Transcript MARCELA JANSSEN 10

Marcela Janssen
End of presentation!
JOURNAL CHAPTER 5
INDEX
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Perpendicular
Bisectors
Angle Bisectors
Concurrency,
concurrency of
perpendicular
bisectors and
circumcenter
concurrency of angle
bisectors of a triangle
theorem and incenter
Median, centroid and
the concurrency of
medians of a triangle
theorem.
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Altitude, orthocenter
and concurrency of
altitudes of a triangle
theorem
Midsegment and
midsegment theorem
Angle-side
relationship theorem
exterior angle
inequality
triangle inequality
Indirect proof
Hinge theorem and
its converse
What is a perpendicular
bisector?
A perpendicular is a line perpendicular to a
segment at the segment’s midpoint.
Perpendicular Bisector
Theorem and its Converse.
Perpendicular Bisector Theorem:
 If a segment is bisected by a
perpendicular line, then any point on the
perpendicular bisector is equidistant to the
endpoint of the segment.
Converse:
 If a point is equidistant to the segment,
then lies on a perpendicular bisector.
Example 1
Given that the line t is the perpendicular
bisector of line JK, JG = x + 12, and KG = 3x –
17, find KG.
Answer: KG = 26.5
Example 2
Given: seg AF is congruent to seg FC, <ABE
is congruent to <EBC
Which line is a perpendicular bisector in
ABC?
Answer – seg. GF
Example 3
Given: L is the perpendicular bisector of seg AB
Prove: XA=XB
statement
Reason
L is the perpendicular bisector
of seg AB
Given
Y is the midpoint of seg AB
Def. of perpendicular bisector
<AYX and < BYX are right
angles
<AYX is congruent to <BYX
Def. of perpendicular
Segment AY is congruent to seg
BY
Def. of a midpoint
Seg XY is congruent to seg XY
Reflexive P.
Triangle AYX is congruent to
triangle BYX
SAS
Seg XA is congruent to seg XB
CPCT
XA=XB
Def. of congruency
What is an angle bisector?
An angle bisector is a ray that divides an
angle into two congruent angles.
Angle bisector theorem and its
converse.
Angle Bisector Theorem:
 If an angle is bisected by a ray/line, then
any point on the line is equidistant from
both sides of the angle.
Converse:
 If a point in the interior of an angle is
equidistant form both sides of the angle,
then it lies on the angle bisector.
Example 1
Given that m<RSQ = m<TSQ and TQ =
1.3, find RQ.
 Answer: RQ = 1.3
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Example 2
Given that RQ = TQ, m<QSR = (9a + 48) o ,
and m<QST = (6a + 50) o, find m<QST.
(9a + 48) o = (6a + 50) o
9a - 6a = 50 – 48
3a = 2
a = 2/3
6/1 (2/3) + 50
12/3 + 50
4 + 50
< QST = 54o
Example 3
Ray MO bisects <LMN, m< LMO = 15x – 28,
and m<NMO = x + 70. Solve for x and find
m<LMN.
15x
15x
14x
X=
+ 8 = x + 70
– 70 = 70 – 8
= 62
62/14
What does concurrent means?
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Concurrent is when three or more lines
intersect at one point.
Concurrency of Perpendicular
bisectors of a triangle theorem
Concurrency of Perpendicular Bisectors of a
Triangle Theorem: The circumcenter of a
triangle is equidistant from the vertices of
the triangle.
What is a circumcenter?
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Circumcenter is the point of concurrency
of the three perpendicular bisectors of a
triangle.
Concurrency of angle bisectors
of a triangle theorem.
The Concurrency of angle bisectors of a
triangle theorem is when a bisector cuts a
triangle in half, therefore making a
perpendicular line, making a 90 degrees
angle.
What is an incenter?
The incenter of a triangle is the point of
concurrency of the three angle bisectors of a
triangle.
What is a median?
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A median is a segment whose endpoints
are a vertex of the triangle and the
midpoint of the opposite side.
What is a centroid?
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A centroid is the point of concurrency of
the three medians of a triangle.
Concurrency of medians of a
triangle theorem.
The centroid of the triangle located two
thirds of the distance from each of the
vertex to the midpoint of the opposite side.
What is altitude?
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Altitude of a triangle is a perpendicular
segment from a vertex to the line
containing the opposite side.
What is orthocenter?
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The orthocenter is the point of
concurrency of the three altitudes of a
triangle.
Concurrency of altitudes of a
triangle theorem
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All the lines that hold the altitudes of a
triangle are concurrent.
What is a midsegment?
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A midsegment of a triangle is a segment
that joins the midpoints of two sides of a
triangle.
Midsegment Theorem
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A midsegment of a triangle is a parallel to
the side of a triangle, and its length of
that side.
Angle-Side Relationship
Theorem
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In any triangle, the side that is opposite to
the biggest angle will have the biggest
length; the side opposite to the smallest
angle will be the smallest length.
Exterior Angle Inequality
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The exterior angle inequality states that
the exterior angle of a
is greater than
the other non-adjacent interior angles.
Triangle Inequality Theorem
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The sum of any two sides length of a
triangle is greater than the third side
length.
AB + BC > AC
BC + AC > AB
AC + AB > BC
Steps of an indirect proof:
1. Assume that what you are trying to
prove is false.
 2. Try to prove it by using previews
knowledge.
 3. When you come to a contradiction, you
have proven the theory true!
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Example
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Given: RS > RQ
Prove: m
RQS > m
S
Proof:
Locate P on RS so that RP=RQ. So RP = RQ
by def. of congruent segments. Then <1 = <2
by isosceles triangle, and m<1 = m<2 by def.
of congruent angles. By the Angle Addition
postulate, m<RQS = .m< 1 + m< 3. So
m<RQS > m<2 by the Comparison Property.
Then m<RQS > m<2 by substitution. By the
Exterior Angle Theorem, m<2 = m<3 + m<S.
So m<2 > m<S by Comparison Property.
Therefore m<RQS > m<S by Trans. property of
inequality
Hinge Theorem and its
converse.
Hinge Theorem:
 If two sides of two triangles are congruent
and the angle between them is not
congruent then the triangle with the larger
angle will have the longer 3rd side.
Converse:
 If the triangle with the larger angle is the
one that haves the longer side, then the
two sides of the triangle are congruent
and the angle between them is not
congruent.
Ready for the Chapter 5 Test?