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# In the figure, triangles ABC and ABD are right triangles. What is the

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Re: In the figure, triangles ABC and ABD are right triangles. What is the [#permalink]
Bunuel wrote:

In the figure, triangles ABC and ABD are right triangles. What is the value of x ?

(A) 20
(B) 30
(C) 50
(D) 70
(E) 90

Source: Nova GMAT
Difficulty Level: 550

Attachment:
triangle (1).jpg

In my observation, problems of this nature are best and fastest solved by rotating the diagram to a orientation that we are best acquainted with. In this case the problem becomes very easy once we rotate the figure to orient the right angle at the bottom left. Have a look at the below diagram.

Also given in the prompt is that BCA is a right triangle.

Ans. D i.e. 70
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Re: In the figure, triangles ABC and ABD are right triangles. What is the [#permalink]
Bunuel wrote:

In the figure, triangles ABC and ABD are right triangles. What is the value of x ?

(A) 20
(B) 30
(C) 50
(D) 70
(E) 90

Solution:

We are given that triangle ABC is a right triangle. If the right angle of triangle ABC is at B (i.e., angle ABC = 90 degrees), then angle ABD of triangle ABD must be an obtuse angle since it’s greater than angle ABC. However, triangle ABD is also a right triangle, so it can’t have an obtuse angle. Therefore, the right angle of triangle ABC can’t be at B. In that case, it must be at C (i.e., angle ACB = 90 degrees), and therefore, angle ABC = 180 - 90 - 20 = 70 degrees. Since angle ABD is greater than 70 degrees, it could be the right angle of triangle ABD. If it’s not, then angle BDA is the right angle of triangle ABD. Notice that if the former is true, then x = 180 - 90 - 20 = 70 degrees, and if the latter is true, then x = 90 degrees. We will prove that only the former (angle ABD is a right angle) can be true. That is, it’s impossible for angle BDA to be a right angle.

Recall that we have established that angle ABD is greater than 70 degrees. Now, if angle BDA were a right angle, then even if angle ABD were just a little bit greater than 70 degrees, for example 71 degrees, then the sum of the measures of the 3 angles of triangle ABD would be 20 + 71 + 90 = 181 degrees, which is not possible since we all know the sum should be 180 degrees. Therefore, angle BDA can’t be a right angle. This leaves us with only angle BDA being the right angle and therefore, x = 70 (mentioned above).