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Dynamic movement, measure, geometric shape

Describe (with an illustrative example/figure) three different meanings of angle (dynamic movement, measure, geometric shape) and the three corresponding meanings of congruent, one for each meaning of angle.
a) Prove the vertical angle theorem. Begin your proof with a clear statement of the theorem, which must include what is given and what is to be proven.
Prove the vertical angle theorem in a different way, using a different meaning of angle. Repeat all parts of question 2.
Prove the vertical angle theorem in a different way, using yet another meaning of angle. Repeat all parts of question 2.
b)Does your proof work on the surface of the sphere? Why or why not?
c)Which of the three meanings of angle from problem 1 are you primarily using in your proof? Briefly explain how this meaning is used in your proof.

Sample Solution

ould think that senators take to these extremes because of corporate money. According to, both Republican and Democratic candidates receive money from oil companies. Republicans receive close to sixteen million dollars while Democrats receive close to four million dollars in an election season. Senator Barack Obama received close to two million dollars during his tenure in the senate, while Hillary Clinton received close to one million during her time in the Senate. Though Republicans usually receive more funding from the oil companies, oil-backed money in politics crosses political lines. Important topics such as climate change cannot be discussed honestly as many people in politics have special interests in their pockets. This inhibits political action, so nations cannot take a stand on the question of climate change. Politics have also engulfed the climate change movement in a terrible way, as well. It seems that the environmental movement needs to be put back in the hands of the scientists and not politicians. Politicians are known to be ruthless and money hungry while scientists might be more inclined to fight for the cause and not their own pocket. Another barrier in the way of resolving climate change is economics. When it comes to maintaining economic stability, it can be hard to limit people’s environmental footprint. For example, if a government wants to put a tax on oil, consumers would have to pay more for many goods. Many people may become upset and begin to resent the movement. This becomes politically problematic in democratic nations in turn. If a politician was to propose a policy such as a gas tax, they may lose many voters. To achieve sustainable environmental goals, economic change is needed. The University of Cambridge in 2015 proposed a plan to rewire the economy with different tasks. This proposed plan was set up in five different sections: government, finance, business, social outcomes, and environmental outcomes. The role of government in this proposal is action. They propose that the government should make official goals to cut greenhouse gas emissions, and they should strictly enforce them. A new tax system would also need to be introduced. These taxes would include waste and pollution taxes, which would promote cleaner production and energy sources. These taxes would also allow the citizenry to be more aware of waste. The government should also encourage companies to meet social and environmental goals, even while they pursue their profits. These goals might include low-carbon energy, zero deforestation, and zero waste. The government would also need to take an active role in limiting the media messages that undermine social and environmental progress.

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