Geometry [Chapter 6] Core Concepts & Theorems

(Theorem 6.11) Triangle Inequality Theorem

The sum of the lengths of any two sides of a triangle is greater than the length of the third side

(Theorem 6.10) Triangle 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

(Theorem 6.9) Triangle 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

(Theorem 6.4) Converse of the Angle Bisector Theorem

if a point is in the interior of an angle and is equidistant from the two side of the angel, then it lies on the bisector of the angle Ex. if line segment DA ⟂ ray BA and line segment DC ⟂ ray BC and DA = DC, then ray BD bisects ∠ABC

(Theorem 6.3) Angle Bisector Theorem

if a point lies on the bisector of an angle, then it is equidistant form the two sides of the angle Ex. if ray CP bisects ∠ACB and line segment PA ⟂ ray CA and line segment PB ⟂ ray CB then PA = PB

(Theorem 6.12) Hinge Theorem

If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second

(Theorem 6.13) Converse of the Hinge Theorem

If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the than the third side of the second, then the included angle of the first is larger than the included angle of the second

Median

Point of Concurrency: CENTROID Property: The centroid of a triangle is two thirds of the distance form each vertex to the midpoint of the opposite side

Perpendicular Bisector

Point of Concurrency: CIRCUMCENTER Property: The circumcenter of a triangle is equidistant from the vertices of the triangle

Angle Bisector

Point of Concurrency: INCENTER Property: The incenter of a triangle is equidistant from the sides of the triangle

Altitude

Point of Concurrency: ORTHOCENTER Property: The lines containing the altitudes of a triangle are concurrent at the orthocenter

Indirect Proof

STEP 1: Identify the statement you want to prove. Assume temporarily that this statement is false by assuming that its opposite is true STEP 2: Reason logically until you reach a contradiction STEP 3: Point out that the desired conclusion must be true because the contradiction proves the temporary assumption false

Midsegment of a triangle

a segment that connects the midpoints of two sides of the triangle. Ex. the midsegments of ΔABC are DE, DF, and FE. the midsegment triangle is ΔDEF

(Theorem 6.2) Converse of the Perpendicular Bisector Theorem

in a plane, if a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of the segment Ex. if DA = DB, then point D lies on the ⟂ bisector of line segment AB

(Theorem 6.1) Perpendicular Bisector Theorem

in a plane, if a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment Ex. if line CP is the ⟂ bisector of line segment AB, then CA = CB

(Theorem 6.7) Centroid Theorem

the centroid of a triangle is two-thirds of the distance from each vertex to the midpoint of the opposite side Ex. the medians of ΔABC meet at point G, and AG = 2/3AD, BG = 2/3BF, CG = 2/3CE

(Theorem 6.5) Circumcenter Theorem

the circumcenter of a triangle is equidistant from the vertices of the triangle Ex. If line segment PD, line segment PE, and line segment PF are perpendicular bisectors, then PA=PB=PC

(Theorem 6.6) Incenter Theorem

the incenter of a triangle is equidistant form the sides of the triangle Ex. if line segment AP, line segment BP, and line segment CP are angle bisectors of ΔABC, the PD=PE=PF

Orthocenter

the lines containing the altitudes of a triangle are concurrent. this point of concurrency is the orthocenter of the triangle

(Theorem 6.8) Triangle Midsegment Theorem

the segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long as that side DE is the midsegment of ΔABC, DE || AC and DE = 1/2AC