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# now you're working with circles

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I've got a few geometry homework questions about arcs, and I'm not too sure about how to solve them.

7: In rectangle PQRS, PS=3 and PQ=4. Let M be the midpoint of line PQ , and let X be the point such that MS=MX and angle MSX=77 degrees, as shown below. Find angle XRS, in degrees. Jan 8, 2020

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Angle XRS works out to 77 - 60 = 17 degrees.

Jan 8, 2020
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I disagree, XRS is 103 (180-77)-SXR (the last angle of triangle SXR that we haven't mentioned yet). Where did you get the numbers 77 and 60?

Guest Jan 8, 2020
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With M as a center,   construct a circle  with radius   MS

This circle will pass through  S , X  and R

Since MS  = MX, then angles  MSX   and  MXS  are equal

So  triangle   MSX  is isosceles  .....so angle  SMX  =  [ 180 - 2(77) ]  =  26°

And this is a central angle in the  circle that intercepts minor  arc   SX

And angle XRS  is an inscribed angle in the circle that intercepts this same arc

So....the measure of angle XRS =  (1/2)26°  = 13°   Jan 8, 2020
edited by CPhill  Jan 8, 2020