Matemática Elementar

Problema 1

An angle is measured in degrees ( $^\circ$ ). A full circle is $360^\circ$ . A right angle is exactly $90^\circ$ .

Classify each angle as acute, right, or obtuse.

a) An angle that measures $45^\circ$ : $\underline{\qquad\qquad}$

b) An angle that measures $90^\circ$ : $\underline{\qquad\qquad}$

c) An angle that measures $120^\circ$ : $\underline{\qquad\qquad}$

d) An angle that measures $175^\circ$ : $\underline{\qquad\qquad}$

(Acute: less than $90^\circ$ ; Right: exactly $90^\circ$ ; Obtuse: greater than $90^\circ$ but less than $180^\circ$ )

Problemas

1. An angle is measured in degrees ( $^\circ$ ). A full circle is $360^\circ$ . A right angle is exactly $90^\circ$ .

Classify each angle as acute, right, or obtuse.

a) An angle that measures $45^\circ$ : $\underline{\qquad\qquad}$

b) An angle that measures $90^\circ$ : $\underline{\qquad\qquad}$

c) An angle that measures $120^\circ$ : $\underline{\qquad\qquad}$

d) An angle that measures $175^\circ$ : $\underline{\qquad\qquad}$

(Acute: less than $90^\circ$ ; Right: exactly $90^\circ$ ; Obtuse: greater than $90^\circ$ but less than $180^\circ$ )

2. Use a protractor to measure the angle described. Then write whether it is acute, right, or obtuse.

An angle opens from one ray pointing right to another ray pointing up-and-to-the-right. The protractor reads $55^\circ$ .

Measure: $\underline{\quad}^\circ$

Type: $\underline{\qquad\qquad}$

A clock shows $3:00$ . The angle between the hour hand and the minute hand is $\underline{\quad}^\circ$ .

Type: $\underline{\qquad\qquad}$

3. An angle measures $70^\circ$ . Another angle is placed next to it without overlapping.

If the two angles together form a right angle ( $90^\circ$ ), what is the measure of the second angle?

$70^\circ + \underline{\quad}^\circ = 90^\circ$

The second angle measures $\underline{\quad}^\circ$ .

Is the second angle acute or obtuse? $\underline{\qquad\qquad}$

4. A line of symmetry divides a shape into two mirror-image halves.

How many lines of symmetry does each shape have?

a) A square: $\underline{\quad}$ lines of symmetry

b) A rectangle (not a square): $\underline{\quad}$ lines of symmetry

c) An equilateral triangle (all sides equal): $\underline{\quad}$ lines of symmetry

d) A circle: $\underline{\quad}$ lines of symmetry

5. Does each letter of the alphabet have a line of symmetry? Write Yes or No, and draw or describe where the line would go.

A — Yes or No? $\underline{\quad}$ Line goes: $\underline{\qquad\qquad}$

B — Yes or No? $\underline{\quad}$ Line goes: $\underline{\qquad\qquad}$

X — Yes or No? $\underline{\quad}$ How many lines? $\underline{\quad}$

Z — Yes or No? $\underline{\quad}$

6. The hands of a clock form angles at different times.

a) At $12:00$ , both hands point straight up and overlap. The angle between them is $\underline{\quad}^\circ$ .

b) At $6:00$ , the hands point in opposite directions. The angle between them is $\underline{\quad}^\circ$ .

c) The minute hand moves $360^\circ$ in $60$ minutes. How many degrees does it move each minute?

$360^\circ \div 60 = \underline{\quad}^\circ \text{ per minute}$

d) How many degrees does the minute hand move in $15$ minutes?

$\underline{\quad} \times 15 = \underline{\quad}^\circ$

Gabarito — Angles and Lines of Symmetry

1. a) $45^\circ < 90^\circ$ → acute

b) $90^\circ = 90^\circ$ → right

c) $90^\circ < 120^\circ < 180^\circ$ → obtuse

d) $90^\circ < 175^\circ < 180^\circ$ → obtuse

2. Measure: $55^\circ$ → Type: acute (less than $90^\circ$ )

At $3:00$ , the minute hand points to $12$ and the hour hand points to $3$ . That is one-quarter of the clock face.

$\frac{1}{4} \times 360^\circ = 90^\circ$

The angle is $90^\circ$ → Type: right

3. $70^\circ + x = 90^\circ$ $x = 90^\circ - 70^\circ = \mathbf{20^\circ}$

The second angle measures $\mathbf{20^\circ}$ .

Since $20^\circ < 90^\circ$ , it is acute.

4. a) Square: 4 lines of symmetry (horizontal, vertical, and 2 diagonals)

b) Rectangle (not a square): 2 lines of symmetry (horizontal and vertical only; diagonals are not lines of symmetry for a non-square rectangle)

c) Equilateral triangle: 3 lines of symmetry (one through each vertex to the opposite midpoint)

d) Circle: infinitely many lines of symmetry (any diameter is a line of symmetry)

5. A — Yes. The line goes vertically through the center of the letter (left and right halves are mirror images).

B — Yes. The line goes horizontally through the middle (top and bottom halves are mirror images).

X — Yes. 2 lines of symmetry (vertical and horizontal).

Z — No. A Z has no line of symmetry (rotating $180^\circ$ maps it to itself, but no reflection does).

6. a) Both hands overlap at $12:00$ : the angle between them is $\mathbf{0^\circ}$ .

b) At $6:00$ the hands point in exactly opposite directions:

$\frac{1}{2} \times 360^\circ = \mathbf{180^\circ}$

c) $360^\circ \div 60 = \mathbf{6^\circ}$ per minute

d) $6^\circ \times 15 = \mathbf{90^\circ}$ (the minute hand moves $90^\circ$ in $15$ minutes — one quarter of the clock face)