Theorems on Angles Formed by Tangents and Secants | Slides Mathematics | Docsity (2024)

Download Theorems on Angles Formed by Tangents and Secants and more Slides Mathematics in PDF only on Docsity! Recall the tangents and secants on a circle; Find the measure of arcs, exterior angles, and interior angles of a circle; Value accumulated knowledge as means of new understanding. Objective s ae aATangents or, Secants? Use Q below to answer the following questions: ʘ 3 At what points does each secant intersect the circle? In and In A Use Q below to answer the following questions: ʘ At what points does each tangent intersects the circle? PointR and point P 4 Use Q below to answer the following questions: ʘ Which angles are formed by two secant lines? 5 ¿𝐈𝐓𝐀 Use Q below to answer the following questions: ʘ Name the intercepted arcs that these angles intercepts: 8 ¿𝐑𝐃𝐈 ¿𝐁𝐐𝐂 ¿𝐁𝐀𝐑¿𝐈𝐓𝐏 RI BC BR IP Find the measure of arcs, exterior angles and interior angles formed by tangents and secants Exterior Angles An exterior angle is formed by two secants, a secant and a tangent or a two tangents drawn from a point outside the circle. The vertex lies outside of the circle. Two secants A secant and a tangent Two tangents BC = DE = 110° 𝑮𝒊𝒗𝒆𝒏 : ¿𝑫𝑨𝑬= 𝟏 𝟐 ¿ DE - BC ) 𝑺𝒐𝒍𝒖𝒕𝒊𝒐𝒏 : 𝟒𝟖°=𝟏 𝟐 ¿ 𝟒𝟖°=𝟓𝟓°−𝟏𝟐 BC ) BC 𝟒𝟖° −𝟓𝟓°=−𝟏𝟐 BC −𝟕 °=−𝟏𝟐 BC −𝟏𝟐 −𝟏𝟐 BC = 14° 14 ED = 200° EA = 40° 𝑮𝒊𝒗𝒆𝒏 : ¿𝑬𝑩𝑫= 𝟏 𝟐 ¿ ED - AD ) 𝑺𝒐𝒍𝒖𝒕𝒊𝒐𝒏 : ¿𝑬𝑩𝑫= 𝟏 𝟐 (𝟐𝟎𝟎° −𝟏𝟐𝟎° ) 40 AD = 120 ¿𝑬𝑩𝑫= 𝟏 𝟐 (𝟖𝟎°) ¿𝑬𝑩𝑫=𝟒𝟎° ABE = 265° AE = ¿ 𝑨𝑫𝑬=¿ 𝑮𝒊𝒗𝒆𝒏 : ¿ 𝑨𝑫𝑬= 𝟏 𝟐 ¿ ABE - AE ) 𝑺𝒐𝒍𝒖𝒕𝒊𝒐𝒏 : ¿ 𝑨𝑫𝑬= 𝟏 𝟐 (𝟐𝟔𝟓° −𝟗𝟓° ) ¿ 𝑨𝑫𝑬= 𝟏 𝟐 (𝟏𝟕𝟎° ) ¿ 𝑨𝑫𝑬=𝟖𝟓° 85 𝟗𝟓° Your turn! CA = (7x+4)° BE = (3x + 2)° ¿𝑪𝑫𝑨=𝟒𝟓° 𝑮𝒊𝒗𝒆𝒏 : ¿𝑪𝑫𝑨= 𝟏 𝟐 ¿ CA - BE ) 𝑺𝒐𝒍𝒖𝒕𝒊𝒐𝒏 : 4 𝑭𝒊𝒏𝒅 𝒙 ,𝒂𝒓𝒄 𝑪𝑨∧𝒂𝒓𝒄 𝑩𝑬 4 4 41 𝒙=𝟐𝟐 CA = [7(22)+4]° CA = CA = 158° BE = (3x + 2)° BE = 68° CA = (7x+4)° BE = [3(22)+2]° BE = 66 + 2 4 44 Interior Angles An interior angle can be formed by two chords or two secants that intersect