Unknown Angle Proofs—Proofs of Known Facts
A proof of a mathematical statement is a detailed explanation of how that statement follows logically from other statements already accepted as true. A theorem is a mathematical statement with a proof.
Once a theorem has been proved, it can be added to our list of known facts and used in proofs of other theorems. For example, in Lesson 9, we proved that vertical angles are of equal measure, and we know (from earlier grades and by paper cutting and folding) that if a transversal intersects two parallel lines, alternate interior angles are of equal measure. How do these facts help us prove that corresponding angles are equal in measure?
In the diagram to the right, if you are given that 𝐴𝐵 ∥ 𝐶𝐷, how can you use your knowledge of how vertical angles and alternate interior angles are equal in measure to prove that 𝑥 = 𝑤?
You now have available the following facts:
Use any or all of these facts to prove that interior angles on the same side of the transversal are supplementary. Add any necessary labels to the diagram below, and then write out a proof including given facts and a statement of what needs to be proved.
Given: 𝐴𝐵 ∥ 𝐶𝐷, transversal 𝐸𝐹 Prove: 𝑚∠𝐵𝐺𝐻 + 𝑚∠𝐷𝐻𝐺 = 180°
Now that you have proven this, you may add this theorem to your available facts.
Use any of these four facts to prove that the three angles of a triangle sum to 180°. For this proof, you will need to draw an auxiliary line, parallel to one of the triangle’s sides and passing through the vertex opposite that side. Add any necessary labels, and write out your proof.
Let’s review the theorems we have now proven:
Side Trip: Take a moment to take a look at one of those really famous Greek guys we hear so much about in geometry, Eratosthenes. Over 2,000 years ago, Eratosthenes used the geometry we have just been working with to find the circumference of Earth. He did not have cell towers, satellites, or any other advanced instruments available to scientists today. The only things Eratosthenes used were his eyes, his feet, and perhaps the ancient Greek equivalent to a protractor.
Watch this video to see how he did it, and try to spot the geometry we have been using throughout this lesson. https://youtu.be/wnElDaV4esg
Construct a proof designed to demonstrate the following:
If two lines are perpendicular to the same line, they are parallel to each other.
(a) Draw and label a diagram, (b) state the given facts and the conjecture to be proved, and (c) write out a clear statement of your reasoning to justify each step.
Discussion Each of the three parallel line theorems has a converse (or reversing) theorem as follows:
Notice the similarities between the statements in the first column and those in the second. Think about when you would need to use the statements in the second column, that is, the times when you are trying to prove two lines are parallel.
In the figure to the right, 𝑥 = 𝑦. Prove that 𝐴𝐵 ∥ 𝐸𝐹.
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