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This lesson focuses on the theorems related to congruent inscribed angles and tangent-chord angles. Key theorems include: Theorem 89, which states that if two angles intercept the same arc, they are congruent; Theorem 90, indicating that two angles intercepting congruent arcs are also congruent; and Theorem 91, which explains that an angle inscribed in a semicircle is a right angle. Additionally, Theorem 92 asserts that the sum of a tangent-tangent angle and its minor arc measures 180º. Explore applications of these theorems with geometric proofs and problem-solving techniques.
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More Angle-Arc Theorems Lesson 10.6
Congruent Inscribed & Tangent-Chord Angles Theorem 89: If two inscribed or tangent-chord angles intercept the same arc, then they are congruent.
Congruent Inscribed & Tangent-Chord Angles Theorem 90: If two inscribed or tangent-chord angles intercept congruent arcs, then they are congruent.
Angles Inscribed in Semi-Circles Theorem 91: An angle inscribed in a semicircle is a right angle. Since the measure of an inscribed angle is one-half the measure of its intercepted arc, and a semi-circle is 180º, C is 90º.
Special Theorem about Tangent-Tangent Angles Theorem 92: The sum of the measures of a tangent-tangent angle and its minor arc is 180º.
A is inscribed in a semicircle, it is a right angle. • Use the Pythagorean Theorem to solve. • (AB)2 + (AC)2 = (BC)2 • (AB)2 + 402 = 412 • AB = 9 mm
Circle O • V S • L N • ΔLVE ~ ΔNSE • EV = ELSE EN • EV • EN = EL • SE Given If two inscribed s intercept the same arc, they are . Same as 2 AA (2, 3) Ratios of corresponding sides of ~ triangles are =. Means-Extremes Products Theorem.
