In Figure 1, Segment CE and Segment LA are diameters of Circle N. The measurement of Arc LE is equal to 120 degrees
What is the measurement of angle 1 - 9
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What is the measurement of angle 1 - 9
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[tex]\quad \huge \quad \quad \boxed{ \tt \:Answer }[/tex]
[tex]\qquad \tt \rightarrow \: \angle1 =30 \degree[/tex]
[tex]\qquad \tt \rightarrow \: \angle 2=60 \degree[/tex]
[tex]\qquad \tt \rightarrow \:\angle 3=60 \degree [/tex]
[tex]\qquad \tt \rightarrow \: \angle4 = 60\degree[/tex]
[tex]\qquad \tt \rightarrow \: \angle 5=120 \degree[/tex]
[tex]\qquad \tt \rightarrow \: \angle6 = 60\degree[/tex]
[tex]\qquad \tt \rightarrow \:\angle7 =30 \degree [/tex]
[tex]\qquad \tt \rightarrow \:\angle8 =60 \degree [/tex]
[tex]\qquad \tt \rightarrow \:\angle 9=60 \degree [/tex]
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[tex] \large \tt Solution \: : [/tex]
[tex] \textsf{Angle LNE = 120° (given)} [/tex]
[tex]\qquad \sf \dashrightarrow \: \angle5 = \angle LNE[/tex]
[tex] \textsf{[ by vertical opposite angle pair ]} [/tex]
[tex]\qquad \sf \dashrightarrow \: \angle5 = 120 \degree[/tex]
[tex] \textsf{Next,} [/tex]
[tex]\qquad \sf \dashrightarrow \: \angle4 + \angle LNE = 180 \degree[/tex]
[tex] \textsf{[ By linear pair ]} [/tex]
[tex]\qquad \sf \dashrightarrow \: \angle4 + 120 \degree = 180 \degree[/tex]
[tex]\qquad \sf \dashrightarrow \: \angle4 = 180 \degree - 120 \degree[/tex]
[tex]\qquad \sf \dashrightarrow \: \angle4 = 60 \degree[/tex]
[tex] \textsf{Next, } [/tex]
[tex]\qquad \sf \dashrightarrow \: \angle6 = \angle4 = 60 \degree [/tex]
[tex] \textsf{[ by vertical opposite angle pair ]} [/tex]
[tex]\qquad \sf \dashrightarrow \: \angle6 = 2 \times \angle1[/tex]
[tex] \textsf{[ angle subtended by arc on centre is double} [/tex] [tex] \textsf{the angle subtended by same arc on part of } [/tex][tex] \textsf{circumference. ]} [/tex]
[tex]\qquad \sf \dashrightarrow \angle1 = \angle6 \div 2[/tex]
[tex]\qquad \sf \dashrightarrow \angle1 = 60 \degree\div 2[/tex]
[tex]\qquad \sf \dashrightarrow \angle1 = 30 \degree[/tex]
[tex] \textsf{similarly, } [/tex]
[tex]\qquad \sf \dashrightarrow \: \angle7 = 30 \degree[/tex]
[tex] \textsf{[ because, by above property : } [/tex][tex]{ \angle 7 } [/tex][tex] \textsf{= 1/2 × } [/tex][tex]{ \angle 4 ]} [/tex]
[tex] \qquad \sf \dashrightarrow \: \angle2 = \angle8 = \dfrac{1}{2} \times \angle LNE[/tex]
[tex] \textsf{[ same property : angle subtended by an arc at centre} [/tex] [tex] \textsf{is double the angle subtended by same arc on part } [/tex][tex] \textsf{of circumference ]} [/tex]
[tex] \qquad \sf \dashrightarrow \: \angle2 = \angle8 = \dfrac{1}{2} \times 120 \degree[/tex]
[tex] \textsf{Hence, } [/tex]
[tex]\qquad \sf \dashrightarrow \: \angle2 = 60 \degree[/tex]
[tex] \textsf{and } [/tex]
[tex]\qquad \sf \dashrightarrow \: \angle8= 60 \degree[/tex]
[tex] \textsf{By the similar method ; } [/tex]
[tex]\qquad \sf \dashrightarrow \: \angle3 = \angle9 = \dfrac{1}{2} \times \angle5[/tex]
[tex]\qquad \sf \dashrightarrow \: \angle3 = \angle9 = \dfrac{1}{2} \times 120 \degree[/tex]
[tex] \textsf{so, } [/tex]
[tex]\qquad \sf \dashrightarrow \: \angle 3 = 60 \degree[/tex]
[tex] \textsf{and } [/tex]
[tex]\qquad \sf \dashrightarrow \: \angle 9= 60 \degree[/tex]
[tex] \textsf{That's all} [/tex] ~