Triangles

Triangles

1. Congruence

   • CPCTC
     • Corresponding Parts of Congruent Triangles are Congruent
   • Third Angles Theorem
     • If 2 angles are congruent to 2 angles of another triangle, then the third angles are congruent
   • Proving triangles are congruent
     • SSS
       • Side-Side_side
       • If 3 sides are congruent to 3 sides of another triangle, then the triangles are congruent
     • SAS
       • Side-Angle-Side
       • If 2 sides and the included angle of 1 triangle are congruent to 2 sides and the included angle of another triagle , then the triangles are congruent
     • ASA
       • Angle-Side-Angle
       • If 2 angles and the included side of 1 triangle are congruent to 2 angles and the included side of another triangle, then the triangles are congruent
     • AAS
       • Angle-Angle-side
       • If 2 angles and the nonincluded side of a triangle are congruent to the corresponding 2 angles and side of another triangle, then the 2 triangles are congruent

2. Angles

   • Theorems
     • Triangle Angle-Sum
       • The sum of the measures of the angles of a triangle is 180
     • Exterior Angle
       • The measure of an exterior angle is equal to the sum of of the measure of the 2 remote interior angles

3. Classifying

   • Angles
     • Acute
       • It has 3 acute angles
       • 3 angles less than 90 degrees
     • Obtuse
       • It has 1 obtuse angle
       • 1 angle that measures more that 90 degrees
     • Right
       • It has 1 right angle
       • an angle that measures 90 degress
     • Equiangular
       • 3 congruent angles
       • The 3 angles are acute, that measure the same
   • Sides
     • Equilateral
       • 3 sides that meaasure the same
     • Isosceles
       • at least 2 sides that measure the same
     • Scalene
       • no congruent sides
   • Theorems
     • Isosceles triangle
       • If 2 sides of a triangle are congruent, then the opposite angles are congruent
     • Converse of Isosceles triangle
       • If 2 angles of a triangle are congruent, then the opposite sides are congruent

4. Transformation

   • Reflection
     • Each point of the preimage and its image are the same distance from the line of reflection
     • Flip
   • Transition
     • Slide
     • A transformation that moves all points of the original figure the same distance in the same direction
   • Rotation
     • Turn
     • Transformation around the center of rotation, through a specific angle, and in a specific direction. Each point of the original figure and its iage are the same distance from the center

5. Special segments

   • Perpendicular Bisector
     • Circumcenter
       • Equidistance from each vertex
       • Interior
         • Acute triangle
       • Exterior
         • Obtuse triangle
       • Side of a triangle
         • Right triangle
       • Where the perpendicular bisectors intersect
     • A bisector that is also perpendicular to the segment
     • Theorems
       • If a point is on the perpendicular bisector, then it´s equidistance from the endpoints of the segment
       • If a point is equidistance from the endpoints, then it´s on the perpendicular bisector
   • Angle bisector
     • Theorems
       • If a point is on the bisector of an angle, then it is equidistance from the sides of the angle
       • If a point in the interior of an angle is equidistance from the sides of the angle, then it´s on the bisector of the angle
     • The locus of points in the interior of the angle equidistant to all the sides
     • Incenter
       • Where the angle bisectors intersect
       • Equidistance to all sides
   • Medians
     • Centroid
       • Where medians of the triangle intersect
       • 2/3 from each vertex
       • 1/3 from each line
     • Segment with endpoints
       • Vertex of a triangle
       • Midpoint of the opposite side
   • Altitudes
     • Orthocenter
       • The lines containing the altitude are concurrent
     • Interior
     • Exterior
     • On the side
     • A segment from a vertex to the line with the opposite side and perpendicular to the line containing that side

6. inequalities

   • Angle inequalities
     • If 1 side of a triangle is longer than another side, then the opposite angle to that side has a greater measurement that the angle opposite the shorter ide
     • If 1 angle of a triangle has a greater measure than another angle, then the opposite side of the greater angle is longer than the side opposite the lesser angle
   • The triangle inequality
     • The sum of the lengths of any 2 sides, must be greater than the ength of the third side
   • Hinge Therom
     • Converse of the Hinge Theorem
       • If 2 sides of a triangle are congruent to 2 sides of another triangle, and the third side in the first is onger 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
     • If 2 sides of sa triangle are congruent to 2 sides of another triangle, and the included angle of the first is larger than included angle of the second triangle, then the third side of the first triangle is longer than the third side of the second triangle