Chapter 4 Part B

By: Britt, Anne, Emily, and Jacob

Chapter 4.5 Notes● Using Congruent Triangles● Given:SD≈TC, CS≈DT● Prove:<SCT≈ <TDS● 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.

● USING MORE THAN 1 PAIR OF TRIANGLES

Given: <GMJ and <HJM are right angles. GF≈HF, <1≈<2

Prove: <GJM≈HJM

4.5 Notes ContStatements

1) GF ≈ HF, <1 ≈ <2

2) FM ≈ FM

3) ∆ FGM ≈ ∆FHM

4) GM ≈ MH

5) <GMJ ≈ <HMJ

6) MJ ≈ MJ

7) ∆GJM ≈ ∆HJM

8) <GJM ≈ <HJM

Reasons

1)Given

2) Reflexive

3) SAS

4) CPCTC

5) Right Angles are Congruent

6) Reflexive

7) SAS

8) Corresponding. Parts of ≈ ∆’s are ≈

4.5 Notes Cont.● Proving a Construction● Given: HJ ≈ HG, GK ≈ JK

4.5 Notes Cont.

Statements

1)HJ≈ HG, GK ≈JK

2) HK≈ HK

3) ∆ HGK ≈ ∆HJK

4) <1 ≈ <2

5) HK is a bisector

Reasons

1) Given

2) Reflexive

3) SSS

4) CPCTC

5) Def. of Bisector

4.6 Notes

● Base Angles- Two angles Adjacent to the base.

● Vertex Angles- The angle Opposite of the base.

4.6 Notes Cont.

● 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.

● Theorem 4.7- Converse 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.

● Corollary to Theorem 4.6-If a triangle is Equilateral, then it is Equiangular.

4.6 Notes Cont.

● Corollary to Theorem 4.7-If a triangle is Equiangular, then it is Equilateral.

4.6 Notes Cont.● 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.

4.7 Notes Cont.

● 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

● Hydrologist

● 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.

● 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.

● 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.

● 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