Multiplication: If a=b, then a*c = b*c

  • Division: If a=b and c≄0, then a/c = b/c

  • Reflexive: a=a

  • Symmetric: If a=b, then b=a

  • Transitive: If a=b and b=c, then a=c.

  • Substitution: If a=b, then b can be replaced with a in any expression and vice versa

  • Distributive: Given a(b+c), then ab + ac.


Section 2.7

  • Theorem 2.7.1 (Equal Complements Theorem) 

    •  Compliments of the same angle, or of congruent angles, are equal in measure.

  • Theorem 2.7.2 (Equal Supplements Theorem) 

    • Supplements of the same angle, or of congruent angles, are equal in measure.

  • Theorem 2.7.3 (Linear Pair Theorem)

    • If 2 angles form a linear pair, then the angles are supplementary.

  • Theorem 2.7.4 (Vertical Angles Theorem)

    • Vertical angles are congruent.

  • Theorem 2.7.5 (Right Angles Congruence Theorem)

    • All right angles are congruent.

  • Theorem 2.7.6 (Equal Supplements Theorem)

    • Two equal supplementary angles are right angles.


Section 3.1

  • Parallel Lines - Coplanar lines that  do not intersect.

  • Skew Lines - Non-coplanar lines that do not intersect and are not parallel

  • Parallel Planes - Planes that do not intersect

  • Transversal - Lines that intersect two or more coplanar lines at different points


Angles Formed by Transversals

  • Alternate Interior Angles - Non Adjacent interior angles that lie on opposite sides of the transversal.

  • Same side Interior Angles - Interior Angles that lie on the same side of the transversal.

  • Corresponding Angles - Angles that lie on the same side of the transversal in corresponding positions

  • Alternate Exterior Angles - Nonadjacent exterior angles that lie on opposite sides of the transversal.


Section 3.2

  • Postulate 3.2.1 (Parallel Post.)

    • Through a point not on a line, there is one and only one line parallel to the given line. 

  • Postulate 3.2.2 (Perp Post.)

    • Through a point not on a line, there is one and only one line perpendicular to the given line. 

  • Postulate 3.2.3 (2 Lines Perp. to 3rd Line)

    • In a line, if 2 lines are perpendicular to the same line, then they are parallel to each other.

  • Theorem 3.2.4 (Alternate Interior Angles Theorem)

    • If 2 lines and a transversal form alternate interior angles that are congruent, then the lines are parallel.

  • Theorem 3.2.5 (Corr. Angles Theorem)

    • If 2 Lines of a transversal form corresponding angles that are congruent then the lines are parallel.

  • Theorem 3.2.6 (Same-side Interior Angles Theorem)

    • If 2 lines and a transversal form a same side interior angles that are supplementary, then the lines are parallel.

  • Theorem 3.2.7 (Alternate Exterior Angles Theorem)

    • If 2 lines and a transversal form alternate exterior angles that are congruent, then the lines are parallel


Section 3.3

  • Theorem 3.3.1 (Alternate Interior Angles Converse)

    • If 2 parallel lines are cut by a transversal, then alternate interior angles are congruent.

  • Theorem 3.3.2 (Corr. Angles Converse)

    • If 2 parallel lines are cut by a transversal, then corresponding angles are congruent.

  • Theorem 3.3.3 (Same-side Interior Angles Converse)

    • If 2 parallel lines are cut by a transversal, then same-side interior angles are supplementary.

  • Theorem 3.3.4 (Alternate Interior Angles Converse)

    • If 2 parallel lines are cut by a transversal, then alternate exterior angles are congruent.


Section 3.4

  • Theorem 3.4.1 (Perpendicular Transversal Theorem)

    • In a plane, let 2 parallel lines be cut by a transversal. If the transversal is perpendicular to one of the parallel lines, then it is perpendicular to the other.

  • Theorem 3.4.2 (2 Lines Parallel to 3rd Line)

    • When 2 lines are parallel to the same line, then all 3 lines are parallel to each other.

