2. Carefully read the proof below. Identify any mistakes or gaps in reasoning in the proof. Write up a formal version of

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2 Carefully Read The Proof Below Identify Any Mistakes Or Gaps In Reasoning In The Proof Write Up A Formal Version Of 1 (157.86 KiB) Viewed 29 times

2. Carefully read the proof below. Identify any mistakes or gaps in reasoning in the proof. Write up a formal version of the proof in which you have corrected all the errors and justified all of the steps. (Try to maintain as much of the original proof as possible.) Then write a few sentences explaining what what incorrectly jus- tified or not justified in the original proof and how you corrected them in your proof. Theorem. Let ABC be a triangle. Extend sides AB and AC to rays AB and AC, forming exterior angles. Let rg be the angle bisector of the exterior angle at B, and let rc be the angle bisector of the exterior angle at C. Then rẻ and rc intersect. Proof. Suppose, seeking contradiction, that rẻ and re do not in- tersect. Then rg and rc must be parallel. Let D be a point on rg and let E be a point on rc. Since r² and rc are parallel, by Propo- sition 28 angle DBC and angle BCE sum to two right angles. The exterior angle at B and the exterior angle at C sum to four right angles. But there's a proposition that says that the exterior angle in a triangle is equal to the sum of the two interior and opposite angles in the triangle. Therefore ext B+ext C = /BAC+ZACB+ZBAC + /BAC = 4 right angles. Therefore, ZBAC equals 2 right angles. This is a contradiction because we know one angle in a triangle cannot equal 2 right angles. Thus our initial assumption must be incorrect, so it must be true that rg and rc intersect.