Question

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Exer 1: On line x, y, given 10 points A₁, A₂, A₃, ..., A_10. From point A outside xy, we connect A to A₁, A₂ to A₃,... . How many angles whose vertices are A?

Answer 1

Solution

Let 10 ray be AA₁, AA₂, AA₃, ..., AA_10.

AA1 and AA₂ or AA₃ . . . . . or AA₁₀ have 9 angles .

AA₂ and AA₃ or AA₄ or . . . . or AA₁₀ have 8 angles.

Then AA₉ and AA₁₀ have 1 angle .

Thus 10 rays have common vertex , have :

1 + 2 + 3 + . . . . + 9 .

= ( 9 + 1 ) * 9 : 2

= 10 * 9 : 2

= 90 : 2

= 45 .

Thus , there are 45 angles .