Download journal5 salvador amaya 9

Salvador Amaya 9-5
Perpendicular Bisector

A line or ray that is at the midpoint of a
segment and is perpendicular to that same
segment.
Perpendicular Bisector Theorem

If a point is on the perpendicular bisector
of a segment, then it is equidistant from
the endpoints of a segment.
Examples
Ex 1: DC=EC
Ex 2: DB=EB
Ex 3: DA=EA
Converse of Perpendicular
Bisector Theorem

Converse: If a point is equidistant from
the endpoints of a segment, then it is on
the perpendicular bisector of a line.
Examples
Ex 1: Since QE=WE, TE is a perpendicular bisector of
QW
Ex 2: Since QR=WR, TE is a perpendicular bisector of
QW
Ex 3: Since QT=WT, TE is a perpendicular bisector of
QW
Angle Bisector

A line or ray that divides an angle in two
congruent angles.
Angle Bisector Theorem

Angle Bisector Theorem: If a point is on
the bisector of an angle, then it is
equidistant from the sides of the angle.
Examples
Ex 1: ZC=XC
Ex 2: ZB=XB
Ex 2: ZV=XV
Converse of Angle Bisector
Theorem

If a point inside an angle is equidistant
from the sides of the angle, then it is on
the bisector of the angle.
Examples
H
Ex 1: Since DS=FS, AS is an angle bisector of <DHF
Ex 2: Since DG=FG, AS is an angle bisector of <DHF
Ex 3: Since DA=FA, AS is an angle bisector of <DHF
Concurrent

Three or more lines that intersect at one
point.
Examples
Concurrency of Perpendicular
Bisectors of a Triangle Theorem

The perpendicular bisectors of a triangle
meet at a point which is equidistant to all
the vertices.
Examples
2
12x
2
2
12x
12x
5
5
5
Circumcenter

Where all three perpendicular bisectors
meet.
Examples
Concurrency of Angle Bisectors
Theorem

The angle bisectors of a triangle meet at
a point which is equidistant to all the
sides.
Examples
3
3
3
8x
8x
8x
6
6
6
Incenter

Where all three angle bisectors meet.
Examples
Median

Segment that has an endpoint in the
vertex of an angle and the midpoint of its
opposite side.
Examples
Centroid

Where all three medians meet.
Examples
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.
Examples
Altitude

Segment from the vertex of a triangle
perpendicular to the opposite segment.
Examples
Orthocenter

Where all three altitudes meet.
Examples
Midsegment

Segment from the midpoint of a
segment to the midpoint of the opposite
segment.
Midsegment Theorem

The midsegment that connects two
sides of a triangle is parallel to the third
side and is half as long.
Examples
Ex 1: Segment f is
parallel to segment a
and half as long.
Ex 2: Segment d is
parallel to segment b
and half as long.
Ex 3: Segment e is
parallel to segment c
and half as long.
Side-Angle Relationships
The angle opposite to the smaller side is
the smaller angle. The angle opposite to
the next smaller side is the next smaller
angle. The angle opposite to the longer
side is the largest angle.
 The side opposite to the smaller angle is
the smaller side. The side opposite to
the next smaller angle is the next
smaller side. The side opposite to the
largest angle is the longest side.

What are the sides in
order from largest to
smallest?
Examples
b, a, c
58°
71°
51°
Examples
What are the angles in
order from largest to
smallest?
10
11
12
A, B, C
Which angle is bigger: B or C?
Examples
35
B because its
opposite side
is bigger.
36
Exterior Angle Inequality

The exterior angle in a triangle is bigger
than any of the non adjacent interior
angles.
Examples
Ex 1: <FAC is greater than
<ABC and <ACB
Ex 2: <ACH is greater than
<ABC and <BAC
Ex 3: <ABG is greater than
<ACB and <BAC
Triangle Inequality

The longest side of a triangle has to be
greater than the sum of the two shorter
sides for the sides to form a triangle.
Examples
Can this sides make a triangle?
 3.01, 3.99, 8
 No because 3.01+3.99=7, which is not
greater than 8.

Examples
Can this sides make a triangle?
 4, 4, 8
 No because 4+4=8 is equal to 8, so
these won’t form a triangle, they will
form a line.

Examples
Can this sides make a triangle?
 2.01, 2.57, 4
 Yes because 2.01+2.57=4.58, which is
greater than 4.

Indirect Proof
1. Write the false of the statement.
 2. That false is the given.
 3. Solve as a normal proof until you
come to a contradiction.

