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.