Houghton Mifflin geometry chapter 4

Congruent

Having the same size and shape

congruent triangles

Two triangle are congruent if and only if their vertices can be matched up so that the corresponding of the triangles are congruent.

Postulate 12-SSS Postulate

If three sides of one triangle are congruent to three sides of another triangle then the triangles are congruent

Postulate 13-SAS postulate

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle then the triangles are congruent

Postulate 14 - ASA postulate

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

A way to prove to segments or two angles are congruent

One) identify two triangles in which the two segments or angles are corresponding parts Two) prove that the triangles are congruent Three) state that the two parts are congruent using the reason corresponding parts of congruent triangles are congruent

Theorem 4-1 "the isosceles triangle theorem"

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

Corollary one

An equilateral triangle is also equiangular

Corollary two

An equilateral triangle has 3 60° angles

Corollary three

The bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint

Theorem 4-2

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

Corollary

And equiangular triangle is also equilateral

Theorem 4-3 AAS Theorem

If two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent

Theorem 4-4 HL theorem

If the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent

Median

A segment from a vertex to the midpoint of the opposite side

Altitude of a triangle

The perpendicular segment from a vertex to the line that contains the opposite side

Perpendicular bisector of a segment

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

Theorem 4-5

If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment

Theorem 4-6

If a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment

Distance from a point to a line

The length of the perpendicular segment from the point to the line

Theorem 4-7

If a point lies on the bisector of an angle then the point is equidistant from the sides of the angle

Theorem 4-8

If a point is equidistant from the sides of an angle, then the point lies on the bisector of the angle