Examples
Prove that: a triangle can’t have two right triangles.
Step
Statement
Reason
Write the false
A triangle can have two right
angles
Given
Solve as normal
m<1=90, m<2=90
Definition of right
angles
m<1+m<2+m<3=180
Definition of a
triangle
90+90+m<3=180
Substitution
m<3=0
Subtraction
Contradiction 
An angle can’t measure 0
Examples
Prove that: The client is not guilty: A lawyer is defending his client, who was
accused of a Murder in Guatemala at 2:00 PM, but another person saw him in
Suchitepequez at the same time.
Step
Statement
Reason
Write the false
The client is guilty
Given
Solve as normal
The client was at Guate
and Suchi at the same
time
Given
Contradiction 
Someone can’t be in two
parts at the same time
Examples
Prove that: A triangle can’t have 2 obtuse angles.
Step
Statement
Reason
Write the false
A triangle can have two obtuse
angles
Given
Solve as normal
m<1=more than 90, m<2=more
than 90
Definition of
obtuse angles
m<1+m<2+m<3=180
Definition of a
triangle
The least an obtuse angle can be
is 91
Definition of
obtuse angles
m<1+m<2+m<3=180
Definition of a
triangle
91+91+m<3=180
Substitution
m<3=-2
Subtraction
Contradiction 
An angle can’t measure negative
Hinge theorem

If two sides of a triangle are congruent
to 2 sides of another triangle, and the
included angle of those 2 sides in the
first triangles is greater than the included
angle of the 2 sides in the second
triangle, then the 3rd side of the 1st
triangle is longer than the 3rd side of the
2nd angle.
Examples

Which side is longer: AB or CD?

CD because the sides in its triangle
have the largest included angle.
Examples

Which side is longer: QW or ZX

ZX because the sides in its triangle have
the largest included angle.
Examples

Which side is longer: ER or DF?

ER because the sides in its triangle
have the largest included angle.
Converse of the Hinge Theorem

If two sides of a triangle are congruent
to 2 sides of another triangle, and the 3rd
sides are not congruent, then the longer
side is across the largest included angle.
Examples

Which angle is larger: a or b?

<a because its opposite side is longer.
Examples

Which angle is larger: c or d?

<d because its opposite side is longer.
Examples
Which angle is
larger: q or w?

<q because
Its opposite side
is longer.

30-60-90 Triangle Relationships






Hypotenuse is twice as long as the short
leg.
Long leg is measure of short leg times √3.
Or…
Short leg is half as long as the hypotenuse.
Or…
Short leg is lenght of long leg divided by
√3.
Examples
What is x and y if the length of the short
leg is 5?
 x= 5(√3)= 5 √3
 y= 5x2= 10

Examples
What is x and y if the length of the
hypotenuse is 14?
 x= 14/2=7
 y= 7(√3)= 7√3

Examples
What is x and y if the length of the long
leg is 12?
 x= 12/√3= 12√3/3= 4√3
 y= 4√3(2)= 8√3

45-45-90 Triangle Relationships
Both legs are congruent.
 Hypotenuse is length of leg times √2.
 Or…
 Leg is lenght of hypotenuse divided by
√2.

Examples
What is x if length of a leg is 13?
 x= 13(√2)= 13√2

Examples
What is x if the length of the hypotenuse
is 8?
 x= 8/√2= 8√2/2=
4√2

Examples
What is x if the length of the hypotenuse
is 9?
 x= 9/√2= 9√2/2=
4.5√2

_____(0-10 pts.) Describe what a perpendicular bisector is. Explain the perpendicular bisector theorem and its converse. Give 3
examples of each.
_____(0-10 pts.) Describe what an angle bisector is. Explain the angle bisector theorem and its converse. Give at least 3 examples
of each.
_____(0-10 pts.) 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.
_____(0-10 pts.) Describe the concurrency of angle bisectors of a triangle theorem. Explain what an incenter is. Give at least 3
examples of each.
_____(0-10 pts.) 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.
_____(0-10 pts.) 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.
_____(0-10 pts.) Describe what a midsegment is. Explain the midsegment theorem. Give at least 3 examples.
_____(0-10 pts.) Describe the relationship between the longer and shorter sides of a triangle and their opposite angles. Give at least
3 examples.
_____(0-10 pts.) Describe the exterior angle inequality. Give at least 3 examples.
_____(0-10 pts.) Describe the triangle inequality. Give at least 3 examples.
_____(0-10 pts.) Describe how to write an indirect proof. Give at least 3 examples.
_____(0-10 pts.) Describe the hinge theorem and its converse. Give at least 3 examples.
_____(0-10 pts.) Describe the special relationships in the special right triangles (30-60-90 and 45-45-90). Give at least 3 examples of
each.
_____(0-5 pts.) Neatness and originality bonus.
______Total points earned (120 possible)