Free Math Worksheets

- Picture a quarter-turn or square corner and compare each opening visually. - The direction the angle faces does not change whether it is a right angle.

1. A right angle is a quarter-turn. Compare the opening between the two rays in each diagram. 2. Diagrams a) and c) show quarter-turns. The other openings are either smaller or larger than a right angle.

a) and c)

- A full circle measures \(360^\circ\). - Equal slices have equal central angles. - Divide \(360^\circ\) by the number of slices.

1. A full circle measures \(360^\circ\). 2. For \(4\) slices, \(360^\circ \div 4=90^\circ\). 3. For \(6\) slices, \(360^\circ \div 6=60^\circ\). 4. For \(9\) slices, \(360^\circ \div 9=40^\circ\). 5. For \(15\) slices, \(360^\circ \div 15=24^\circ\).

a) \(90^\circ\) b) \(60^\circ\) c) \(40^\circ\) d) \(24^\circ\)

- Divide the full circle into \(12\) equal sections. - Find the angle for one step between neighboring numbers. - Count the number of steps for each part.

1. The clock face divides \(360^\circ\) into \(12\) equal sections, so each section measures \(360^\circ \div 12=30^\circ\). 2. From \(12\) to \(1\) is one section, so the angle is \(30^\circ\). 3. From \(12\) to \(4\) is four sections, so the angle is \(4 \times 30^\circ=120^\circ\). 4. From \(12\) to \(6\) is six sections, so the angle is \(6 \times 30^\circ=180^\circ\). This is a straight angle.

a) \(30^\circ\) b) \(120^\circ\) c) \(180^\circ\), a straight angle

- Each hour section measures \(30^\circ\). - Count the smaller number of sections between the hands. - Compare each result with \(90^\circ\).

1. The \(12\) hour marks divide \(360^\circ\) into equal sections of \(360^\circ \div 12=30^\circ\). 2. At \(2{:}00\), the hands are two sections apart: \(2 \times 30^\circ=60^\circ\), an acute angle. 3. At \(5{:}00\), the hands are five sections apart: \(5 \times 30^\circ=150^\circ\), an obtuse angle. 4. At \(9{:}00\), the smaller separation is three sections: \(3 \times 30^\circ=90^\circ\), a right angle.

a) \(60^\circ\), acute b) \(150^\circ\), obtuse c) \(90^\circ\), right

- Find the number of elapsed hours. - Divide \(360^\circ\) by \(12\) to find the turn in one hour. - Multiply by the elapsed time.

1. The elapsed time is \(5\) hours. 2. The hour hand turns \(360^\circ \div 12=30^\circ\) each hour. 3. In \(5\) hours, it turns \(5 \times 30^\circ=150^\circ\).

\(150^\circ\)

- Check the turns at \(B\), \(C\), and \(D\) one at a time. - A right-angle turn is a quarter-turn; look for the turn whose opening is different.

1. Compare the three turns visually with a quarter-turn. 2. The turns at \(B\) and \(D\) are right-angle quarter-turns. The turn at \(C\) is smaller than a right angle. 3. Therefore, the turn at \(C\) is not a right angle.

The turn at \(C\) is not a right angle.

- There are three pairs of segments to compare. - Look for the pair whose opening matches a quarter-turn, even though the angle is rotated.

1. Compare the three pairs of segments that meet at \(B\). 2. The opening between \(\overline{BA}\) and \(\overline{BC}\) is a quarter-turn. The other two openings are not right angles. 3. Therefore, \(\overline{BA}\) and \(\overline{BC}\) form the right angle.

Segments \(\overline{BA}\) and \(\overline{BC}\) form the right angle.

- Check every turn in each route visually. - A route is allowed only if every change in direction is a quarter-turn.

1. Check every change in direction and compare it with a quarter-turn. 2. In a), all three turns are right-angle turns. 3. In b), the turns are not right angles, so the route is not allowed. 4. In c), both turns are right-angle turns. Therefore, the allowed routes are a) and c).

The robot can follow routes a) and c) from start to finish.

- How many degrees are in a straight angle? - How many degrees are in a full turn? - To find a fraction of a quantity, multiply. - To write an angle as a fraction of a full turn, place the angle measure over \(360\).

1. A straight angle measures \(180^\circ\). 2. For part a, \(\frac{4}{9} \times 180^\circ = 80^\circ\). 3. A full turn measures \(360^\circ\). 4. For part b, the fraction is \(\frac{135}{360}\). 5. Simplify: \(\frac{135}{360} = \frac{3}{8}\).

a) \(80^\circ\) b) \(\frac{3}{8}\) of a full turn

- Think of the wheel as a circle divided into equal sections. - Divide \(360^\circ\) by the number of cars in part a. - In part b, find how many \(45^\circ\) sections fit in \(360^\circ\).

1. A full circle measures \(360^\circ\). 2. For \(12\) evenly spaced cars, the angle is \(360^\circ \div 12=30^\circ\). 3. If each angle is \(45^\circ\), the number of cars is \(360^\circ \div 45^\circ=8\).

a) \(30^\circ\) b) \(8\) cars

- Find how many degrees the minute hand turns in one minute. - A full turn is \(360^\circ\). - Each step from one clock number to the next is the same angle.

1. The minute hand turns \(360^\circ \div 60=6^\circ\) each minute. 2. In \(10\) minutes, it turns \(10 \times 6^\circ=60^\circ\). 3. A turn of \(150^\circ\) takes \(150^\circ \div 6^\circ=25\) minutes. 4. Moving from \(2\) to \(6\) covers four clock sections. Each section is \(30^\circ\), so the angle is \(4 \times 30^\circ=120^\circ\).

a) \(60^\circ\) b) \(25\) minutes c) \(120^\circ\)