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Chapter 5 summary, Ghalia Al-Khuraisi 10A - Coggle Diagram

- - - - Hinge theorem: If two sides of a triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second triangle, then the third side of the first triangle is longer than the third side of the second triangle.
      - Converse of the Hinge theorem: If two sides of a triangle are congruent to two sides of another triangle, and the third side in the first is longer than the third side in the second triangle, then the included angle measure of the first triangle is greater than the included angle measure in the second triangle.
  - - - identify the conclusion you are asked to prove, make the assumption that this conclusion is false by assuming that the opposite is true.
      - use logical reasoning to show that this assumption leads to a contradiction of the hypothesis or some other fact, such as a definition, postulate, theorem, or corollary.
      - point out that since the assumption leads to a contradiction, the original conclusion, what you were asked to prove, must be true.
- - - - comparison property of inequality
        
        a < b, a = b, or a > b
      - transitive property of inequality
        
        if a < b and b < c, then a < c
        
        if a > b and b > c then a > c
      - addition property of inequality
        
        if a > b, then a + c > b + c
        
        if a < b, then a + c < b + c
      - subtraction property of inequality
        
        if a > b, then a - c > b -c
        
        if a < b, then a - c < b - c
    - - The measure of an exterior angle of a triangle is greater than the measure of either of its corresponding remote interior angles.
    - - If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side
      - If one angle of triangle has great measure than another angle, then the side opposite the greater angle is longer then the side opposite the lesser angle.