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Understanding Statements, Reasoning, and Geometry Fundamentals

This guide covers essential concepts in geometry, including statements, reasoning, and definitions. It emphasizes the importance of deductive reasoning in geometric proofs, distinguishing between deduction and induction. Key geometric terms are defined, such as points, lines, angles, triangles, and congruence. The document outlines postulates necessary for constructing geometric principles and explores relationships between angles, including complementary and supplementary angles. It also highlights properties of equality and congruence that are foundational in geometric proof construction, particularly concerning parallel lines.

september
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Understanding Statements, Reasoning, and Geometry Fundamentals

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  1. 1.1 Statements and Reasoning • Statement – group of words/symbols which is either true or false. • Examples of geometric statements: • mA = 80º • mB + mC = 80º • ABC is a right triangle • Line l is parallel to line m

  2. 1.1 Statements and Reasoning • Deduction – the truth of the conclusion is guaranteed. Example: • If p then q • p • Therefore q • Induction – the truth is not guaranteed. Example: • 3, 5, and 7 are odd numbers that are prime • Therefore all odd numbers are prime • Geometric proofs use deductive logic

  3. 1.2 Informal Geometry and Measurement  Point – represented by a dot Line – with arrows on each end Ray – with an arrow on one end Collinear – 3 points are collinear if they are on the same line.In between – x is in between A and B A x B

  4. 1.2 Informal Geometry and Measurement C 1 B Angle – may be referred to as ABC, B, or 1 A D Triangle – referred to as DEF E F Line segment – referred to as , BC = length of the segment C B Midpoint – if AB = BC, since B isbetween A and C, B is the midpoint A B C

  5. 1.2 Informal Geometry and Measurement • Congruence (denoted by ) • Two segments are congruent if they have the same length • Two angles are congruent if they have the same measure • Bisect – to divide into 2 equal parts

  6. 1.2 Informal Geometry and Measurement • Bisecting a segment into 2 congruent segments • Bisecting an angle into 2 congruent angles

  7. 1.3 Early Definitions and Postulates • Definitions – terminology of the mathematical system is defined.Examples: • Isosceles triangle – a triangle that has 2 congruent sides • Line segment – consists of the 2 points (endpoints) and all the points between them

  8. 1.3 Early Definitions and Postulates • Postulates – assumptions necessary to build the mathematical system.Examples: • Postulate 1: Through 2 distinct points, there is exactly one line. • Postulate 2:The measure of any line segment is a unique positive number

  9. 1.3 Early Definitions and Postulates A B C • Segment addition – AB + BC = AC • If AC = 8 and BC = 5, what is length AB?

  10. 1.4 Angles and their relationships • Angle addition - mABD + mDBC = mABC • If ABC = 130º and mDBC = 50º , what is mABD? A D B C

  11. 1.4 Angles and their relationships • Acute angle – 0 < x < 90 • Right angle - 90 • Obtuse angle – 90 < x < 180 • Straight angle - 180

  12. 1.4 Geometry Terminology – Pairs of Angles • Complementary angles – add up to 90 • Supplementary angles – add up to 180 • Vertical angles – the angles opposite each other are congruent

  13. 1.5 Introduction to Geometric Proof • Form of a geometric proof:

  14. 1.5 Introduction to Geometric Proof • Examples of Reasons: • Given (use first) • Definitions (like “definition of bisector”) • Properties (like “corresponding angles are congruent”) • Postulates and theorems (like “segment addition”)

  15. 1.5 Introduction to Geometric Proof • Properties of equality: • Reflexive (also referred to as “identity”): a = a • Symmetric: if a = b then b = a • Transitive: if a = b and b = c, then a = c

  16. 1.5 Introduction to Geometric Proof • Properties of congruence: • Reflexive (also referred to as “identity”):11 • Symmetric: if 12 then 21 • Transitive: if 12 and 23 , then 1 3

  17. 2.1 Parallel Lines – Special Angles • Intersection – 2 lines intersect if they have one point in common. • Perpendicular – 2 lines are perpendicular if they intersect and form right angles • Parallel – 2 lines are parallel if they are in the same plane but do not intersect

  18. 2.1 Parallel Lines – Special Angles 1 2 3 4 • When 2 parallel lines are cut by a transversal the following congruent pairs of angles are formed: • Corresponding angles:1 & 5, 2 & 6, 3 & 7, 4 & 8 • Alternate interior angles: 4 & 5, 3 & 6 • Alternate exterior angles: 1 & 8, 2 & 7 5 6 7 8

  19. 2.1 Parallel Lines – Special Angles 1 2 3 4 • When 2 parallel lines are cut by a transversal the following supplementary pairs of angles are formed: • Same side interior angles: 3 & 5, 4 & 6 • Same side exterior angles: 1 & 7, 2 & 8 5 6 7 8

  20. 2.1 Parallel Lines – Special Angles • Terminology: • Corresponding angles – in the same relative “quadrant” (upper right, lower left, etc.) • Alternate – on opposite sides of the transversal • Same side – on the same side of the transversal • Interior – in between the 2 parallel lines • Exterior – outside the 2 parallel lines

  21. 2.3 Parallel Lines – Review 1 2 3 4 • What type of angles are: • 1 & 8 • 4 & 6 • 4 & 5 • 2 & 6 • 1 & 7 5 6 7 8

