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Geometry and Trigonometry Difficulty: Hard

In the figure, $A C = C D$ . The measure of angle $E B C$ is $45 degree$ , and the measure of angle $A C D$ is $104 degree$ . What is the value of $x$ ?

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Explanation

The correct answer is $83$ . It's given that in the figure, $A C = C D$ . Thus, triangle $A C D$ is an isosceles triangle and the measure of angle $C D A$ is equal to the measure of angle $C A D$ . The sum of the measures of the interior angles of a triangle is $180 degree$ . Thus, the sum of the measures of the interior angles of triangle $A C D$ is $180 degree$ . It's given that the measure of angle $A C D$ is $104 degree$ . It follows that the sum of the measures of angles $C D A$ and $C A D$ is $left parenthesis 180 minus 104 right parenthesis degree$ , or $76 degree$ . Since the measure of angle $C D A$ is equal to the measure of angle $C A D$ , the measure of angle $C D A$ is half of $76 degree$ , or $38 degree$ . The sum of the measures of the interior angles of triangle $B D E$ is $180 degree$ . It's given that the measure of angle $E B C$ is $45 degree$ . Since the measure of angle $B D E$ , which is the same angle as angle $C D A$ , is $38 degree$ , it follows that the measure of angle $D E B$ is $(180 - 45 - 38) °$ , or $97 degree$ . Since angle $D E B$ and angle $A E B$ form a straight line, the sum of the measures of these angles is $180 degree$ . It's given in the figure that the measure of angle $A E B$ is $x °$ . It follows that $97 + x = 180$ . Subtracting $97$ from both sides of this equation yields $x equals 83$ .