Shu believes that a theorem is missing from the lesson. His conjecture is that if two coplanar lines are cut by a transversal so that a pair of same-side exterior angles are supplementary, then the two lines are parallel. Complete the two-column proof with the statements and reasons provided. Given: 21 and 23 are supplementary. Prove: m || n Proof: Statements Reasons 1. 21 and Z3 are supplementary 1. 2 2. Linear Pair Theorem 3. Z1 22 4 Converse of Corresponding Angles Postulate

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8. Let ABC be a triangle, and let D, E, and F be the points where the incircle is tangent to sides BC, CA and AB respectively. (a) Prove that the cevians AD, BE and CF are concurrent using Ceva's Theorem. (b) Demonstrate the concurrency of the cevians by constructing the triangle, its incircle, and the cevians in GeoGebra.

1) Consider the following: A C D B Let 8 denote the defect of a triangle. One of the following two statements can be proved in Neutral Geometry and the other cannot. Prove the one that can be proved and explain why the other cannot be proved in Neutral Geometry: S: 8(ACD) < 8(ABC) T: 8(ACD) ≤ 8(ABC)