inside of the circle. T T AB = 52° DE = x° ¿ 𝑨𝑪𝑩=𝟒𝟎° 𝑮𝒊𝒗𝒆𝒏 : ¿ 𝑨𝑪𝑩= 𝟏 𝟐 ¿ BA + DE ) 𝑺𝒐𝒍𝒖𝒕𝒊𝒐𝒏 : 𝟒𝟎°=𝟏 𝟐 (𝟓𝟐°+𝒙 °) 𝟒𝟎°=𝟐𝟔°+𝟏𝟐 𝒙 ° 𝟒𝟎° −𝟐𝟔°=+𝟏 𝟐 𝒙 ° 𝟏𝟒°=𝟏 𝟐 𝒙 ° 𝟏 𝟐 𝟏 𝟐 𝒙=𝟐𝟖° ¿𝑬𝑪𝑫=¿ 𝟒𝟎° NB = 107° AM = 19° ¿ 𝑨𝑿𝑴=¿ 𝑮𝒊𝒗𝒆𝒏 : ¿ 𝑨𝑿𝑴= 𝟏 𝟐 ¿ NB + AM ) 𝑺𝒐𝒍𝒖𝒕𝒊𝒐𝒏 : ¿𝑵𝑿𝑩=¿ ¿ 𝑨𝑿𝑴= 𝟏 𝟐 (𝟏𝟐𝟔° ) ¿ 𝑨𝑿𝑴=𝟔𝟑° 63 ¿𝑴𝑿𝑩=𝟏𝟖𝟎° −𝟔𝟑° ¿𝑴𝑿𝑩=𝟏𝟏𝟕° 117¿𝑴𝑿𝑩=¿ ¿ 𝑨𝑿𝑵=¿63 117 ¿ 𝑨𝑿𝑴= 𝟏 𝟐 ¿ 107 + 19 ) AC = (3x+18)° DB = (x+10)° ¿𝟏=𝟖𝟎° 𝑮𝒊𝒗𝒆𝒏 : ( AC + DB ) 𝑺𝒐𝒍𝒖𝒕𝒊𝒐𝒏 : 𝟖𝟎°=𝟏 𝟐 [(𝟑 𝒙+𝟏𝟖)°+(𝒙+𝟏𝟎)° ] 𝒙=¿𝟑𝟑¿𝟐=¿ 𝟖𝟎°=𝟏 𝟐 (𝟒 𝒙+𝟐𝟖)° 𝟖𝟎°=𝟐 𝒙 °+𝟏𝟒° 𝟐 𝒙=𝟔𝟔° 𝒙=𝟑𝟑 AC = DB =𝟏𝟎𝟎° AC = (3x+18)° AC = [3(33)+18]° AC = (99+18)° AC = 117° DB = (x+10)° DB = 43° DB = (33+10)° 𝟏𝟏𝟕° 𝟒𝟑° ¿𝟏= 𝟏 𝟐 AC = (2x+17)° DB = (2x+3)° ¿𝟏=𝟔𝟎° 𝑮𝒊𝒗𝒆𝒏 : AC + DB ) 𝑺𝒐𝒍𝒖𝒕𝒊𝒐𝒏 : 6 𝒙=¿𝟐𝟓¿𝟐=¿ 6 6 𝟓𝟎=𝟐 𝒙 25 AC = DB =𝟏𝟐𝟎° AC = (2x+17)° AC = [2(25)+17]° AC = (50+17)° AC = 67° DB = (2x+3)° DB = 53° DB = (50+3)° 𝟔𝟕° 𝟓𝟑° ¿𝟏= 𝟏 𝟐( 6 DB = [2(25)+3)° Check your understanding _____ 1. If two secants intersect in the interior of the circle, then the angle formed is one-half the positive difference of the measures of the intercepted arcs. _____ 2. If two secants intersect in the exterior of the circle, then the angle formed is one-half the positive difference of the measures of the intercepted arcs. _____ 3. If two tangents intersect in the exterior of the circle, then the angle formed is one-half of the sums of the measures of the intercepted arcs. False Tell whether the following statement is true or false. True False Check your understanding _____ 4. An interior angle can be formed by two chords or two secants that intersect inside of the circle. _____ 5. The measure of the angle formed by the two chords or two secants is equal to ½ the sum of the measures of the intercepted arcs. _____ 6. An exterior angle is formed by two secants, a secant and a tangent or a two tangents drawn from a point outside the circle. Tell whether the following statement is true or false. True True True