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How can we draw a 60 degree angle without any protacter?

This page shows how to construct (draw) a 60 degree angle with compass and straightedge or ruler. This construction works by creating an equilateral triangle. Recall that an equilateral triangle has all three interior angles 60°. We use one of those angles to get the desired 60 degree result. See the proof below for more details.

Step-by-step Instructions  
After doing this  
1. Draw a line segment which will become one side of the angle. (Skip this step if you are given this line.) The exact length is not important. Label it PQ. P will be the angle's vertex. Geometry construction with compass and straightedge or ruler or ruler
2. Set the compass on P, and set its width to any convenient setting. Geometry construction with compass and straightedge or ruler or ruler
3. Draw an arc across PQ and up over above the point P. Geometry construction with compass and straightedge or ruler or ruler
4. Without changing the compass width, move the compass to the point where the arc crosses PQ, and make an arc that crosses the first one. Geometry construction with compass and straightedge or ruler or ruler
5. Draw a line from P, through the intersection of the two arcs. Geometry construction with compass and straightedge or ruler or ruler
6. Done. The angle QPR has a measure of 60° Geometry construction with compass and straightedge or ruler or ruler

Proof

This construction works by creating an equilateral triangle. Recall that an equilateral triangle has all three interior angles 60°. The image below is the final drawing above with the red items added.

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View Full Answer

ok this is very easy, so pay close attention.

first establish the facts:
the hour hand moves through 360 degrees every 12 hours, i.e. 30 degrees every hour or 1/2 a degree every minute
the minute hand moves through 360 degrees every hour, i.e. 6 degrees every minute.

Now you're set!

Initial condition: at 3 pm, the hour hand is at 90 degrees (assuming 12 to be zero reference) and the minute hand at 0.
in 24 minutes (using the above facts), the hour hand moves 0.5 x 24 = 12 degrees and the minute hand moves 6 x 24 = 144 degrees. So the new positions are:

hour hand: 90 + 12 = 102 degrees
minute hand: 0 + 144 = 144 degrees

therefore the difference between the clock hands can be
1) 144 - 102 = 42 degrees, or
2) 360 - (144 - 102) = 360 - 42 = 318 degrees

depending on which way you look at it.
Go through the solution carefully and try to understand it. It really is very logical, and no need for a protractor whatsoever!

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