  22. 2.3 Parallel Lines – Review • If 2 lines are parallel and cut by a transversal: • Corresponding angles, alternate interior angles, and alternate exterior angles are congruent • Same-side interior angles and same-side exterior angles are supplementary

  23. 2.3 Proving Lines Are Parallel • Given two lines cut by a transversal, if any one of the following are true: • Corresponding angles, alternate interior angles, or alternate exterior angles are congruent • Same-side interior angles or same-side exterior angles are supplementary • Then the two lines are parallel

  24. 2.3 Proving Lines Are Parallel:2 More Theorems • Two lines parallel to the same line must be parallel • Two lines perpendicular to the same line must be parallel

  25. 2.4 The Angles of a Triangle • Triangles classified by number of congruent sides

  26. 2.4 The Angles of a Triangle • Triangles classified by angles

  27. 2.4 Angles of a Triangle • In a triangle, the sum of the interior angle measures is 180º (mA + mB + mC = 180º) A B C

  28. 2.4 The Angles of a Triangle • The measure of an exterior angle of a triangle equals the sum of the measures of the 2 non-adjacent interior angles - m1 + m2 = m4 2 1 3 4

  29. Question: What do you call a parrot who just died?

  30. 2.5 Convex Polygons • Polygon - a closed plane figure whose sides are line segments that intersect only at the endpoints • Regular Polygon – a polygon with all sides equal length and all interior angles equal measure

  31. 2.5 Convex Polygons • Concave polygons: A line segment can be drawn between 2 points and the segment is outside the polygon • Convex polygons: A polygon that is not concave

  32. 2.5 Convex Polygons • Classified by number of sides

  33. 2.5 Convex Polygons • Formulas for polygons

  34. 2.5 Convex Polygons • Formulas for regular polygons

  35. 3.1 Congruent Triangles • ABC  DEF if all 3 angles are congruent and all 3 sides are congruent. • This means • AB = DE, BC = EF, and AC = DF • ABC   DEF, BAC   EDF and ACB   DFE

  36. 3.1 Congruent Triangles • Included/opposite sides and angles for ABC are: • A is opposite side BC • A is included by sides AB and AC • Side AB is opposite C • Side AB is included by A and B C A B

  37. 3.1 Congruent Triangles • SSS – If the 3 sides of a triangle are congruent to the 3 sides of a second triangle, then the triangles are congruent • SAS – If 2 sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the triangles are congruent.

  38. 3.1 Congruent Triangles • ASA - If 2 angles and the included side of a triangle are congruent to the two angles and included side of a second triangle, then the triangles are congruent. • AAS - If two angles and the non-included side of a triangle are congruent to 2 angles and the non-included side of another triangle, the triangles are congruent

  39. 3.1 Congruent Triangles • Right Triangle – In a right triangle, the side opposite the right angle is the hypotenuse and the sides of the right angle are the legs of the right triangle. • HL (hypotenuse-leg) – If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and leg of another right triangle, then the triangles are congruent.Note: In the book this is introduced in section 3.2.

  40. 3.1 Congruent Triangles • To show congruence of triangles:

  41. 3.2 Corresponding Parts of Congruent Triangles are Congruent • CPCTC – Corresponding Parts of Congruent Triangles are Congruent • Proofs using CPCTC: • Recognize that what you are trying to prove involves corresponding parts of 2 triangles • Show the triangles are congruent by SSS, SAS, ASA, AAS, etc. • State the conclusion with reason “CPCTC”

  42. 3.3 Isosceles Triangles Vertex • Parts of the isosceles triangle: Vertex Angle Leg Leg Base Base Angles

  43. 3.3 Isosceles Triangles • 2 sides (legs) of an Isosceles triangle are  (by definition) • 2 angles (base angles) of a Isosceles triangle are 

  44. 3.3 Equilateral Triangles • An equilateral triangle is also equiangular • An equiangular triangle is also equilateral • Each angle of an equilateral triangle measures 60 60 60 60

  45. 3.3 Triangle Terminology • Angle bisector: divides an angle of the triangle into two equal angles • Median: segment that connects a vertex of a triangle to the midpoint of the other side

  46. 3.3 Triangle Terminology • Altitude: line segment drawn from the vertex of a triangle that is perpendicular to the opposite side (note: the altitude can be outside the triangle) • Perpendicular bisector: (of a side of a triangle) is the line that intersects the midpoint of the side and is perpendicular to the side

  47. 3.4 Three Basic Constructions • Construct the perpendicular bisector (first half of problem 15) • Construct the angle bisector (problem 9 and second half of problem 15) • Construct an angle with the same measure with a given ray/segment as one of the sides (problem 7) • Note: trick to get a 60 degree angle is to construct an equilateral triangle

  48. 3.5 Inequalities in a Triangle • The angle opposite the larger side is the bigger angle.In ABC, if AB > AC then m C > m B A B C

  49. 3.5 Inequalities in a Triangle • The side opposite the larger angle is the bigger side.In ABC, if m C > m B then AB > AC A B C

  50. 3.5 Inequalities in a Triangle • Triangle Inequality: The sum of the lengths of any two sides of a triangle is greater than the length of the third side In ABC, CA + AB > BC A B C

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