Unit 3: Congruent Triangles

Polygon

The union of three or more line segments in the same plane such that each segment intersects exactly two others, each one at one of its endpoints, and no two segments with a common endpoint are collinear

Triangle

A polygon with three sides

Quadrilateral

A polygon with four sides

SAS theorem

If two corresponding sides of distinct triangles and their included angle are congruent then the triangles are congruent

ASA theorem

If two corresponding angles and their included side of distinct triangles are congruent then the triangles are congruent

Side-angle theorem

If the two sides of a triangle are congruent then the angles opposite these sides are congruent

Side-angle theorem converse

If two angles of a triangle are congruent then the sides opposite these angles are congruent

Scalene triangle

A triangle in which no two sides are congruent

Isosceles triangle

A triangle in which at least two sides are congruent

Equilateral triangle

A triangle in which all sides are congruent

Equiangular triangle

A triangle in which all angles are congruent

Right triangle

A triangle in which one of the angles is a right angle

Acute triangle

A triangle in which all angles are acute

Obtuse triangle

A triangle in which one of the angles is obtuse

Legs of an isosceles triangle

The two congruent sides of an isosceles triangle

Base of an isosceles triangle

The side opposite the vertex angle of an isosceles triangle

Base angles of an isosceles triangle

Two congruent angles whose collinear rays form the base side of an isosceles triangle

Vertex angle of an isosceles triangle

The angle whose rays are formed by the legs of an isosceles triangle

Legs of a right triangle

The two sides collinear to the rays forming the right angle of a right triangle

Hypotenuse

The side opposite the right angle in a right triangle

Cevian

A line segment drawn from a vertex of a triangle to a point on the opposite side

Median

A line segment that joins any vertex of a triangle to the midpoint of the opposite side

Altitude

A line segment drawn from any vertex of a triangle perpendicular to and ending on the opposite side, or an extension of the opposite side

Angle bisector of a triangle

A line segment which bisects any angle of the triangle and terminates on the side opposite the angle

If a line segment bisects the vertex angle of an isosceles triangle…

then it is also the altitude and median to the base

If a line segment is the altitude to the base of an isosceles triangle…

then it is also the median to the base and bisector of the vertex angle

If a line segment is the median to the base of an isosceles triangle…

then it is also the altitude to the base and bisector of the vertex angle

If a triangle is equilateral, then it is also…

equiangular

If a triangle is equiangular, then it is also…

equilateral

Perpendicular bisector

A line that is perpendicular to the line segment at its midpoint

If two lines intersect to form congruent adjacent angles… (the forgotten theorem)

then they are perpendicular

If two __ intersect to form congruent angles then the lines are __

lines, perpendicular

Through a point on the line, there exists __ __ line that is __ to the line

exactly one, perpendicular

Through a point not on a line, there exists __ __ line that is __ to the lines

exactly one, perpendicular

If two points are equidistant from the endpoints of a line segment… (equidistance theorem)

then the line joining them will be the perpendicular bisector of the line segment

If a point is on the perpendicular bisector of a line segment… (equidistance theorem)

then it is equidistant from the endpoints of a line segment

If a point is equidistant from the endpoints of a line segment… (equidistance theorem)

then it lies on the perpendicular bisector of the line segment

SSS theorem

If three sides of one triangle are congruent to the corresponding sides of a second triangle then the triangels are congruent

Circle

A set of points equidistant from a fixed point called the center

Radius

A line segment that joins the center of a circle to a point on the circle

Radii congruence theorem

All radii of a given circle are congruent