document.write( "Question 1057927: ∠ABC is formed by two tangents intersecting outside of a circle. If minor arc AC = 110°, what is the measure of ∠ABC? \n" ); document.write( "

Algebra.Com's Answer #672996 by ikleyn(52787) $\"\"$ $\"About$

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\n" ); document.write( "∠ABC is formed by two tangents intersecting outside of a circle. If minor arc AC = 110°, what is the measure of ∠ABC?
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document.write( "The plot is shown in the Figure on the right (although not in the scale).    \r\n" );
document.write( "The point O is the center of the circle. \r\n" );
document.write( "It is connected with the point C by the segment OC.\r\n" );
document.write( "OA and OB are the radii drawn to the tangent points A and C respectively.\r\n" );
document.write( "We are given that the measure of the minor arc AC is 110°.\r\n" );
document.write( "It means that the measure of the central angle AOC is 110°.\r\n" );
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document.write( "Consider the triangles OAB and OCB.\r\n" );
document.write( "These triangles are right angled triangles.\r\n" );
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document.write( "They have the common hypotenuse OB and congruent legs OA and OC.\r\n" );
document.write( "Hence, the triangles are congruent.\r\n" );
document.write( "It implies that the angles AOB and COB are congruent.\r\n" );
document.write( "Then the measure of each of these angles is the half o the measure \r\n" );
document.write( "of the angle AOC, i.e. 55°.\r\n" );
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document.write( "The angle ABO is complement to the angle AOB and, therefore, is \r\n" );
document.write( "equal to 90°-55° = 35°.\r\n" );
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document.write( "Similarly, the angle CBO is complement to the angle COB and, therefore, \r\n" );
document.write( "is equal to 90°-55° = 35°.\r\n" );
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document.write( "                Figure. \r\n" );
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document.write( "  Then the angle ABC is 35° + 35° = 70°.\r\n" );
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document.write( "Answer.  Angle ABC is 70°.\r\n" );
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