Chapter 5-1

Perpindiculars and Bisectors and Altitudes

Perpendicular Bisector Theorem: a point that lies on the perpendicular bisector of a segment is equidistant from endpoints of the segment.

Converse of the Perpendicular Bisector Theorem: if a point is equidistant from the endpoints of the segment, then it is on the perpendicular bisector of the segment.

Concurrent Lines: three or more lines (or rays or segments) that intersect at the same point.

Perpendicular Bisector of a Triangle: a line (or ray or segment) that is perpendicular to the side of a triangle at the midpoint of a side.

Circumcenter Theorem: the circumcenter is equidistant from the vertices.

Angle Bisector of a triangle: bisector of an angle of the triangle

point of concuraccy: incenter

Incenter theorem: the incenter is equidistant from the sides

Median of a Triangle: segment whose endpoints are a vertex and the midpoint of the opposite side

Point of concurracy: centroid

Centroid Theorem: from the centroid to the side is half the distance to the angle.

Altitude of a triangle: the perpendicular segment from a vertex to the opposite side, or the extension of that side

Point of concurracty: orthocenter

Sec. 5.2 Notes

Longer Side Theorem
-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.
Larger Angle Theorem
-If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle.
Exterior Angle Inequality
-The measure of an exterior angle is greater than the measure of either 2 nonadjacent interior angles.

Section 4.3

In class today we covered Section 4.3. In this section, we learned many things regarding to congruent triangles. To be more specific, we learned about: CPCTC, the Properties of Congruent Triangles, and Congruence Transformations.

CPCTC is a rule that allows us to prove our answers, usually when doing a proof. It stands for Corresponding Parts of Congruent Triangles that are Congruent. This rule tells us that corresponding angles are congruent and corresponding side are congruent. Here’s how that would look.

Properties of Congruent Triangles: are properties that we seen before, but now they can be used to classify steps when doing a proof for a triangle. The properties are:

Reflexive Property of Congruent Triangles: where every triangle is congruent to itself

Symmetric Property of Congruent Triangles: if triangle DOG is congruent to triangle CAT, then triangle CAT is congruent to triangle DOG

Transitive Property of Congruent Triangles: if triangle CAT is congruent to triangle DOG & triangle DOG is congruent to triangle ELK, then triangle CAT is congruent to triangle ELK.

Finally, we learned about the three Congruence Transformations. These show us that you can slide, flip, reflect, or turn a triangle. When doing this, remember that the size and shape will never change.

Triangle Congruence Theorems

THE FOUR:
SSS (side-side-side)
SAS (side-angle-side)
ASA (angle-side-angle)
AAS (angle-angle-side)

CPCTC: Corresponding parts of congruent triangles are congruent
-only to be used after two triangles have been proved congruent
-CANNOT PROVE TWO TRIANGLES CONGRUENT USING CPCTC

November 6 Blog

The Base Angles are the to two adjacent angles to the base,

The Vertex Angle is the angle opposite to the base.

Theorems

Isosceles Triangle Theorem
If two sides of a triangle are congruent, then the angles opposite them are congruent.
(ex. If line BU is congruent to line GU, then angle G is congruent to angle B.)

Converse of Isosceles Triangle Theorem
If two angles of a triangle are congruent, then the sides opposite them are congruent.
(ex. Is angle G is congruent to angle B, then line BU is congruent to line GU)



Corollaries

A triangle is equilateral, if and only if it is equiangular.

Angles of equilateral and equiangular triangles = 60 degrees.