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Subject Test Math 1

Question 3 of 32

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SAT Subject Test

The figure presents a polygon containing 5 line segments. Segment A D is horizontal with point D to the right of point A. Segment A B extends upward and to the left from point A to point B. Segment B D is drawn. Segment A C extends upward and to the right from point A, intersects B D, and continues to a point C, which lies above and to the left of point D. Segment C D is drawn. The angle above A C and above B D is labeled x degrees. Angle A B D is labeled y degrees, and angle A C D is labeled z degrees.

In the figure above stack A B with bar on top and stack C D with bar on top are parallel. What is x in terms of y and Error converting from MathML to accessible text.?

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y plus z

Correct Answer: 
Yes

2 y plus z

Correct Answer: 
No

2 y minus z

Correct Answer: 
No

180 minus y minus z

Correct Answer: 
No

180 plus y minus z

Correct Answer: 
No

The correct answer is A.
The figure presents a polygon containing 5 line segments. Segment A D is horizontal with point D to the right of point A. Segment A B extends upward and to the left from point A to point B. Segment B D is drawn. Segment A C extends upward and to the right from point A, intersects B D at a point labeled E, and continues to point C, which lies above and to the left of point D. Segment C D is drawn. Angle B E C is labeled x degrees, angle A B D is labeled y degrees, and angle A C D is labeled z degrees.

Let E be the point of intersection of lines stack A C with left right arrow on top and stack B D with left right arrow on top invisible times.

One way to determine x in terms of y and z is to find the measure of each of the angles of ▵ E D C in terms of x, y and z and then apply the triangle sum theorem. Since angle C E D is supplementary to angle B E C and the measure of angle B E C is given to be x degree, the measure of angle C E D is left parenthesis 180 minus x right parenthesis degree. Since line stack A C with left right arrow on top is a transversal to the parallel lines stack A B with left right arrow on top and stack C D with left right arrow on top invisible times, the alternate interior angles angle B D C and angle D B A are of equal measure. Thus the measure of angle B D C is y degree invisible times, which is also the measure of angle E D C. The measure of angle D C E is given to be z degree invisible times. Therefore, the sum of the angle measures of ▵ E D C, in degrees, is left parenthesis 180 minus x right parenthesis plus y plus z. The triangle sum theorem applied to ▵ E D C gives the equation left parenthesis 180 minus x right parenthesis plus y plus z equals 180, which can be solved for x to arrive at x equals y plus z.

Alternatively, one can apply the interior angle sum theorem to pentagon A B E C D. Since line stack A D with left right arrow on top is a transversal to the parallel lines stack A B with left right arrow on top and stack C D with left right arrow on top, it follows that angle B A D and angle A D C are supplementary; that is, the sum of the measures of these two angles is 180 degree invisible times. The measure of angle B E C, interior to polygon A B E C D, is left parenthesis 360 minus x right parenthesis degree. The measure of angle E B A is given to  be y degree invisible times, and the measure of angle D C E is given to be z degree. Therefore, the sum of the measures of the interior angles of pentagon A B E C D is 180 plus left parenthesis 360 minus x right parenthesis plus y plus z. The interior angle sum theorem applied to pentagon A B E C D gives the equation 180 plus left parenthesis 360 minus x right parenthesis plus y plus z equals 540, which can be solved for x to arrive at x equals y plus z.

Question Difficulty: 
easy