Ninth Grade Mathematics: Constructing Angles: 60, 90, 45, 30, 75, 105 Degrees
A protractor allows approximate measurement of any angle, but in pure Euclidean geometry, angles of specific degree measures can be constructed with absolute theoretical exactness using only a compass and straightedge. In this chapter, you will master the construction of the master 60-degree angle, explore successive bisections to produce 30-degree and 15-degree angles, erect perpendicular 90-degree and 45-degree angles, and synthesize composite angles such as 75 degrees and 105 degrees.
The Master 60-Degree Angle: The Equilateral Triangle Principle
The starting point for all Euclidean angle construction is the 60-degree angle, which is derived directly from the construction of an equilateral triangle.
Construction Algorithm for 60-Degree Angle:
1. Draw a baseline ray with initial point A: Ray AB.
2. Place the compass point at A. With any convenient radius r, draw a large arc
cutting Ray AB at point P.
3. WITHOUT CHANGING THE COMPASS RADIUS, place the compass point at P.
4. Draw an arc intersecting the first arc at point Q.
5. Draw the ray originating at A and passing through Q: Ray AQ.
Q (Second Arc)
*
/ \
r / \ r
/ \
*-------*---------------------> B
A r P
Angle QAB = 60 degrees!
Proof of the 60-Degree Construction
Join PQ with a straight line segment. By construction: AP = r (first arc radius), AQ = r (first arc radius), and PQ = r (second arc radius from P). Therefore, Triangle APQ has AP = AQ = PQ = r, making it an equilateral triangle. In any equilateral triangle, all three interior angles are equal: 3 * Angle A = 180 degrees, so Angle QAB = 60 degrees.
Constructing 30-Degree and 15-Degree Angles
Once a 60-degree angle is established, successive angle bisections generate its submultiples:
60 degrees ---(Bisect)---> 30 degrees ---(Bisect)---> 15 degrees
To construct 30 degrees: construct the 60-degree angle QAB, and then apply the standard angle bisector construction to bisect Angle QAB into two 30-degree angles. Bisecting the 30-degree angle yields an exact 15-degree angle.
Constructing 90-Degree and 45-Degree Angles
A 90-degree angle is constructed by either bisecting a 180-degree straight angle or erecting a perpendicular through the midpoint of two 60-degree arc steps:
Method: 60 + 30 = 90 degrees
1. Draw an initial arc from center A cutting ray AB at P (0 degrees).
2. From P, swing an arc to mark Q (60 degrees).
3. From Q, swing the same radius to mark R (120 degrees).
4. From Q and R, draw intersecting arcs of equal radius above the circle to mark S.
5. Ray AS bisects the interval between 60 and 120 degrees:
60 + (120 - 60)/2 = 60 + 30 = 90 degrees!
S (90 deg)
*
/ | \
/ | \
R * / | \ * Q (60 deg)
(120) \ / | \ /
*-------+-------*-----------------> B
A P (0 deg)
Bisecting the 90-degree angle yields an exact 45-degree angle.
Constructing Composite Angles: 75, 105, 120, and 135 Degrees
By combining known angle benchmarks through addition and bisection, an entire family of angles that are multiples of 15 degrees can be constructed:
Target Angle Algebraic Decomposition Construction Strategy
---------------------------------------------------------------------------------
120 degrees 60 + 60 Two consecutive 60-deg arcs
75 degrees 60 + 15 (or (60 + 90) / 2) Bisect the angle between 60 and 90 deg
105 degrees 90 + 15 (or (90 + 120) / 2) Bisect the angle between 90 and 120 deg
135 degrees 90 + 45 (or (90 + 180) / 2) Bisect the angle between 90 and 180 deg
150 degrees 120 + 30 (or (120 + 180) / 2) Bisect the angle between 120 and 180 deg
Constructing 75 Degrees:
1. Construct Ray AB, the 60-degree ray, and the 90-degree ray.
2. The angle between the 60-degree ray and the 90-degree ray measures 30 degrees.
3. Construct the angle bisector of this 30-degree region: 30 / 2 = 15 degrees.
4. The resulting ray makes an angle of 60 + 15 = 75 degrees with baseline AB.
Chapter Exercises
Exercise 1: Describe step-by-step how to construct an exact 120-degree angle using only a compass and straightedge, including the geometric justification.
Exercise 2: Explain in detail how to construct a 105-degree angle by bisecting the angle between a 90-degree ray and a 120-degree ray.
Exercise 3: State the mathematical rule that determines which integer degree angles can be constructed using only a compass and straightedge, and explain why an angle of 20 degrees cannot be constructed.
Solutions and Explanations
Solution 1: To construct a 120-degree angle: draw a baseline ray AB originating at vertex A. With A as center and any radius r, draw a large circular arc cutting ray AB at point P (representing 0 degrees). Without changing the compass radius, place the compass point at P and swing an arc intersecting the original arc at point Q; ray AQ forms a 60-degree angle. Next, place the compass point at Q with the exact same radius r and swing an arc further along the main arc to mark point R. Draw ray AR from vertex A through R. Because arc PQ subtends 60 degrees and arc QR subtends an additional 60 degrees from the same radius, the total angle Angle RAB is 60 + 60 = 120 degrees.
Solution 2: To construct a 105-degree angle: begin by drawing a baseline ray AB. Construct a 90-degree perpendicular ray AD and a 120-degree ray AE using the standard compass arc protocol. The angle between the 90-degree ray and the 120-degree ray measures 120 - 90 = 30 degrees. Let the 90-degree ray intersect the main construction arc at point M, and let the 120-degree ray intersect the main arc at point N. With centers M and N and equal radius, draw two intersecting arcs in the interior of Angle DAE, meeting at point P. Draw ray AP from vertex A through P. Ray AP bisects the 30-degree angle into two 15-degree angles. The total angle between ray AP and baseline ray AB is 90 + 15 = 105 degrees.
Solution 3: Using a compass and straightedge, the smallest positive integer degree angle that can be constructed is 3 degrees (derived from the difference between an interior angle of a regular pentagon, 108 deg, and a 60-deg angle, or from 15 deg and other submultiples). Consequently, an integer angle can be constructed with compass and straightedge if and only if its degree measure is a multiple of 3 degrees (such as 3, 6, 9, 12, 15, ..., 60, 75, 90, 105 degrees). An angle of 20 degrees cannot be constructed because 20 is not a multiple of 3. Furthermore, constructing a 20-degree angle would be equivalent to trisecting a 60-degree angle (60 / 3 = 20), which Pierre Wantzel proved is algebraically impossible with compass and straightedge.