randRange( 0, 360 ) randRange( 10, 80 ) * 2 ( START + 180 ) % 360 4 CENTRAL "blue" "orange"

If the GIVEN_LABEL angle measures GIVEN degrees, what does the ASKED_LABEL 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 + CENTRAL ); graph.circle.drawCentralAngle( START, START + CENTRAL ); style({ stroke: ORANGE, fill: ORANGE }); graph.circle.drawInscribedAngle( SUBTENDED_POINT, START, START + CENTRAL ); graph.circle.drawMovablePoint( SUBTENDED_POINT, START + CENTRAL, START );
CENTRAL / 2 degrees

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

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

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

style({ stroke: PINK }, function() { graph.circle.drawInscribedArc( START + CENTRAL, SUBTENDED_POINT, START + CENTRAL + 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 + CENTRAL ); graph.circle.drawRadius( SUBTENDED_POINT ); });

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 - CENTRAL^{\circ}}

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

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

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

CENTRAL / 2 "orange" "blue"
CENTRAL degrees

The pink sides of the triangle are radii, so they must be equal. This means the triangle is isosceles and that the base angles, or the pink and orange angles, are equal.

style({ stroke: PINK }, function() { graph.circle.drawRadius( START + CENTRAL ); graph.circle.drawRadius( SUBTENDED_POINT ); }); style({ stroke: PINK }, function() { graph.circle.drawInscribedArc( START + CENTRAL, SUBTENDED_POINT, START + CENTRAL + 180 ); });

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

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

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

The green and blue angles are supplementary.

\color{GREEN}{\text{green angle}} + \color{BLUE}{\text{blue angle}} = 180^{\circ}

\color{BLUE}{\text{blue angle}} = 180^{\circ} - \color{GREEN}{\text{green angle}}

\color{BLUE}{\text{blue angle}} = 180^{\circ} - \color{GREEN}{\text{180 - CENTRAL}^{\circ}}

\color{BLUE}{\text{blue angle}} = \color{BLUE}{CENTRAL^{\circ}}