Download loo - Mr. Turner`s Wiki

Journal Chapter 5
Daniela Lopez L.
Perpendicular Bisector: a line that
cuts a segment into two equal
parts and forms a right angle.
Perpendicular Bisector Theorem:
The Perpendicular bisectors of the
sides of a triangle are concurrent
(meet)) at a point that is
equidistant from the vertices of the
triangle. The point where they meet
is called a circumcenter.
Converse: If a point is equidistant
from the endpoints of a segment
then that point is found on the
perpendicular bisector of the
segment.
PERPENDICULAR BISECTORS:
PERPENDICULAR BISECTOR
THEOREM:
Angle bisector: Line/ray
that divides an angle into
two congruent sides.
Angle Bisector Theorem:
If a point is on the
bisector of an angle then
it is equidistant from the
triangles sides.
Converse: If a point on
the inside of a triangle is
equidistant from both
sides then it is the angle
bisector.
Concurrent: when three or more lines
intersect they are called concurrent. The
point of concurrency is the exact place
where they meet.
The point where three perpendicular
bisector of a triangle meet is called the
circumcenter.
Concurrency of perpendicular bisectors
theorem:
perpendicular bisectors of a triangle
intersect at a point that is equidistant
from the vertices of the triangle.
A triangle’s angle bisectors
are also concurrent, the
point where they meet is
called the incenter of the
triangle.
Incenter theorem: the
incenter of the triangle is
equally distant from the
three sides.
Median:
segment
that has
endpoints at
one vertex
and the
midpoint of
the side
opposite to
it.
Centroid: point where medians
of a triangle concur.
O
CONCURRENCY OF MEDIANS OF A TRIANGLE THEOREM: The
medians of a triangle intersect at a point that is two thirds of the
distance from each vertex to the midpoint of the opposite side.
Point where altitudes meet is
called the orthocenter.
Altitude of a triangle:
perpendicular segment from a
vertex to the line on the
opposite side. Can be on the
outside or inside of a triangle.
Concurrency of altitudes of a
triangle theorem: Lines
containing the altitudes are
concurrent.
Midsegment: Segment that
joins the midpoints of two
sides of a triangle, when the
three midsegments of a
triangle join, they form the
midsegment triangle.
Triangle midsegment
theorem:
midsegment of a
triangle is parallel to
a side of the triangle
and half the length of
that side.
_____(0-10 pts.) Describe the exterior angle
inequality. Give at least 3 examples.
O An exterior angle of a triangle is greater than
either of the non-adjacent interior angles
<BAD is the exterior angle
<BAD is >(greater than) <BCA
<BAD is > (greater than) <CBA
_____(0-10 pts.) Describe the triangle inequality.
Give at least 3 examples.
O For three lines to form a triangle the sum of two of them must be
greater than the measure of the remaining one.
Writing Indirect Proofs
1. Identify what must be proven.
2. Assume the opposite (negation) of the conjecture.
3. Use direct reasoning to show that negation is contradicted.
4. Since negation has been proven wrong the original conjecture from
step one must be true.
Given: m<P>m<R Prove: QR>QP
Assume QR is less than or equal to QP.
(negation)
2. If QR is less than QP then m<P is less than
<m<R because the larger angle is opposite
to the longer side. This means QR is
greater than or equal to QP.
1.
_____(0-10 pts.) Describe the hinge theorem and its
converse. Give at least 3 examples.
O 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 triangle is longer than the third side of the
second triangle
en.wikipedia.org/wiki/Hinge_theorem
_____(0-10 pts.) Describe the relationship between the longer and
shorter sides of a triangle and their opposite angles. Give at least 3
examples.
A2 + B2 > C2 the
triangle is acute
A2 + B2 = C2 the
triangle is right
A2 + B2 < C2 the
triangle is obtuse