  • Theorem 3.4.3

    • If 2 lines are perpendicular, then they intersect to form 4 right angles.

  • Theorem 3.4.4

    • If 2 lines intersect to form a linear pair of congruent angles, then the lines are perpendicular to each other.


Section 3.6 

  • Slope - measures the tilt of a line. Ratio of vertical change to horizontal change.

    • Slope of a vertical line is undefined.

    • Slope of a horizontal line is 0.

  • Slope Intercept Form

    • y = mx + b, where m is the slope and b is the y-intercept.

  • Point-slope Form

    • y - y1 = m (x - x1), where m is the slope and x1, y1 is the point on the line.

  • Perpendicular Lines - Have slopes that are opposite reciprocals. Have a product of -1.


Section 3.7 

  • Standard Form: ax + by = c. Where a, b, and c are integers.


  • Ex: Find equation of a line w/ slope -4 containing point (-2,5) in slope-intercept form.

    • Y - y1 = m (x - x1)

Y - 5 = -4 (x + 2)

Y - 5 = -4x - 8

Y = -4x -3


Section 4.1

  • Triangle - figure formed by 3 noncollinear points connected by segments

  • Adjacent Sides - Two sides that share a common vertex


Triangle Classification

  • By sides

    • Scalene = No congruent sides

    • Isosceles = At least 2 sides are congruent

      • Equilateral = All 3 sides are congruent

  • By angles

    • Acute = All angles are less than 90 degrees

      • Equiangular = Contains three 60 degree angles

    • Right = Contains a 90 degree angle

    • Obtuse = Contains an angle above 90 degrees


  • Theorem 4.1.1 (Triangle-sum Theorem)

    • The sum of the measures of the interior angles of a triangle is 180


  • Corollary - special name for a theorem easy to prove as a direct result from another proven theorem

  • Corollary 4.1.2 (Exterior Angles)

    • The measure of each exterior angle of a triangle equals the sum of its measures of its two non-adjacent angles.


Section 4.2

  • Congruent Figures - Figures that have the exact same slope and size. Has corresponding sides and congruent angles.

  • Theorem 4.2.1 (3rd Angles Theorem)

    • If 2 angles of a triangle are congruent to 2 angles of another triangle, then the 3rd pair of angles are congruent.


Section 4.3 

  • Postulate 4.3.1 (Side-Side-Side Post.) SSS

    • If the 3 sides of a triangle are congruent to the 3 sides of another triangle, then they are congruent.

  • Postulate 4.3.2 (Side-Angle-Side Post.) SAS

    • If 2 sides and the included angle of a triangle are congruent to 2 sides and the included angle of another, the the triangles are congruent


Section 4.4

  • Postulate 4.4.1 (Angle-Side-Angle Post.) ASA

    • If 2 angles and the included side of a triangle are congruent to 2 angles and the included side of another, the the triangles are congruent.

  • Postulate 4.4.2 (Angle-Angle-Side Post.) AAS

    • If 2 angles and the included side of a triangle are congruent to 2 angles and the included side of another, the the triangles are congruent.


Section 4.5 

  • CPCTC - Stands for “Corresponding Parts of Congruent Triangles are Congruent”


Section 4.6 

  • Isosceles Triangle

    • Legs - 2 congruent sides

    • Base - 3rd side

    • Vertex angle - Angle opposite the base

    • Base angles - 2 angles adjacent to the base

  • Theorem 4.6.1 (Isosceles Base Angle)

    • When 2 sides of a triangle are congruent then the angle opposite those sides are congruent.

  • Theorem 4.6.2 (Converse 4.6.1)

    • When 2 angles of a triangle are congruent then the sides opposite those angles are congruent.

