Editing Inscribed angle (section)

====Inscribed angles where one chord is a diameter====
[[File:InscribedAngle 1ChordDiam.svg|thumb|class=skin-invert-image|Case: One chord is a diameter]]
Let {{mvar|O}} be the center of a circle, as in the diagram at right.  Choose two points on the circle, and call them {{mvar|V}} and {{mvar|A}}.  Designate point {{mvar|B}} to be [[diametrically opposite]] point {{mvar|V}}.  Draw chord {{Mvar|VB}}, a diameter containing point {{Mvar|O}}.  Draw chord {{Mvar|VA}}.  Angle {{Math|∠''BVA''}} is an inscribed angle that intercepts arc {{Mvar|{{overarc|AB}}}}; denote it as {{Mvar|ψ}}.  Draw line {{mvar|OA}}.  Angle {{math|∠''BOA''}} is a [[central angle]] that also intercepts arc {{Mvar|{{overarc|AB}}}}; denote it as {{mvar|θ}}.

Lines {{mvar|OV}} and {{mvar|OA}} are both [[radius|radii]] of the circle, so they have equal lengths.  Therefore, triangle {{math|△''VOA''}} is [[isosceles]], so angle {{math|∠''BVA''}} and angle {{math|∠''VAO''}} are equal.

Angles {{math|∠''BOA''}} and {{math|∠''AOV''}} are [[supplementary angle|supplementary]], summing to a [[straight angle]] (180°), so angle {{math|∠''AOV''}} measures {{math|180° &minus; ''θ''}}.

The three angles of triangle {{math|△''VOA''}} [[sum of angles of a triangle|must sum to {{math|180°}}]]:

<math display=block>(180^\circ - \theta) + \psi + \psi = 180^\circ.</math>

Adding <math>\theta - 180^\circ</math> to both sides yields

<math display=block>2\psi = \theta.</math>