Triangle upper A upper C upper D partially overlaps triangle upper E upper B upper D. Point upper C is

Question

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 $x$ ?

Accepted Answer

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 &#176; . Thus, the sum of the measures of the interior angles of triangle A C D is 180 &#176; . It's given that the measure of angle A C D is 104 &#176; . It follows that the sum of the measures of angles C D A and C A D is 180 - 104 &#176; , or 76 &#176; . 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 &#176; , or 38 &#176; . The sum of the measures of the interior angles of triangle B D E is 180 &#176; . It's given that the measure of angle E B C is 45 &#176; . Since the measure of angle B D E , which is the same angle as angle C D A , is 38 &#176; , it follows that the measure of angle D E B is 180 - 45 - 38 &#176; , or 97 &#176; . Since angle D E B and angle A E B form a straight line, the sum of the measures of these angles is 180 &#176; . It's given in the figure that the measure of angle A E B is x &#176; . It follows that 97 + x = 180 . Subtracting 97 from both sides of this equation yields x = 83 .

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