randRange( 0, 360 ) randRange( 10, 80 ) * 2 ( START + 180 ) % 360 4

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

This is a special case where the blue and orange angles' sides share the same line. The blue angle is called a central angle, and the orange angle is called an inscribed 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

The green and blue angles are supplementary. Because the blue angle is MEASURE degrees, the green angle must be 180 - MEASURE degrees.

style({ stroke: GREEN }, function() { graph.circle.drawCentralArc( START + MEASURE, INSCRIBED ); })

We know that the angles in a triangle sum to 180 degrees.

style({ stroke: PINK }, function() { graph.circle.drawInscribedArc( START + MEASURE, INSCRIBED, START + MEASURE + 180 ); });

\color{GREEN}{\text{green angle}} + \color{PINK}{\text{pink angle}} + \color{ORANGE}{\text{orange angle}} = 180^{\circ}

The pink sides of the triangle are radii, so they must be equal.

style({ stroke: PINK }, function() { graph.circle.drawRadius( START + MEASURE ); graph.circle.drawRadius( INSCRIBED ); })

This means that the triangle is isosceles and that the base angles, or the pink and orange angles, are equal.

\color{GREEN}{\text{green angle}} + 2 \cdot \color{ORANGE}{\text{orange angle}} = 180^{\circ}

2 \cdot \color{ORANGE}{\text{orange angle}} = 180^{\circ} - \color{GREEN}{180 - MEASURE^{\circ}}

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

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

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