Question 1043931
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ABCD is a cyclic quadrilateral, angle DAB =80 and angle ACB =50 .
Prove that AB=AD.
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<pre>
Let me remind you that a <U>cyclic quadrilateral</U> is a quadrilateral inscribed into the circle.

So, ABSD is inscribed into a circle. 
Then the measure of the angle DAB plus the measure of the opposite angle BCD is 180°.


    (If a quadrilateral is inscribed into a circle, then the sum of the measures of its opposite angles is 180°.  

    See the lesson <A HREF=https://www.algebra.com/algebra/homework/Polygons/Quadrilateral-inscibed-in-a-circle.lesson>Quadrilateral inscribed in a circle</A> in this site).


So,  m<I>L</I>DAB + m<I>L</I>BCD = 180°.

But  m<I>L</I>BCD = m<I>L</I>ACB + m<I>L</I>ACD.

Therefore,  m<I>L</I>DAB + m<I>L</I>ACB + m<I>L</I>ACD = 180°.

But  m<I>L</I>DAB + m<I>L</I>ACB = 80° + 50° = 130°.

Therefore,  m<I>L</I>ACD = 180° - 130° = 50°.

Thus the inscribed angles  <I>L</I>ACB  and  <I>L</I>ACD  have equal measures.

Hence, they lean equal arcs: the length of the arc AB is the same as the length of the AD.

It implies that the chords AB and AD have equal lengths.   QED.
</pre>

Solved.


See the lessons associated with it:


&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Polygons/Quadrilateral-inscibed-in-a-circle.lesson>Quadrilateral inscribed in a circle</A>


and 


&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/A-circle-its-chords-tangent-and-secant-lines-the-major-definitions.lesson>A circle, its chords, tangent and secant lines - the major definitions</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/The-longer-is-the-chord-the-larger-its-central-angle-is.lesson>The longer is the chord the larger its central angle is</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/An-inscribed-angle.lesson>An inscribed angle in a circle</A> 

in this site.