Transcript Chapter 4 Part B

Chapter 4 Part B
By: Britt, Anne, Emily, and Jacob
Chapter 4.5 Notes
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Using Congruent Triangles
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Given:SD≈TC, CS≈DT
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Prove:<SCT≈ <TDS
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Plan for Proof: Use CPCTC
(corresponding parts of
congruent Triangles are
Congruent. By the Reflexive
Property, ST≈ST. You can
use the SSS congruence
postulate to conclude that
∆CST≈ ∆DTS Because of
CPCTC, it follows <SCT≈
<TDS.
CH. 4.5 Notes Cont.
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USING MORE THAN 1 PAIR OF TRIANGLES
Given: <GMJ and <HJM are right angles.
GF≈HF, <1≈<2
Prove: <GJM≈HJM
4.5 Notes Cont
Statements
Reasons
1) GF ≈ HF, <1 ≈ <2
1)Given
2) FM ≈ FM
2) Reflexive
3) ∆ FGM ≈ ∆FHM
3) SAS
4) GM ≈ MH
4) CPCTC
5) <GMJ ≈ <HMJ
5) Right Angles are Congruent
6) MJ ≈ MJ
6) Reflexive
7) ∆GJM ≈ ∆HJM
7) SAS
8) <GJM ≈ <HJM
8) Corresponding. Parts of ≈ ∆’s
are ≈
4.5 Notes Cont.
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Proving a Construction
Given: HJ ≈ HG, GK ≈ JK
4.5 Notes Cont.
Statements
Reasons
1)HJ≈ HG, GK ≈JK
2) HK≈ HK
3) ∆ HGK ≈ ∆HJK
4) <1 ≈ <2
5) HK is a bisector
1) Given
2) Reflexive
3) SSS
4) CPCTC
5) Def. of Bisector
4.6 Notes
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Base Angles- Two
angles Adjacent to the
base.
Vertex Angles- The
angle Opposite of the
base.
4.6 Notes Cont.
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Theorem 4.6-Base Angles
Theorem-If two sides of a
triangle are congruent, then
the angles opposite of them
are Congruent.
4.6 Notes Cont.
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Theorem 4.7Converse of the Base
Angles Theorem- If
two angles of an
opposite triangle are
congruent, then the
two sides opposite
them are congruent.
4.6 Notes Cont.
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Corollary to Theorem 4.6-If a triangle is
Equilateral, then it is Equiangular.
4.6 Notes Cont.
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Corollary to Theorem 4.7-If a triangle is
Equiangular, then it is Equilateral.
4.6 Notes Cont.
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Theorem 4.8- Hypotenuse-Leg (HL) Congruence
Theorem-If the hypotenuse and a leg of a Right triangle
are congruent to the hypotenuse and a leg of a second
Right triangle, then the two triangles are congruent.
4.7 Notes
Coordinate Proof- is the placing of geometric
figures in a coordinate plane.
Using the Distance Formula
A right triangle has legs of nine units and twelve
units. Place the triangle in a coordinate plane.
Label the coordinates of the vertices and find
the length of they hypotenuse.
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4.7 Notes Cont.
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Solution- One possible placement is shown.
Notice that one leg is vertical and the other leg
is horizontal, which assures that the same
vertical segments have the same slope, and the
points on the same horizontal segment have the
same slope. You can use the distance formula
to find the length of the hypotenuse.
Career In Geometry
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Hydrologist
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Hydrologists study underground and surface
water sources as well as the distribution and
circulation of water in the atmosphere. These
water sources include rivers, ponds, lakes,
oceans, underground water supplies, and glaciers.
The circulation of water in the atmosphere
includes rain, snow, and other forms of
precipitation.
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Hydrologists commonly conduct research on climate assessment,
erosion and sedimentation, hydroelectric power plants, irrigation
systems, flood readiness, distribution and use of public water supplies,
and the environmental impact of pollution on water quality.
Hydrologists may be assigned to a specific project such as the
development of an environmentally safe drainage plan for wastewater.
Others may conduct research on a larger scale, aimed at developing
new methods and techniques in hydrologic studies. Many hydrologists
oversee a team of technologists and technicians.
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Hydrologists may be considered under the general heading of
Geologist. In addition to studying water, geologists concern
themselves with soil, earthquake activity, hazardous waste sites,
petroleum deposits, and other natural formations. A geologist's
area of specialization defines his or her specific title. For
example, petroleum geologists map the surface of Earth both
underwater and on land for the existence of oil and natural gas.
Mineralogists analyze, identify, and classify minerals according
to their composition and structure.
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Education and Training
In the U.S., hydrologists may need the following
education and training:
a bachelor's degree in environmental science,
geology, or a related field for most entry-level positions
a master's degree, or preferably a doctorate degree, in
hydrology for positions in research or at a college or
university
On the Job
Hydrologists often work with a team of scientists,
technologists, and technicians. Much of their time is
spent collecting data and performing experiments.
This aspect of a hydrologist's work, called fieldwork,
often requires extensive travel. The remainder of their
time is spent analyzing data, drawing conclusions, and
preparing technical reports.
Math on the Job
Most hydrologic studies require the ability to analyze
data and draw conclusions. The information that
hydrologists analyze is often numerical. For example,
hydrologists may collect data on the movement and
quantity of groundwater over time.
Related Careers
geologist
● geophysicist
● meteorologist
● mineralogist
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