Triangles

Triangles

1. Chapter 5

   • Chapter 4
     • Classifying
       • Classifying triangles 4-1
         • Classifications of triangles by angles
           • Acute Triangle: 3 Acute angles
           • Equiangular Triangle: 3 Congruent acute angles
           • Obtuse Triangle: 1 obtuse angle
           • Right angle: 1 Right angle
         • Classifications of triangles by sides
           • Equilateral triangle: 3 Congruent sides
           • Isosceles Triangle: At least 2 congruent sides
           • Scalene Triangle: No congruent sides
       • Isosceles & equilateral triangles 4-6
         • Properties of isosceles Triangles: The parts of an isosceles triangle have special names
           • Legs of an isosceles triangle: Two congruent sides. Vertex angle: Angle with sides that are legs. Base angles: Two angles formed by the base and the congruent sides.
         • Isosceles Triangle theorem: If two sides of a triangles are congruent, then the angles opposite those sides are congruent.
         • Converse of Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the sides opposite those angles are congruent.
         • Isosceles triangle
         • Equilateral Triangle
           • A triangle is equilateral if and only if it is equiangular
             • Each angle of an equilateral triangle measures 60.
     • Angles
       • Angles of triangles 4-2
         • Triangle Angle-Sum Theorem
           • The sum of the measures of the angles of a triangle is 180. EX- m<A + m<B + m<C = 180
             • Auxiliary line: An extra line or segment drawn in a figure to help analyze geometric relationships
         • Exterior Angle Theorem
           • The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles EX- m<A + m<B = m<1
     • Congruence
       • Congruent Triangles 4-3
         • Congruence and Corresponding Parts:
           • If two geometric figures have exactly the same shape and size, they are "congruent".
           • In two congruent polygons, all of the parts of one polygon are congruent to the corresponding parts(matching parts) of the other polygon.
           • Two polygons are congruent if and only if their corresponding parts are congruent.
         • Prove Triangles Congruent
           • If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent.
       • Proving Triangles Congruent SSS,SAS 4-4
         • SSS: If three sides of one triangle are congruent to three sides of a second triangle, then the triangles are congruent.
         • SAS: If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the triangles are congruent.
       • Proving Triangles Congruent ASA,AAS 4-5
         • ASA Postulate: An included side is the side located between two consecutive angles of a polygon.
           • ASA: If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.
         • AAS Theorem: The congruence of two angles and the non included side are also sufficient to prove two triangles congruent. This Is a theorem because it can be proved using the Third Angles Theorem.
           • AAS: If two angles and the nonicluded side of one triangle are congruent to the corresponding two angles and side of a second triangle, then the two triangles are congruent
     • Tranformation
       • Congruence Transformations 4-7
         • A transformation is an operation that maps an original geometric figure. A transformation can change the position, size, or shape of a figure
           • Congruence transformation or Isometry: one in which the position of the image may differ from that of the preimage, but the two figures remain congruent
         • Reflections, Translations, & Rotations
           • Reflection: A tranformation over a line called the line of reflection
           • Translation: A transformation that moves all points of the original figure the same distance in the same direction.
           • Rotation: A transformation around a fixed point called the center rotation.
   • Special Segments
     • Bisectors of Triangles 5-1
       • Perpendicular bisector: When a bisector is also perpendicular to the segment Point of concurrency: Point where concurrent lines intersect. Circumcenter: Point of concurrency of the perpendicular bisectors
       • Perpendicular bisectors
         • Perpendicular Bisector Theorem: If a point is on the perpendicular bisector of a segment, then it is equidistant form 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.
       • Circumcenter Theorem: The perpendicular bisectors of a triangle intellect at a point called the circumcenter that is equidistant form the vertices.
       • Angle Bisectors
         • Angle Bisector Theorem: If a point is on the bisector of an angle, then it is equidistant from the sides of the angle.
           • Converse of the Angle Bisector Theorem: If a point in the interior of an angle is equidistant 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
     • Medians & Altitudes of Triangles 5-2
       • Median: segment with endpoints Centroid:The point of concurrency of the medians of a triangle
         • Centroid Theorem: The medians of a triangle intersect at a point called the centroid that is two thirds of the distance from each vertex to the midpoint of the midpoint of the opposite side
       • Altitude: segment from the vertex to the line containing the opposite side and perpendicular to the line containing that side
         • Orthocenter: The lines containing the altitudes of a triangle are congruent, intersecting at a point
   • Inequalities
     • Inequalities in one Triangle 5-3
       • Inequality: For any real numbers a & b, a> b if and only if there is a positive number c such that a = b + c
         • Exterior angle Inequality: The measure of an exterior angle of a triangle is greater than the measure or either of its corresponding remote interior angle. m<1 > m<A, m<1 > m<B
       • Angle-Side Relationships in Triangles
         • 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 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
     • The Triangle Inequality 5-5
       • 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. PQ + QR > PR
     • Inequalities in two triangles 5-6
       • • 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, 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.