Measure and draw angles using degrees

Learning goal: Explain degree measure using a circle, measure angles with a protractor and draw angles of specified whole-number degree measures. An angle is formed by two rays with a common endpoint, called the vertex. Each ray extends indefinitely in one direction. A drawing shows only part of each ray, but the opening between their directions determines the angle.

Imagine a circle centered at the vertex. The two rays meet the circle at two points. Turning from one ray to the other sweeps a fraction of the circular arc. One full turn is divided into 360 equal one-degree turns. An angle sweeping one of those equal turns measures one degree, written 1°. An angle sweeping sixty such turns measures 60°.

Degree measure counts equal turns rather than measuring the length of a drawn side. A full turn is 360°, a half turn is 180°, and a quarter turn is 90°. A 90° angle is a right angle. An angle smaller than 90° and greater than zero is acute; an angle between 90° and 180° is obtuse. These benchmarks help estimate before reading a tool.

A larger circle produces a longer arc for the same fraction of a turn, but the degree measure stays the same. Extending the drawn rays also leaves their directions and opening unchanged. Two angles with differently drawn ray lengths can both measure sixty degrees. Do not judge angle size by how much ink appears in the drawing.

A semicircular protractor marks angles from zero to 180 degrees. Place its center mark exactly at the angle's vertex. Align its straight zero baseline with one ray. Read where the other ray crosses the numbered arc. Use the scale that begins at zero on the aligned starting ray; many protractors have two scales running in opposite directions.

For an angle opening sixty degrees above a ray pointing right, use the scale that reads zero on the right. The other scale may show 120 at the same crossing. The opening is visibly less than a right angle, so the acute estimate supports sixty rather than 120. Estimation helps detect a wrong scale but does not replace careful measurement.

If a drawn ray is too short to reach the protractor scale, extend its straight direction with a ruler. Do not shift the protractor center away from the vertex to make the ray reach. Rotating the whole page may make alignment easier without changing the angle. Read the whole-number degree mark carefully and record the degree symbol.

To draw a specified angle, draw a starting ray and mark its endpoint as the vertex. Center the protractor there, align zero with the ray and mark the required degree position. Remove the protractor and draw a second ray from the vertex through that mark. For a 125° angle, check that the opening is obtuse and less than a straight 180° angle. A mark at the desired number without a second ray is not a completed angle.

Worked measurement: A ray begins at the right-hand zero baseline and the other crosses the protractor at thirty degrees on that scale. The measure is 30°, an acute angle. A reading of 150° would come from the opposite scale and conflict with the visibly small opening.

Worked construction: Draw a 90° angle. Mark the vertex, draw a ray to the right, align the protractor center and baseline, mark ninety degrees, then draw the second ray through the mark. The rays are perpendicular. Recheck the finished drawing with the protractor rather than assuming the mark remained aligned.

Guided practice: Use a physical protractor to measure acute, right and obtuse angles in several rotations. First estimate the type, then place the tool and record the measure. Partners should observe vertex alignment, zero-scale selection and the point where the second ray crosses the scale.

Independent evidence: Measure angles A, B and C in the lower part of the illustration with a real protractor before checking the answer to question 7, then draw 40°, 90° and 125° angles with a protractor and ruler. Print the page or hold the protractor against the screen; zooming changes the drawing's size but not its angle measures. Explain how a degree relates to a full circle and why longer rays do not make a greater angle. Written answers alone cannot establish accurate tool use; keep the finished drawings and the observer's notes.