This is a theoretical exposition of a solution to a question, posed below, that related to the sides and angles of triangles. The problem posed is: Given a positive integer a and a line segment AB having length a, describe all triangles ABC having integer sides, with( B) ̂=90°? The same question can be posed when B=60°. We describe a process that answers the question in the affirmative, while producing all such triangles, for any The subsequent discussion and proofs considers the very specific cases of triangles involving 90° and 60° angles. Much of the discussions will include the idea of a \"θ°\" - integral triangle. We define, for the purpose of this paper, the \"θ°\" - integral triangle as a triangle having sides a, b and c, where a, b and c are positive integers. Then (a, b, c) is called a \"θ°\" - integral triangle if \"θ\" is the angle included by sides a and b. A 90\"°\" - integral triangle is also known as a Pythagorean triple.

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