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4.1 Lines, Angles, and Triangles - Triangle sum theorem
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  • From left to right, horizontal line segment upper P upper V includes the following points:
    • Upper P
    • Upper Q
    • Upper R
    • Upper S
    • Upper T
    • Upper V
  • Points upper X and upper U are each below line segment upper P upper V.
  • From left to right, the following 2 overlapping triangles are formed between points on line segment upper P upper V and points upper X and upper U:
    • Triangle upper Q upper X upper S
    • Triangle upper R upper U upper T
  • Line segments upper R upper U and upper S upper X intersect at point upper W.
  • A note indies the figure is not drawn to scale.

In the figure shown, points Q , R , S , and T lie on line segment P V , and line segment R U intersects line segment S X at point W . The measure of SQX is 48°, the measure of SXQ is 86°, the measure of SWU is 85°, and the measure of VTU is 162°. What is the measure, in degrees, of TUR

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From left to right, horizontal line segment upper P upper V includes the following points: Upper

Hard-difficulty · SAT Math · Lines, Angles, and Triangles — Triangle sum theorem. Read the question above, select your answer, and check the full explanation below to understand exactly why the correct choice works.

Answer explanation

The correct answer is 123 . The triangle angle sum theorem states that the sum of the measures of the interior angles of a triangle is 180  degrees. It's given that the measure of SQX is 48° and the measure of SXQ is 86°. Since points S , Q , and X form a triangle, it follows from the triangle angle sum theorem that the measure, in degrees, of QSX is 180-48-86, or 46 . It's also given that the measure of SWU is 85°. Since SWU and SWR are supplementary angles, the sum of their measures is 180 degrees. It follows that the measure, in degrees, of SWR is 180-85, or 95 . Since points R , S , and W form a triangle, and RSW is the same angle as QSX, it follows from the triangle angle sum theorem that the measure, in degrees, of WRS is 180-46-95, or 39 . It's given that the measure of VTU is 162°. Since VTU and STU are supplementary angles, the sum of their measures is 180 degrees. It follows that the measure, in degrees, of STU is 180-162, or 18. Since points R , T , and U form a triangle, and URT is the same angle as WRS, it follows from the triangle angle sum theorem that the measure, in degrees, of TUR is 180-39-18, or 123.