Chapter 5 Overview Flashcards
midsegments of a triangle
a segment connecting the midpoints of 2 sides of
the triangle
Theorem 5-1 – Triangle Midsegment Theorem
If a segment joins the midpoints of 2 sides of a triangle, then the segment is
parallel to the third side and is half as long.
equidistant
same distance from an object
Theorem 5-2 Perpendicular Bisector Theorem
If a point is on the ⊥ bisector of a segment, then it is equidistant from the
endpoints of the segment.
Theorem 5-3 Converse of the Perpendicular Bisector Theorem
If a point is equidistant from the endpoints of a segment, then it is on the ⊥
bisector of the segment.
distance from a point to a line
length of the ⊥ segment from the pt to the line
Theorem 5-4 Angle Bisector Theorem
If a point is on the bisector of an angle, then the pt is equidistant from the sides
of the angle.
Theorem 5-5 Converse of the Angle Bisector Theorem
If a pt in the interior of an angle is equidistant from the sides of the angle, then
the pt is on the angle bisector.
concurrent
when 3 or more lines intersect at one pt
point of concurrency
pt at which 3 or more lines intersect
Theorem 5-6 – Concurrency of Perpendicular Bisectors Theorem
The ⊥ bisectors of the sides of a ∆ are concurrent at a pt equidistant from the
vertices.
circumcenter of the triangle
pt of concurrency of the ⊥ bisectors of a ∆
Theorem 5-7 – Concurrency of Angle Bisectors Theorem
The bisectors of the angles of a ∆ are concurrent at a pt equidistant from the
sides of the triangle.
incenter of the triangle
pt of concurrency of the ∠ bisectors of a ∆
**always inside the triangle
median of a triangle
a segment whose endpoints are a vertex and the
midpoint of the opposite side
Theorem 5-8 – Concurrency of Medians Theorem
The medians of a ∆ are concurrent at a point that is two-thirds of the distance
from each vertex to the midpoint of the opposite side.
centroid of the triangle
the pt of concurrency of the medians
**always inside the ∆
altitude of a triangle
the perpendicular segment from the vertex of a ∆ to the
line containing the opposite side
- it can be inside or outside of the ∆ or it can be a side of the ∆
Theorem 5-9 – Concurrency of Altitudes Theorem
The lines that contain the altitudes of a ∆ are concurrent.
orthocenter of the triangle
pt of concurrency of the altitudes of a ∆
indirect reasoning
all possibilities are considered and then all but one are
proved false
indirect proof
a proof involving indirect reasoning
Step 1 – State as a temporary assumption the opposite (negation) of what you want to prove.
Step 2 – Show that this temporary assumption leads to a contradiction.
Step 3 – Conclude that the temporary assumption must be false and that what you want to prove must be true.
Comparison Property of Inequality
If a = b + c and c > 0, then a > b.
Corrolary to the Triangle Exterior Angle Theorem
The measure of an exterior ∠ of a ∆ is greater than the measure of each of its
remote interior angles.
Theorem 5-10
If 2 sides of a ∆ are not congruent, then the larger angle lies opposite the longer
side.
Theorem 5-11
If 2 angles of a ∆ are not congruent, then the longer side lies opposite the larger
angle.
Theorem 5-12 Triangle Inequality Theorem
The sum of the lengths of any 2 sides of a ∆ is greater than the length of the 3rd
side.
Theorem 5-13 – The Hinge Theorem (SAS Inequality Theorem)
If 2 sides of one ∆ are ≅ to two sides of another ∆, and the included angles are
not ≅, then the longer third side is opposite the larger included angle.
Theorem 5-14 – Converse of the Hinge Theorem (SSS Inequality)
If 2 sides of one ∆ are ≅ to two sides of another ∆, and the 3rd sides are not ≅,
then the larger included angle is opposite the longer 3rd side.
