Chapter 4: Lessons 1,2,3, & 6

Chapter 4: Lessons 1,2,3, & 6 By Mai Mohammad

Lesson 1: Coordinates & Distance Quadrants: I, II, III, IV Axes: x-axis, y-axis Origin: O (0,0) Coordinates: A (6,3), B (-8,7)C (-3,-5), D (3,-2) A one-dimensional coordinate system is used to choose an origin A two-dimensional coordinate system to locate points in the plane

The Pythagorean Theorem gives us the distance formula: The length of AB and BC are given (using the grid) AB² + BC² = AC² AC is the distance The Distance Formula: The distance formula is used to find the distance from one point to another using their coordinates

Lesson 2: Polygons and Congruence Definition of a polygon:A connected set of at least three line segments in the same plane such that each segment intersects exactly two others, one at each endpoint Not polygons: Polygons:

Definition of congruent triangles:Two triangles are congruentiff there is a correspondence between their vertices such that all of their corresponding sides and angles are equal Corollary to the definition of congruent triangles:Two triangles congruent to a third triangle are congruent to each other

Lesson 3: ASA and SAS Congruence The ASA Postulate: If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent (a side included by 2 angles) The SAS Postulate: If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent (an angle included by 2 sides)

Lesson 6: SSS Congruence The SSS Theorem: If the three sides of one triangle are equal to the three sides of another triangle, the triangles are congruent

Lab: Proving Triangles Congruent At least three pieces of the criteria are necessary to prove congruence (two angles and a segment, two segments and an angle, three segments, etc.) Proves why ASA, AAS, SSS work and other combinations, like AAA, do not

Summary: To find the distance between two points, the Pythagorean Theorem or the distance formula can be used Polygons are made up of at least three line segments of the same plane that intersect exactly two other segments, one at each endpoint (triangle, square, pentagon, etc.) ASA, SAS, SSS, and AAS prove triangle congruence

CHAPTER 4LESSONS 4, 5, 7 & PROOFS By Clare Strickland

Lesson 4: Congruence Proofs Two triangles are congruent iff there is a correspondence between their vertices such that all of their corresponding sides and angles are equal: Corresponding Parts of Congruent Triangles are Equal (CPCTE) Generally proved using SAS, ASA, or SSS Can go in many different orders

EXAMPLE PROOF

Lesson 5: Isosceles and Equilateral Triangles • A triangle is: • Scaleneiff it has no equal sides • Isoscelesiff it has at least 2 equal sides • Equilateraliff all of its sides are equal

Lesson 5: Isosceles and Equilateral Triangles • A triangle is • Obtuseiff it has an obtuse angle • Right iff it has a right angle • Acuteiff all of its angles are acute • Equiangulariff all of its angles are equal

Lesson 5: Isosceles and Equilateral Triangles • Theorems: • If two sides of a triangle are equal, the angles opposite them are equal. • If two angles of a triangle are equal, the sides opposite them are equal.

Lesson 5: Isosceles and Equilateral Triangles • Corollaries: • An equilateral triangle is equiangular • An equiangular triangle is equilateral

Lesson 7: Constructions • How to copy a line segment: • Set the radius of the compass to the length of AB. Draw line l and mark point P. With P as center, draw an arc of radius AB that intersects line l and draw point Q.

Lesson 7: Constructions • How to copy an angle: • Draw PQ as one ray of the angle. With point A as its center, draw an arc to create points B and C. Using that same radius on the compass, draw an arc on line PQ. Set the radius on your compass to length BC. Use that compass setting to draw an arc with point R at its center. Mark the intersection of the arcs as point S. Draw line segment PS

Lesson 7: Constructions • How to copy a triangle: • Construct line segment XY equal to AB. Set the compass length of CB, and with point Y as its center construct an arc of that length. Set the compass length of CA, and with point X as its center construct an arc of that length. Mark the point of intersection of the two arcs as point Z. Use a straightedge to construct XZ and YZ

Proofs • Tips for Proofs: • Set up the two columns (Statements & Reasons) and number each step • Mark up your figure with your given • Identify what you’re looking for • When you name an angle, use three letters • Be careful of when you’re using arrows versus • Use different colors to help visualize

Reasons for Two Column Proofs

Given: BD is a bisector of AC, BD is perpendicular to AC Prove: ABC is isosceles Statements: 1. BD is a bisector of AC, BD is perpendicular to AC 2. AD=AC ADB & CDB are right angles ADB= CDB 5. BD = BD ADB = CDB 7. AB=CB 8. ABC is isosceles Reasons: 1. Given 2. Bisector  2 = parts 3. Perp right angles 4. All right angles  = 5. Reflexive Property 6. SAS (steps 2, 4, 5) 7. CPCTE 8. Def. of isosceles

Chapter 4: Lessons 1,2,3, & 6