Transcript journal chap 5

Sara Wunderlich
Describe what a perpendicular bisector is. Explain the
perpendicular bisector theorem and its converse. Give 3 examples
of each.
Perpendicular bisector- line that intersects a segment, forming a 90° angle, at
the segment’s midpoint. Basically, It is when you bisect and it crosses a segment
exactly at a right angle
Perpendicular bisector theorem-If a segment is bisected by a perpendicular line,
then any point on the perpendicular bisector is equidistant of the endpoints of
the segment.
Converse-if a point is equidistant from the endpoints of a segment, then it is on
the perpendicular bisector of the segment.
EXAMPLE
1
Perpendicular bisector theorem
Given: L is the perpendicular bisector of
AB.
Prove: AYX = BYX
Since L is the perpendicular bisector of AB,
L
AB and Y is the midpoint of AB. By
the def. of perp, <AYX and <BYX are right
angles and <AYX = <BYX. BY the definition
of midpoint, AY= BY. And by the reflexive
property, XY = XB so, AYX = BYX by
SAS.
example2
Example 3
constructing it
Describe what an angle bisector is. Explain the angle bisector
theorem and its converse. Give at least 3 examples of each.
Angle bisector- is the line that divides the angle into two
equal parts, or congruent angles.
Angle bisector theorem: if a point is on the bisector of
an angle, then it is equidistant from the sides of the
angle.
Converse: if a point in the interior of an angle is
equidistant from the sides of the angle, then it is the
bisector.
Example 2
Constructing it, example 3
Describe what concurrent means. Explain the concurrency of
Perpendicular bisectors of a triangle theorem. Explain what a
circumcenter is. Give at least 3 examples of each.
When three or more lines intersect in a single point
they are said to be concurrent.
In a triangle, the three medians, three
perpendicular bisectors, three angle
bisectors, and three altitudes are each
concurrent.
Concurrency of Perpendicular Bisectors
of a Triangle Theorem The
perpendicular bisectors of a triangle
intersect at a point that is equidistant
from the vertices of the triangle.
CIRCUMCENTER: the point of congruency, where three
or more perpendicular lines intersect.
Example 1
Describe the concurrency of angle bisectors of a triangle theorem.
Explain what an incenter is. Give at least 3 examples of each.
Incenter – the point where the three angle bisectors meet,
making it a point of concurrency.
the angle bisectors of a triangle intersect at a point that is
equidistant from the sides of the triangle.
Describe what a median is. Explain what a centroid is. Explain
the concurrency of medians of a triangle theorem. Give at least
3 examples of each.
The median of a triangle is a line segment that goes from a
vertex to the midpoint of the opposite side.
Centroid- the point where the medians of the triangle intersect,
it is always inside the triangle.
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.
Describe what an altitude of a triangle is. Explain what an
orthocenter is. Explain the concurrency of altitudes of a triangle
theorem. Give at least 3 examples.
Altitude of a triangle- is a line segment from a vertex to the opposite
side so that the line segment is perpendicular to the side. Since a
triangle has 3 vertices, it has 3 altitudes. It can be inside outside or on
triangle.
Orthocenter – the point of intersection of the three altitudes, making
the concurrent.
Concurrency of Altitudes of a Triangle
Theorem The lines containing the
altitudes of a triangle are concurrent.
Describe what a Midsegment is. Explain the Midsegment
theorem. Give at least 3 examples.
Midsegment - is a line segment whose endpoints are the midpoints of
a triangle. Every triangle has three Midsegment.
Midsegment theorem:
A Midsegment of a triangle is parallel to a side of the triangle and is
half as long.
Example 1
Example 2
Example 3
Describe the relationship between the longer and shorter
sides of a triangle and their opposite angles. Give at
least 3 examples.
In any triangle the side opposite to the biggest angle will
always have the longest length. The side opposite to the
smallest angle will have smallest length.
Describe the exterior angle inequality. Give at least 3 examples.
Describe the triangle inequality. Give at least 3 examples.
In any triangle, any 2 sides of a triangle must add up to
more than third side.
AB+BC>AC
BC+AC>AB
AC+BC>BC
Example 1
Example 2
Example 3
Describe how to write an indirect proof. Give at least 3
examples.
To write and indirect proof:
1: assume that what you are trying to prove is false.
2: Try to prove using steps of a proof.
3: when you come to a point of contradiction, you have
proof that the theory is true
Example 1
A scalene triangle cannot have two congruent angles.
A scalene triangle can have 2 congruent angles
 Base angle theorem, it would be isosceles
 isosceles cannot be scalene by def. of isosceles
 a scalene triangle cannot have 2 congruent angles.
Example 2
A triangle cannot have two obtuse angles
A triangle can have two obtuse angles
 <1,<2 ,<1+<2+<3 has to equal 180
 <1 and <2 do not equal 180 therefore a triangle cannot have 2
obtuse angles.
Example 3
An isosceles triangle cannot have a base < that is a right angle.
An isosceles triangle can have a base < that is a right angle.
<1+<2+<3 have to equal 180
<1+<2 = 180, so therefore an isosceles triangle cannot have a base <
that is right.
Describe the hinge theorem and its converse. Give at least 3
examples.
Hinge Theorem - If 2 sides of 2 triangles are congruent and the
angle between them is not congruent, then the triangle with the
larger angle will have a longer third side.
Converse- if two sides of one triangle are congruent to two
sides of another triangle and the third sides are not
congruent, then the larger included angle is across from the
longer third side.
Example 1
a is congruent to a1 and
b to b1
Example 2
Example 3