Download Concept Summary on Triangles

Concept Summary on Triangles
Whenever two figures have the same shape and size, they are called congruent figures.
Two triangles are congruent if their vertices can be matched up so that the
corresponding parts of the triangle are equal.
Corresponding parts of congruent triangles are congruent.
If three sides of one triangle are equal to three sides of another triangle, then the
triangles are congruent. (SSS)
If two sides and the included angle of one triangle are equal to two sides and the
included angle of another triangle, then the triangles are congruent. (SAS)
If two angles and the included side of one triangle are equal to two angles and the
included side of another triangle, then the triangles are congruent. (ASA)
ASA leads to the AAS and SAA corollaries. If two angles on a triangle are known, then
the third is also known.
Right Triangles
If two legs of one right triangle are congruent to the corresponding legs of another right
triangle, the triangles are congruent. (LL)
If the hypotenuse and an acute angle of one right triangle are congruent to the
hypotenuse and corresponding angle of another right triangle, then the triangles
are congruent. (HA)
If one leg and an acute angle of a right triangle are congruent to the corresponding leg
and angle of another right triangle, then the triangles are congruent. (LA)
If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and
corresponding leg of another right triangle, then the triangles are congruent.
(HL)
If two triangles are congruent, then the corresponding parts of the two congruent
triangles are congruent. (CPCTC)
Points of Concurrency
Altitude of a Triangle: The perpendicular distance between a vertex of a triangle and
the side opposite that vertex. Sometimes called the height of a triangle. Also, sometimes
the line segment itself is referred to as the altitude.
Median of a Triangle: A segment is a median of a triangle if and only if its endpoints
are a vertex of the triangle and the midpoint of the side opposite the vertex.
Bisect: To cut in half exactly
Angle Bisector: A segment or ray that shares a common endpoint with an angle and
divides the angle into two equal parts.
Perpendicular Bisector: A perpendicular line or segment that passes through the
midpoint of a segment.
Incenter- formed by angle bisectors of the 3 interior angles; the center of an inscribed
circle
Centroid- formed by the medians; the balance point of a triangle
Circumcenter- formed by the perpendicular bisector of each side; center of a
circumscribed circle; can be inside, outside or on the perimeter of the triangle
Orthocenter- formed by the altitudes of each side; visually takes a plane figure and
makes it a solid figure; can be inside or outside of the triangle
Isosceles and Equilateral triangles
If two sides of a triangle are equal, then the angles opposite those sides are equal.
An equilateral triangle is also equiangular.
An equilateral triangle has three 60o angles.
The bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its
midpoint.
If two angles of a triangle are equal, then the sides opposite those angles are equal.
An equiangular triangle is also equilateral.
If a point lies on the bisector of an angle, then the point is equidistant from the sides of
the angle.
If a point is equidistant from the sides of an angle, then the point lies on the bisector of
the angle.
If a point lies on the perpendicular bisector of a segment, then the point is equidistant
from the endpoints of the segment.
If a point is equidistant from the endpoints of a segment, then the point lies on the
perpendicular bisector of the segment.
Triangle Inequality Theorems
If one side of a triangle is longer than another side, then the angle opposite the longer
side is larger than the angle opposite the shorter side.
If one side of a triangle is larger than another angle, then the side opposite the larger
angle is longer than the side opposite the smaller angle.
The sum of the length of any two side of a triangle is greater than the length of the third
side.
The measure of an exterior angle of a triangle is greater than the measure of either of the
nonadjacent interior angles.