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Chapter 4: Congruence

By: Nicholas Cheung & Olivia Giorlandino. Chapter 4: Congruence. Lesson 1: Coordinates and Distance. The origin is the starting point of any coordinate plane A one-dimensional coordinate system is a coordinate plane that only has one dimension and is made up of one horizontal line

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Chapter 4: Congruence

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  1. By: Nicholas Cheung & Olivia Giorlandino Chapter 4: Congruence

  2. Lesson 1: Coordinates and Distance • The origin is the starting point of any coordinate plane • A one-dimensional coordinate systemis a coordinate plane that only has one dimension and is made up of one horizontal line • A coordinate is a number assigned to a point on the line p 0 1 2 3 4 5

  3. Lesson 1: Coordinates and Distance • A two-dimensional coordinate system is a coordinate plane that has two dimensions and is made up of one vertical line, known as the y-axis, and one horizontal line, known as the x-axis. The origin is located where the two axis meet and divide it into 4 quadrants • The coordinates of a two-dimensional coordinate plane are found by drawing perpendicular lines from the point to the axes • Coordinates of a two-dimensional system appear in the form (x, y) meaning the x coordinate and the y coordinate’s intersection. coordinate origin

  4. Lesson 1: Coordinates and Distance • The Distance Formula: the difference between one point (x1, y1) and another point (x2, y2) √(x1-x2)2 + (y1-y2)2

  5. Lesson 2: Polygons and Congruence • 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. Polygons: Not Polygons:

  6. Lesson 2: Polygons and Congruence • Congruent: any two things that has the same size and shape as each other • Two triangles are congruent iff there is a correspondence between their vertices such that all of their corresponding sides and angles are equal • Two triangles congruent to a third triangle are congruent to each other

  7. 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. • 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 m

  8. Lesson 4: Congruence Proofs • Definition of Congruent Triangles: All 6 sets of corresponding parts of two triangles are equal iff the two triangles are congruent  Corresponding parts of congruent triangles are equal (CPCTE) FDE CAB DE AB DFE ACB EF BC DEF ABC FD AC

  9. Lesson 5: Isosceles and Equilateral Triangles • A Triangle is • Scalene iff it has no equal sides • Isosceles iff it has at least two equal sides • Equilateral iffall of its sides are equal

  10. Lesson 5: Isosceles and Equilateral Triangles • A Triangle is • Obtuse iff it has an obtuse angle • Right iff it has a right angle • Acute iffall of its angles are acute • Equiangular iffall of its angles are equal

  11. Lesson 5: Isosceles and Equilateral Triangles • Theorems using sides and angles with each other: • Isosceles Triangle Theorem (ITT): If two sides of a triangle are equal, the angles opposite them are equal • Converse to Isosceles Triangle theorem: If two angles of a triangle are equal, the sides opposite of them are equal • Corollaries: • An equilateral triangle is equiangular • An equiangular triangle is equilateral

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

  13. Lesson 7: Constructions • The constructions in this chapter allow us to accurately copy a triangles' sides, angles, and even the whole triangle itself, which allowed us to create congruent triangles from scratch • Construction 3: To copy a line segment • Construction 4: To copy an angle • Construction 5: To copy a triangle

  14. Extra Lesson The extra lesson of this chapter was a lab where we attempted to determine the minimum criteria for proving two triangles congruent. We used toothpicks and twist ties to physically determine the minimum criteria, and the results looked a little like this….

  15. 3 Corresponding Sides = Congruence?

  16. 2 Corresponding Sides + 1 Corresponding Angle = Congruence?

  17. 2 Corresponding Sides + Corresponging Included angle = Congruence?

  18. 3 Corresponding Equal Angles= Congruence?

  19. 2 Corresponding Angles+ Corresponding Included Side = Congruence?

  20. Chapter Summary The main focus of chapter four was learning what congruence is, and then learning the basic and necessary theorems and postulates to enable us to prove triangle congruence. The SAS Postulate, ASA Postulate, and SSS Theorem are introduced in this chapter, as well as key theorems concerning isosceles and equilateral triangles. Overall, chapter 4 greatly broadened our ability to complete and solve triangle proofs

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