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1.1 Statements and Reasoning

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

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1.1 Statements and Reasoning

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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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