Triangles

Triangle

A figure formed by three segments that connect three non-colinear points.

Opposite

Across from

Included

between

Triangle inequality theorem

When two shorter side lengths are added they must be greater than the longest length.

Angle Sum Theorem

The sum of three interior angles in any triangle always adds up to 180 degrees

Third angel Theorem

If two angles of a triangle are congruent to two angles of a second triangle, then the third angles of both triangles are also congruent.

Interior angles

Angles inside a triangle

Exterior angles

Angles on the outside of a triangles that form linear pairs with interior angles of the triangle.

Remote interior angles

interior angles that don’t touch the exterior angle

exterior angle theorem

the value of the exterior angle is equal to the sum of the two interior angles

Congruent Triangles

Triangles with corresponding angles and lengths that are the same size and shape

CPCTC

Corresponding Parts of Congruent Triangles are Congruent

Reflexive property

Every triangle is congruent to itself

Symmetric property

Every congruence statement can be reversed

Transitive property

In a chain of congruence statements, the first triangle is congruent to the last triangle

SSS postulate

If the sides of one triangle are congruent to the sides of a second triangle, then they are CONGRUENT.

SAS postulate

If two sides and the included angles of one triangle are congruent to two sides and the included angles of another triangle, then the triangles are CONGRUENT

ASA postulate

If two angles and the included side of one triangle are congruent to another’s, then the triangles are congruent.

AAS theorem

If two angles and a Non included side of one triangle are congruent to the corresponding ones of another, then the triangles are congruent

Angle and side combonations that are not congruent

SSA and AAA

Similar

Figures with the same shape

Dilate

To change the size but not the shape of a geometric figure

Center of Dilation

The point in which a figure is scaled up or down in a dilation

Similar triangles have corresponding angles that are…

congruent

Corresponding side lengths of a SIMILAR triangle are..

proportional

Scale Ratio

The ratio of the lengths of corresponding sides in similar figures

Cross-conduct property

when you cross multiply two equal ratios, the products are equal

AA similarity postulate

If two angles of one triangle are congruent to another, then the triangles are similar

SSS similarity theorem

If the lengths of the corresponding sides of two triangles are proportional, then the triangles are similar

SAS similarity theorem

If an angle of one triangle is congruent to an angle of another & the sides including these angles are proportional, the triangles are similar

Isosceles Triangle Theorem

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

The converse of Isosceles Triangle Theorem

If two angles of a triangle are congruent, then the sides opposite those angles are congruent.(AAS)

Equilateral Triangles are…

equiangular, all angles measure 60 degrees

The longest side of a triangle is always…

Opposite the angles with the largest measure

Median of a Triangle

A line segment that passes through a vertex and bisects the side opposite that vertex

Centroid

The point where all the medians intersect

Altitude of a Triangle

A line/segment that passes through a vertex and is perpendicular to the line containing the opposite side

Orthocenter

The intersection of all the altitudes of a triangle (may lay outside the triangle)

Isosceles triangles have median and altitudes that are…

Congruent, makes the triangle isosceles

Angle bisector

A segment that divides an angle of a triangle equally & intersects the opposite side

Incenter

The point where all three angle bisectors of a triangle intersect

perpendicular bisector

A perpendicular segment that divides a given side of a triangle equally.

Circumcenter of a triangle

The point shared by the perpendicular bisectors of a triangle’s sides (can fall out of the triangle if obtuse)

midsegment

A segment connecting two sides of a triangles at their midpoints

Triangle proportionality theorem

A line parallel to one side of a triangle divides the other two sides proportionally (CONVERSE IS TRUE)