  • Theorem 4.6.3 (Perp. Bisector of Base)

    • If the line bisects the vertex angle of the isosceles triangle, then the line is also the perpendicular bisector of the base

  • Corollary 4.6.4 

    • If a triangle is equilateral, then it is equiangular

  • Corollary 4.6.5

    • If a triangle is equiangular, then it is equilateral

  • Right Triangles

    • Hypotenuse - Side opposite of the angle

    • Legs - 2 sides that are not the hypotenuse

  • Theorem 4.6.6 (Hypotenuse-Leg Theorem)

    • If the hypotenuse and leg of 1 right triangle are congruent to the hypotenuse and leg of another right triangle,, then the triangles are congruent.


Section 5.1

  • Perpendicular bisector - A line, segment, ray or angle that is perpendicular to the segment at its midpoint.

  • Theorem 5.1.2 (Perp Bisector)

    • If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment

  • Theorem 5.1.3 (Conv. Perp Bisector)

    • If a point is equidistant from the endpoints of a segment, then it is in the perpendicular bisector of the segment.

  •  Theorem 5.1.5 (Angle Bisector)

    • If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle.

  • Theorem 5.1.6 (Conv.  Angle Bisector)

    • If a point in the interior of an angle is equidistant from the sides of the angle, then the point is on the angle bisector.


Section 5.2

  •  Concurrent - Describes 3 or more lines that intersect at 1 point

  • Point of congruence - Point at which 3 or more lines intersect.

  • Theorem 5.2.1

    • The perp bisector of the sides of the triangle are concurrent of a point equidistant from the vertices. 

  • Circumcenter: Point of concurrency of the perpendicular bisector of a triangle.

    • Acute Triangle: Inside the triangle

    • Right Triangle: Midpoint of the hypotenuse

    • Obtuse Triangle: Outside the triangle

  • Theorem 5.2.2 (Concurrency of Angle Bisectors)

    • The bisectors of angles of a triangle are concurrent at a point equidistant from the sides of a triangle.

  • Incenter - Point of concurrency of the angle bisectors of a triangle

Section 5.3

  • Medians - segment whose endpoints are the vertex and the midpoint of the opposite side

  • Centroid - Point of concurrency of the medians of a triangle 

    • Center of Balance

  • Theorem 5.3.1 (Concurrency of Medians)

    • Medians of the triangle are concurrent at a point ⅔  the distance from each vertex to the midpoint of the opposite side.

  • Centroid of a triangle

    • The centroid of a triangle with vertices (x1, y1) (x2, y2) (x3, y3) is formed by:

  • Altitude - Perpendicular segment forming the center of a triangle to the line containing the opposite side.

  • Orthocenter - Point of concurrency of the altitude of a triangle

    • Acute - Inside the triangle

    • Right - Point of the right angle

    • Obtuse - Outside the triangle


Section 5.4 

  • Midsegment - Segment connecting the two midpoints of 2 sides of a triangle 

  • Theorem 5.4.1 (Triangle Midsegment)

    • If a segment joins the midpoints of 2 sides of a triangle, then the segment is parallel to the third side and ½ as long.


Section 5.5

  • Property 5.5.1 (Comparison Prop. of Inequality)

    • If a = b+c and c>0, the a > b

  • Corollary 5.5.2

    • The measure of an exterior of an angle is greater than the measures of either of its remote interior angles.

  • Theorem 5.5.3 (Triangle Inequality Theorem)

    • If 2 sides of a triangle are not cong, the the angle opposite the longer side is greater than the angle opposite the shorter side.

  • Theorem 5.5.4 (Converse Angle Inequality)

    • If 2 angles of a triangle are not congruent, then the side opposite the greater angle is longer than the side opposite the lesser angle

  • Theorem 5.5.5 (Triangle Inequality for Side Length)

    • The sum of the lengths of any 2 sides of a triangle is greater than the length of the 3rd side


Section 5.6

  • Theorem 5.6.1 (Hinge Theorem) 

    • If 2 sides of a triangle are congruent to 2 sides of another triangle, and the included angles are not congruent, then the longer 3rd side is opposite the included angle.

  • Theorem 5.6.2 (Converse Hinge Theorem)

    • If 2 sides of a triangle are congruent to 2 sides of another triangle, and the 3rd sides are not congruent, then the larger included angle is opposite the longer side.