**CORRESPONDENCE AND CONGRUENT TRIANGLES**

A CORRESPONDENCE is a matching up of the parts of the segments and
angles of one triangle with the segments and angles of another triangle.

**CORRESPONDENCE AND CONGRUENT TRIANGLES.**

CORRESPONDING PARTS OF CONGRUENT TRIANGLES ARE CONGRUENT.

**PROVING TRIANGLES CONGRUENT.**

PROVING TRIANGLES CONGRUENT--SSS POSTULATE

POSTULATE : SSS Postulate: If three sides of one triangle are congruent
respectively to three sides of another triangle, then the two triangles are
congruent.

**PROVING TRIANGLES CONGRUENT--SAS POSTULATE**

POSTULATE : SAS Postulate: If two sides and the incluled angle of
one triangle are congruent respectively to two sides and the included angle
of a second triangle, then the two triangles are congruent.

**PROVING TRIANGLES CONGRUENT--ASA POSTULATE**

POSTULATE : ASA Posulate: If two angles and the included side of
one triangle are congruent respectively to two angles and the included side
of another triangle, then the two triangles arer congruent.

USING CONGRUENT TRIANGLES--TYPICAL TRIANGE PROOF

ISOSCELES TRIANGLE THEOREMS

PARTS OF AN ISOSCELES TRIANGLE.

**ISOSCELES TRIANGLE THEOREMS**

THEOREM : THE ISOSCELES TRIANGLE THEOREM: If two sides of a triangle
are congruent (in other words, if it's and isosceles triangle), then the angles
opposite those sides are also congruent.

COROLLARY : An equilateral triangle is also equiangular.

COROLLARY : An equilateral triangle has three 60° angles.

COROLLARY : The bisector of the vertex angle of an isosceles triangle is also
the perpendicular bisector of the base of the triangle.

ISOSCELES TRIANGLE THEOREMS

THEOREM : CONVERSE ISOSCELES TRIANGLE THEOREM: If two angles of a
triangle are congruent, then the sides opposite those angles are also congruent.

COROLLARY: An equiangular triangle is also equilateral

**OTHER METHODS OF PROVING TRIANGLES CONGRUENT**

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

**OTHER WAYS OF PROVING TRIANGLES CONGRUENT**

THEOREM : HL THEOREM: (USED ONLY WITH RIGHT TRIANGLES): If the hypotenuse
and a leg of one right triangle are congruent to the corresponding parts of
another right triangle, then the two triangles are congruent.

**SUMMARY: WAYS TO PROVE TWO TRIANGLES CONGRUENT**

ALL TRIANGLES:

SSS

SAS

ASA

AAS

RIGHT TRIANGLES:

HL

**MORE THAN ONE PAIR OF CONGRUENT TRIANGLES**

**PROVING QUADRILATERALS ARE PARALLELOGRAMS**

THEOREM : If both pairs of opposite sides of a quadrilateral are
congruent, then it is a parallelogram.

THEOREM : If one pair of opposite sides of a quadrilateral are both parallel
and congruent, then it is a parallelogram.

**PROVING QUADRILATERALS ARE PARALLELOGRAMS**

THEOREM : If both pairs fo opposite angles of a quadrilateral are
congruent, then it is a parallelogram.

THEOREM : If the diagonals of a quadrilateral bisect each other, then it is
a parallelogram.

**PROVING QUADRILATERALS ARE PARALLELOGRAMS**

1. Show that both pairs of opposite sides are parallel.

2. Show that both pairs of opposite sides are congruent.

3. Show that one pair of opposite sides are both congruent and parallel.

4. Show that both pairs of opposite angles are congruent.

5. Show that the diagonals bisect each other.

**THEOREMS INVOLVING PARALLEL LINES**

THEOREM : A line that contains the midpoint of one side of a triangle
and is parallel to another side passes through the midpoint of the third side.

** **

THEOREMS INVOLVING PARALELL LINES

THEOREM : The segment that joins the midpoint of two sides of a triangle:

1) Is parallel to the third side.

2) is half as long as the third side

**SPECIAL PARALLELOGRAMS**

THEOREM : The midpoint of the hypotenuse of a right triangle is equidistant
from the three vertices.

THEOREM : If an angle of a parallelogram is a right angle, then the parallelogram
is a ____________________.

THEOREM : If two consecutive sides of a parallelogram are congruent, then the
parallelogram is a rhombus.

**TRAPEZOIDS**

DEF. OF TRAPEZIOD: A quadrilateral with exactly one pair of parallel
lines.

ISOSCELES TRAPEZIOD: A trapeziod with congruent legs.

**TRAPEZIODS**

THEOREM : The base angles of an isosceles trapezoid are congruent.

**TRAPEZOIDS**

THEOREM : The median of a trapezoid:

1) is parallel to the bases.

2) has a length equal to the average of the base lengths.

INEQUALITIES IN ONE TRIANGLE

THEOREM : If two sides of a triangle are not congruent, then the

smaller angle is opposite the smaller side.

THEOREM : If two angle of a triangle are not congruent, then the

smaller side is opposite the smaller angle.

** **

INEQUALITIES IN ONETRIANGLE

COROLLARIES: The perpendicular segment from a point to a line (or
to a plane) is the shortest segment from the point to the line (or to the plane).

THEOREM : THE TRIANGLE INEQUALITY: The sum of the lengths of any two sides of
a triangle is greater than the length of the third side.

INEQUALITIES IN TWO TRIANGLES

THEOREM : SAS INEQUALITY THEOREM: In two triangles with two sides
congruent but included angles not congruent, then the third sides are not congruent,
and the smaller side is opposite the smaller included angle.

THEOREM : SSS INEQUALITY THEOREM: In two triangles with two sides congruent,
but the third sides not congruent, then the smaller included angle is opposite
the smaller side.

**BASIC CONSTRUCTIONS**

CONSTRUCTION 1: Given an segment, construct a segment congruent to the given
segment.

CONSTRUCTION 2: Given an angle, construct an angle congruent to the given angle.

CONSTRUCTION 3: Given an angle, construct the bisector of the angle.

**CONSTRUCTIONS OF PERPENDICULAR AND PARALELL LINES**

CONSTRUCTION 4: Given a segment, construct the perpendicular bisector
of the segment.

CONSTRUCTION 5: Given a point on a line, construct the perpendicular to the
line at the given point.

CONSTRUCTION 6: Given a point outside a line, construct the perpendicular to
the line from the given point.

CONSTRUCTION 7: Given a point outside a line, construct the parallel to the
given line through the given line.