randRange( 0, 360 ) randRange( 10, 80 ) * 2 randFromArray( [ randRange( START + MEASURE + 5, START + 180 - 5 ), randRange( START + MEASURE + 180 + 5, START + 360 - 5 ) ] ) % 360 4

If the blue angle measures MEASURE degrees, what does the orange angle measure?

This is a special case where the center of the circle is outside the inscribed orange angle. The blue angle is called a central angle.

init({ range: [ [ -RADIUS - 1, RADIUS + 1 ], [ -RADIUS - 1, RADIUS + 1 ] ] }); addMouseLayer(); graph.circle = new Circle( RADIUS ); style({ stroke: BLUE, fill: BLUE }); graph.circle.drawCenter(); graph.circle.drawPoint( START ); graph.circle.drawPoint( START + MEASURE ); graph.circle.drawCentralAngle( START, START + MEASURE ); style({ stroke: ORANGE, fill: ORANGE }); graph.circle.drawInscribedAngle( INSCRIBED, START, START + MEASURE ); graph.circle.drawMovablePoint( INSCRIBED, START + MEASURE, START );
MEASURE / 2 degrees

What do we know about the angles formed by the dashed diameter shown above?

style({stroke: BLUE, "stroke-dasharray": "-"}, function() { graph.circle.drawChord( INSCRIBED, INSCRIBED + 180 ); });

From the previous inscribed angles exercises, we know the following about the green and pink angles.

\color{GREEN}{\text{green angle}} = \dfrac{1}{2} \cdot \color{PINK}{\text{pink angle}}

style({stroke: BLUE, fill: BLUE}, function() { graph.circle.drawPoint( INSCRIBED + 180 ); }); style({stroke: PINK}); var start = min( ( INSCRIBED + 180 ) % 360, START ), end = max( ( INSCRIBED + 180 ) % 360, START ); graph.central = graph.circle.drawCentralAngle( start, end, 0.7 ); style({stroke: GREEN}); graph.inscribed = graph.circle.drawInscribedAngle( INSCRIBED, start, end, 0.7 );

We can see another pair of these special case inscribed and central angles, with the same relationship between green and pink angles.

graph.central.arc.animate({opacity: 0.4}); graph.central.radii[0].remove(); graph.central.radii[1].remove(); graph.inscribed.arc.animate({opacity: 0.4}); graph.inscribed.chords[0].remove(); graph.inscribed.chords[1].remove(); var start = min( ( INSCRIBED + 180 ) % 360, ( START + MEASURE ) % 360 ), end = max( ( INSCRIBED + 180 ) % 360, ( START + MEASURE ) % 360 ); style({stroke: PINK}); graph.central = graph.circle.drawCentralAngle( start, end, 0.9 ); style({stroke: GREEN}); graph.inscribed = graph.circle.drawInscribedAngle( INSCRIBED, start, end, 0.9 );

Looking at the picture, we can see the following is true:

\color{GREEN}{\text{small green angle}} + \color{ORANGE}{\text{orange angle}} = \color{GREEN}{\text{big green angle}}

graph.central.arc.animate({opacity: 0.4}); graph.central.radii[0].remove(); graph.central.radii[1].remove(); graph.inscribed.arc.animate({opacity: 0.4}); graph.inscribed.chords[0].remove(); graph.inscribed.chords[1].remove();

Substituting what we know about green and pink angles, we get the following:

\dfrac{1}{2} \cdot \color{PINK}{\text{small pink angle}} + \color{ORANGE}{\text{orange angle}} = \dfrac{1}{2} \cdot \color{PINK}{\text{big pink angle}}

\color{ORANGE}{\text{orange angle}} = \dfrac{1}{2}( \color{PINK}{\text{big pink angle}} - \color{PINK}{\text{small pink angle}})

We can see from the picture that the following is also true:

\color{PINK}{\text{small pink angle}} + \color{BLUE}{\text{blue angle}} = \color{PINK}{\text{big pink angle}}

\color{BLUE}{\text{blue angle}} = \color{PINK}{\text{big pink angle}} - \color{PINK}{\text{small pink angle}}

Combining what we know about blue and orange angles:

\color{ORANGE}{\text{orange angle}} = \dfrac{1}{2} \cdot \color{BLUE}{\text{blue angle}}

\color{ORANGE}{\text{orange angle}} = \dfrac{1}{2} \cdot \color{BLUE}{MEASURE^{\circ}}

\color{ORANGE}{\text{orange angle}} = \color{BLUE}{MEASURE / 2^{\circ}}