Free Math Worksheets

- Think about which sides would never meet if they were extended. - Look for sides that are opposite each other.

A rectangle has four sides. Each pair of opposite sides is parallel. The top and bottom sides form one pair, and the left and right sides form the other pair. Therefore, the rectangle has two pairs of parallel sides.

The frame has two pairs of parallel sides. The top and bottom sides form one pair, and the left and right sides form the other pair.

- Look at the sides that are opposite each other. - Imagine extending those sides. Would they ever meet?

1. A square has two pairs of parallel sides, and a rectangle also has two pairs of parallel sides. 2. In both shapes, each pair of opposite sides is parallel.

a) Each shape has \(2\) pairs of parallel sides. b) The opposite sides are parallel.

- Check the vertices one at a time. - Look for the vertex where the two sides form a square corner.

1. Check the interior angle at each vertex. 2. Only at \(A\) do the two sides form a square corner, so the right angle is at \(A\).

The right angle is at \(A\).

- Count the vertices of the figure. - Think about whether any vertex is different from the others.

1. A rectangle has four vertices. 2. At each vertex, the two sides meet at a right angle. Therefore, the rectangle has \(4\) right angles.

The rectangle has \(4\) right angles.

- First identify angles that look like square corners. - Compare the other angles with a right angle.

1. Angle \(\alpha\) at \(A\) is a right angle. 2. Angle \(\beta\) at \(B\) is greater than \(90^\circ\), so it is obtuse. 3. Angle \(\gamma\) at \(C\) is less than \(90^\circ\), so it is acute. 4. Angle \(\delta\) at \(D\) is a right angle.

\(\alpha\): right \(\beta\): obtuse \(\gamma\): acute \(\delta\): right

- Do not let the rotation of the second figure mislead you. - Compare the shape of the angles, not their direction on the screen.

1. Both figures are squares; the second square is simply rotated. 2. Each of the four vertices is a right angle. Therefore, a) has \(4\) right angles, and b) also has \(4\) right angles.

a) has \(4\) right angles. b) has \(4\) right angles.

- Count the right angles in each figure separately. - Pay attention to the word “exactly.”

1. Figure a) has right angles at its two right-hand vertices, so it has \(2\) right angles. 2. Figure b) has \(4\) right angles, and figure c) has no right angles. Therefore, the answer is a).

Figure a) has exactly \(2\) right angles.

- At each vertex, consider only the smaller angle. - Trace the entire boundary once.

1. Follow the boundary and check all six vertices. 2. At each vertex, a horizontal segment and a vertical segment meet. The smaller angle is a right angle each time, so there are \(6\) right angles.

There are \(6\) right angles.

- Check each vertex of the outline separately. - A tilted angle can still be a right angle.

1. At \(A\) and \(B\), the horizontal base and the vertical walls meet at right angles. 2. At \(D\), the two roof segments also meet at a right angle. The angles at \(C\) and \(E\) are not right angles.

The right angles are at \(A\), \(B\), and \(D\).

- Examine each diagram independently. - Begin with the figure whose number of right angles is easiest to identify.

1. Figure a) is a circle and has no vertices, so it has \(0\) right angles. 2. Figure b) is a right triangle with exactly \(1\) right angle. 3. Figure c) is a rectangle with \(4\) right angles, so it belongs in the “more than \(1\) right angle” group.

No right angles: a). Exactly \(1\) right angle: b). More than \(1\) right angle: c).

- Trace the entire boundary of each figure once as you count. - At an indentation, consider the smaller angle at the vertex.

1. In a), the smaller angle is a right angle at all \(8\) vertices. 2. In b), \(5\) of the smaller vertex angles are right angles. 3. In c), \(3\) vertex angles are right angles.

a) \(8\) b) \(5\) c) \(3\)

- Look at both upper corners in each diagram. - A diagram matches only if both upper corners look like right angles.

1. Compare the two upper corners in each diagram. 2. In a) and c), the slanted top side makes the two upper angles different from right angles. 3. In b), both upper corners are square corners, so both are right angles.

Diagram b)

- Pair each face with the face directly opposite it. - Remember that every square is also a rectangle. - Compare a cube with a rectangular prism that has unequal edge lengths.

1. The \(6\) faces form \(3\) pairs of congruent opposite faces. 2. Every face is a rectangle. A square is a special type of rectangle. 3. A rectangular prism that is not a cube can have exactly \(2\) square faces. The two squares are opposite each other, and the other four faces are non-square rectangles.

a) \(3\) pairs b) Rectangles c) Yes. It can have exactly \(2\) square faces.

- The middle letter names the vertex. - Look first at the intersection of the diagonals. - Compare other angles with a right angle.

1. The diagonals are perpendicular, so \(\angle ASB\) and \(\angle BSC\) are right angles. 2. Angles \(\angle BCD\) and \(\angle CAD\) are acute. 3. Angles \(\angle DAB\) and \(\angle ABC\) are obtuse.

a) \(\angle ASB\), \(\angle BSC\) b) \(\angle BCD\), \(\angle CAD\) c) \(\angle DAB\), \(\angle ABC\)

- Treat each pane as a separate rectangle. - A shared corner may be counted once for each pane that meets there because each pane corner is counted separately.

1. Each of the \(4\) rectangular panes has \(4\) right angles. 2. \(4 \times 4 = 16\). The four panes have \(16\) right angles altogether.

The four panes have \(16\) right angles altogether.

- What is true about opposite faces of a rectangular prism? - If one face is a square, what must be true about the opposite face? - Think about how matching opposite faces are paired.

1. A rectangular prism has three pairs of opposite faces that are the same size and shape. 2. If one face is a square, the opposite face must also be a square of the same size. 3. Therefore, square faces occur in pairs, so a rectangular prism cannot have exactly one square face.

Opposite faces of a rectangular prism are the same size and shape. If one face is a square, its opposite face must also be a square, so exactly one square face is impossible.

- List the dimensions of the three pairs of faces. - A rectangle is a square only when its two side lengths are equal. - If two different pairs of edge lengths are equal, what must be true about all three edge lengths?

1. The three pairs of faces have dimensions length \(\times\) width, width \(\times\) height, and length \(\times\) height. If all three edge lengths are different, none of these faces is a square, so there are \(0\) square faces. 2. For one pair of faces to be square, two edge lengths must be equal. 3. For a second pair of faces to be square, a second pair of edge lengths must be equal. 4. These equalities force the length, width, and height to be equal. The third pair of faces is then also square, so the solid is a cube with \(6\) square faces.

a) \(0\) square faces b) Two square face-pairs force all three edge lengths to be equal. Then the third face-pair is also square, so the prism has \(6\) square faces, not \(4\).

- A quadrilateral has exactly four vertices and four sides. - Check whether every required side is actually drawn. - A marked point may lie on a side without being a vertex of the quadrilateral. - Three consecutive vertices cannot lie on one straight line.

1. A quadrilateral must be a closed figure with exactly four vertices, and all four of its sides must be drawn. 2. Quadrilateral \(ADSE\) is present because \(\overline{AD}\), \(\overline{DS}\), \(\overline{SE}\), and \(\overline{EA}\) are all drawn. 3. Quadrilateral \(ADCE\) is also present because \(\overline{AD}\), \(\overline{DC}\), \(\overline{CE}\), and \(\overline{EA}\) are all drawn. Point \(S\) lies on side \(\overline{DC}\), but it does not have to be a vertex of this quadrilateral. 4. No other quadrilateral is present. Other candidates either require a side that is not drawn or use three consecutive collinear points.

The quadrilaterals shown are \(ADSE\) and \(ADCE\).