TRIANGLES

TRIANGLES

1. CHAPTER 4

   • LESSON 1
     • An acute triangle is a triangle with three acute angles.
     • An equiangular triangle is a triangle where all three interior angles are equal in measure.
     • An obtuse triangle is a triangle in which one of the angles is an obtuse angle.
     • A right triangle has one right angle
     • A triangle with all three sides of equal length.
     • An isosceles triangle is a triangle with two equal sides
     • A scalene triangle is a triangle that has three unequal sides
   • LESSON 2
     • An auxiliary line is an extra line needed to complete proof in plane geometry.
     • The exterior angle is the angle between a side of a rectilinear figure and an adjacent side extended outward.
     • Remote interior angles are just the two angles that are inside the triangle and opposite from the exterior angle.
   • LESSON 3
     • Congruent means two things are identical in form
     • Congruent polygons have the same number of sides, and all corresponding sides and interior angles are congruent
     • Corresponding parts are parts that match between two congruent triangles
     • CPCTC: The word corresponding refers to parts that match between two congruent triangles. You can identify corresponding angles and corresponding sides. They will have the same length in congruent triangles.
   • LESSON 4
     • Sss states that triangles are congruent if all tree sides in one triangle are congruent to the corresponding sides in the other
     • Sas states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangle are congruent
   • LESSON 5
     • Asa states that when two triangles have corresponding angles and sides that are congruent as shown below, the triangles themselves are congruent
     • Aas states that triangles are congruent if two pairs of corresponding angles and a pair of opposite sides are equal in both triangles
     • The included side is the common leg of two angles
   • LESSON 6
     • The vertex angle is a point about which an angle is measured
     • Base angles are either of the angles of a triangle that has one side in common with the base.
     • The isosceles triangle theorem states that if two sides of a triangle are congruent, then the angles opposite to these sides are congruent
     • The converse of isosceles triangle theorem states that if two angles of a triangle are congruent, then the sides opposite to these angles are congruent
     • The equilateral triangle property states that if all sides are equal, all angles are equal
     • A leg of a triangle is one of its sides
   • LESSON 7
     • The transformation is a thorough or dramatic change in form or appearance.
     • The preimage is the inverse of an image
     • The image is the new figure created after a transformation
     • Congruence transformation is another term for isometry
     • Isometry is a distance-preserving transformation between metric spaces, usually assumed to be bijective
     • A reflection is a geometric figure is reflected across a line, creating a mirror image.
     • In a translation, every point of the object must be moved in the same direction and for the same distance.
     • The rotation is the process or act of turning or circling around something

2. CHAPTER 5

   • LESSON 1
     • The perpendicular bisector theorem states that the locus of all points that are equidistant from its endpoints
     • The converse of the perpendicular bisector theorem states that if a point is on the perpendicular bisector of a segment, then it is equidistant from the segment's endpoints
     • The point of concurrency is a place where three or more, but at least three lines, rays, segments or planes intersect in one spot
     • The circumcenter is the point at which the perpendicular bisectors of the sides of a triangle intersect and which is equidistant from the three vertices
     • The circumcenter theorem states that the vertices of a triangle are equidistant from the circumcenter
     • An angle bisector of an angle of a triangle divides the opposite side into two segments that are proportional to the other two sides of the triangle
     • The converse of angle bisector theorem states that if a point is in the interior of an angle and equidistant from the sides, then it lies on the bisector of the angle
     • The incenter is the center of the incircle of a triangle or another figure
     • The incenter theorem states that the incenter of a triangle is equidistant from each side of the triangle
   • LESSON 2
     • The median is the median value range of a range of values
     • The centroid is the center of mass of a geometric object
     • The centroid theorem is the point where three medians concide
     • The altitude is the height of an object or point in relation to ground level
     • The orthocenter is one of the triangle's points of concurrency formed by the intersection of the triangle's 3 altitudes
   • LESSON 3
     • The inequality is the relation between two expression that are not equal
     • The exterior angle inequality theorem states that an exterior angle of a triangle is greater than either of the non-adjacent interior angles.
     • angle-side relationships
       • In any triangle, the smallest side and smallest angle are opposite one another.
       • In any triangle, the largest side and largest angle are opposite one another.
       • If two sides are congruent, then the corresponding two angles will be congruent.
   • LESSON 5
     • The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side.
   • LESSON 6
     • The hinge theorem states that if two sides of one 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.
     • The converse of hinge theorem states that if two sides of one 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.