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Geometry – Chapter 14 Lesson Plans Section 14.5 – Segment Measures Enduring Understandings: The student shall be able to: 1. Find measures of chords, secants, and tangents. Standards: 31. Circles Apples geometric relationships to solving problems, such as relationships between lines and segments associated with circles, the angles they form, and the arcs they subtend; and the measures of these arcs, angles, and segments. Essential Questions: How can we calculate the lengths of the parts of intersecting secants? Warm up/Opener: Activities: Thm 14-13: If two chords of a circle intersect, then the product of the measures of the segments of one chord equals the product of the measures of the segments of the other chord. B A D E C Proof of Theorem 14-13: Statements Draw AD and BC forming ABE and DCE A D, B C Reasons Construction 14-2: If inscribed angles intercept the same arc, then the angles are congruent AA CPCTC Multiplication by common denominator ABE ~ DCE AE/DE = EB/EC AE * EC = ED * EB Defn: A segment is an External Secant Segment iff it is the part of a secant segment that is outside a circle. Thm 14-14: If two secant segments are drawn to a circle from an exterior point, then the product of the measures of one secant segment and its external secant segment equals the product of the measures of the other secant segment and its external secant segment. C K L J D Proof of Theorem 14-14: Statement Reason Draw KL and CD forming JCD and JLK Construction 14-2: If inscribed angles intercept the same C L arc, then the angles are congruent Reflexive property J J AA JCD ~ JLK JC/JL = JD/JK CPCTC JC * JK = JL * JD Multiplication by common denominator A special case of theorem 14-14 is when one segment is a tangent segment. This is covered in Theorem 14-15: If a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment equals the product of the measures of the secant segment and its external secant segment. E F G Proof of Theorem 14-15: Statement Draw EG and EH forming FEG and FHE FEG EHG F F FEG ~ FHE FE/FH = FG/FE FE2 = FH * FG H Reason Construction They intersect the same arc, and by 14-11 if a secant-tangent angle has its vertex on the circle, then its degree measure equals one-half the intercepted arc, and by 14-1 the degree measure of an intercepted arc is one-half the intercepted arc. Reflexive Property AA CPCTC Multiplication by common denominator Assessments: Do the “Check for Understanding” CW WS 14.5 HW pg 616 - 617, # 8 - 26 all (19) if I only cover this section HW pg 616 – 617, # 9 – 25 odd (9) if I cover this section with another section