1. Chapter 4: Lessons 1,2,3, & 6 BY MAI MOHAMMAD

2. 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

3. 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

4. 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:

5. Definition of congruent triangles: Two triangles are congruent iff 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

6. 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)

7. 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

8. 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

9. 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

10. BY CLARE STRICKLAND CHAPTER 4 LESSONS 4, 5, 7 & PROOFS

11. 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

12. EXAMPLE PROOF

13. Lesson 5: Isosceles and Equilateral Triangles A triangle is:  Scalene iff it has no equal sides  Isosceles iff it has at least 2 equal sides  Equilateral iff all of its sides are equal

14. Lesson 5: Isosceles and Equilateral Triangles A triangle is  Obtuse iff it has an obtuse angle  Right iff it has a right angle  Acute iff all of its angles are acute  Equiangular iff all of its angles are equal

15. 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.

16. Lesson 5: Isosceles and Equilateral Triangles Corollaries:  An equilateral triangle is equiangular  An equiangular triangle is equilateral

17. 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.

18. 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

19. 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

20. 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

21. Reasons for Two Column Proofs Segments Definition, Postulate, or Theorem Definition of midpointMidpt = parts Definition of betweenness of pointsDef. of BOP Definition of segment bisectorSegment bisector = parts Ruler PostulateRuler Post. Betweeness of Points TheormBOP Thm. A line segment had exactly one midpointSegment 1 midpt.

22. Reasons for Two Column Proofs Angles Definition, Postulate, or Theorem Definition of Betweeness of RaysDef. of BOR Definition of Perpendicular Lines right angle Definition of straight angleStraight 180º Definition of right angleRight 90º Definition of angle bisector bisector = parts Definition of a linear pairLin. Pr. Opp rays & Definition of supplementary anglesSuppl. Sum = 180º Definition of complementary anglesCompl. Sum = 90º Protractor PostulateProtractor Post.

23. Reasons for Two Column Proofs Angles Definition, Postulate, or Theorem An angle has exactly one ray that bisects it 1 bisector Betweeness of Rays TheoremBOR Theorem If 2 angles are complementary to the same angle, they are equal compl same = If 2 angles are supplementary to the same angle, they are equal suppl same = If two angles form a linear pair, they are supplementary Lin pr suppl Vertical angles are equalVertical = If lines are perpendicular, they form 4 right angles 4 right All right angles are equalRight = If two angles in a linear pair are equal, their sides are perpendicular Lin pr = sides

24. 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 3.ADB & CDB are right angles 4.ADB= CDB 5. BD = BD 6.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