Question 830235
<pre>
{{{drawing(520/3,200,-1.3,1.3,-2.3,.7,
locate(-.05,-2,T), locate(-1.15,.1,A),locate(1.01,.1,H),locate(-.05,.65,M),
line(-1,0,0,sqrt(2)-1),line(0,sqrt(2)-1,1,0),line(0,-2.005689708,-1,0), line(0,-2.005689708,1,0),locate(-.2,.31,"135°"), red(arc(0,sqrt(2)-1,.8,-.8,202,338)
locate(-.24,-1.5,"53°"), green(line(-1,0,1,0)),
arc(0,-2.005689708,1.2,-1.2,63,117)arc(0,-2.005689708,1.4,-1.4,63,117))

 )}}}

given: kite MATH, m&#8736;M=135°, m&#8736;T=53°
to find: m&#8736;MAT  (same as &#8736;M before we drew the green diagonal below) 

m(MA) = m(MH)
&#916;MAH is isosceles
m&#8736;MAH = m&#8736;MHA
m&#8736;M+m&#8736;MAH+m&#8736;MHA = 180°
m&#8736;M = 135°
135°+m&#8736;MAH+m&#8736;MHA = 180°
m&#8736;MAH+m&#8736;MHA = 180°-135°
m&#8736;MAH+m&#8736;MAH = 45°
2m&#8736;MAH = 45°
 m&#8736;MHA = 22.5°

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m(TA) = m(TH)
&#916;TA is isosceles
m&#8736;TAH = m&#8736;THA
m&#8736;T+m&#8736;TAH+m&#8736;THA = 180°
m&#8736;T = 53°
53°+m&#8736;TAH+m&#8736;THA = 180°
m&#8736;TAH+m&#8736;THA = 180°-53°
m&#8736;TAH+m&#8736;TAH = 127°
2m&#8736;TAH = 127°
 m&#8736;TAH = 63.5°
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&#8736;MAT = m&#8736;MAH+m&#8736;TAH = 22.5°+63.5° = 86°

Edwin</pre>