Triangles

Triangles

1. Inequalities

   • For any real number a and b, a > b if and only if there is a positive number c such that a= b + c
   • Theorems
     • Exterior Angle Inequality
       • The measure of an exterior angle of a triangle is greater than the measure of either of its corresponding remote interior angles
     • Triangle Inequality Theorem
       • The sum of the lengths of any two sides of a triangle must be greater than the length of the third side
     • Hinge Theorem
       • If 2 sides of a triangle are congruent to 2 sides of another triangle, and the included angle of the first is larger than the included angle of the 2nd triangle, then the 3rd side of the 1st triangle is longer than the 3rd side of the 2nd triangle
     • Converse Hinge Theorem
       • If 2 sides of a triangle are congruent to 2 sides of another triangle, and the 3rd side of another triangle, and the 3rd side in the 1st is longer than the 3rd side in the 2nd triangle, then the included angle measure of the 1st triangle is greater than the included angle measure in the 2nd triangle
   • Angle-Side Relationships in Triangle
     • If one side of triangle is longer than another side, then the opposite the longer side has a greater measure than the angle opposite the shorter side
     • If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle

2. Congruence

   • Congruent
     • When two figures have exactly the same shape and size
     • Congruent Polygons
       • All parts of a polygon are congruent to the corresponding parts of another polygon
       • Two polygons are only congruent if their corresponding parts are congruent
         • CPCTC= Corresponding Parts of Congruent Triangles are Congruent
     • Proving Congruence
       • Side-Side-Side (SSS) Congruence
         • 3 sides of 1st triangle≅ 3 sides of 2nd triangles
       • Side-Angle-Side (SAS) Congruence
         • Two sides and a included angle of 1st triangle≅two sides and a included angle of 2nd triangle
       • Angle-Side-Angle (SAS) Congruence
         • 2 angles and a included side of te 1st triangle ≅ 2 angles and a included side of the 2nd angle
       • Angle-Angle-Side (AAS) Congruence
         • 2 angles and non included side the 1st triangle ≅ 2 angles and the non included side of the 2nd triangle
           • Then the triangles are congruent
   • Corresponding parts
     • Matching parts
     • Corresponding sides
     • Corresponding angles
   • Third angle theorem
     • 2 angles of 1st triangle are congruent to 2 angles of 2nd triangle
       • Then the 3rd angles of the triangles are congruent
   • Properies
     • Reflexive property
     • Symmetric Property
     • Transitive Property

3. Classyfying

   • By angles
     • Acute triangle
       • 3 acute angles
     • Equiangular Triangle
       • 3 congruent acute angles
     • Obtuse Triangle
       • 1 obtuse angle
     • Right triangle
       • 1 right angle
   • By sides
     • Equilateral triangle
       • 3 congruent sides
     • Isosceles
       • at least 2 congruent sides
     • Scalene triangle
       • No congruent sides
   • Theorems
     • Isosceles Triangle Theorem
       • If 2 sides of a triangle are congruent, then the angle opposite of those sides are congruent
     • Converse of Isosceles Triangle Theorem
       • If 2 angles of a triangle are congruent, then the side opposite of those angles are congruent
   • Isosceles Properties
     • Legs
       • 2 congruent sides
     • Vertex angle
       • Angle with sides that are legs
     • Base Angles
       • 2 angles formed by the base and the congruent sides
   • Equilateral Properties
     • A triangle is only equilateral if it is equiangular
     • Each angle measures 60

4. Special Segments

   • Perpendicular Bisector
     • Circumcenter
       • The circumcenter is equidistant from each vertex
   • Angle Bisector
     • Incenter
       • The incenter is equidistant from each side of the triangle
   • Median
     • Centroid
       • The centroid is 2/3 of the distance from each midpoint of the opposite side
   • Altitude
     • orthocenter
       • The lines containing the altitudes are concurrent at the orthocenter
   • Bisectors of Triangles
     • Perpendicular to the segment
       • Theorems
         • Perpendicular Bisector Theorem
           • If a point is on the perpendicular bisector of segment, then it is equidistant from the endpoints of the segment
         • Converse of the Perpendicular Bisector Theorem
           • If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment
         • Angle Bisector Theorem
           • The points that are on the bisector of an angle, then it is equisdant from the sides of the angle
         • Converse of the Angle Bisector Theorem
           • The point in the inferior of an angle is equisdant from the sides of the angle, then it is on the bisector of the angle
         • Incenter Theorem
           • The angle bisectors of a triangle intersect at a point called the incenter that is equidistant from the sides of the triangle

5. Transformation

   • Transformation
     • operation that maps an original geometric figure
     • can change the position, size or shape of figure
   • Congruence Transformation
     • The position of the image is different from the preimage
       • Isometry
         • Rigid Transformation
   • Reflection
     • Transformation over a line called the line of reflection
       • Each point of the preimage and image are the same distance from the line of reflection
   • Translation
     • transformation that moves all the points of the image the same distance and the same direction
   • Rotation
     • transformation around a fixed point, through a specific angle and in a specific direction

6. Angles

   • Triangle Angle-Sum Theorem
     • The sum of the measures of the triangle is 180.
       • m<a + m<b +m<c =180
     • Auxilary Line
       • Extra segment included to help analyze geometric relationships
   • Exterior Angle Theorem
     • Exterior angle measure= sum of the measures of the two remote interior angles
       • M<a + M<b = M<1
         • Exterior angles
           • Formed by one side of the triangle and the extension of an adjacent side
         • Remote interior angles
           • Each triangle has 2
             • Not adjacent to exterior angle

7. Triangles

   • Vanessa Olvera 9-4

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