Free math worksheets

- Identify how \(\overline{AB}\) and \(\overline{CD}\) are positioned in the rectangle. - Recall the relationship between opposite sides of a rectangle.

1. In rectangle \(ABCD\), \(\overline{AB}\) and \(\overline{CD}\) are opposite sides. 2. Opposite sides of a rectangle are parallel. 3. Therefore \(\overline{AB} \parallel \overline{CD}\).

Yes. \(\overline{AB} \parallel \overline{CD}\) because opposite sides of a rectangle are parallel.

- 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 \(A\).

- Both figures have the same number of vertices, so vertex count does not decide the answer. - Check the angle made by each pair of neighboring sides. - Look for four square-corner turns.

1. Figure a) is slanted, and none of its four corners is a right angle. 2. In figure b), each pair of neighboring sides meets at a right angle. Therefore figure b) has four right angles.

b)

- Compare opposite sides in each quadrilateral. - Exactly one pair means one pair has the same direction and the other pair does not. - Reject a figure if it has either zero or two pairs of parallel sides.

1. In a), the top and bottom sides are parallel, while the two slanted sides are not parallel. It has exactly one pair. 2. In b), both pairs of opposite sides are parallel. 3. In c), no pair of sides is parallel. Therefore a) is the only match.

a)

- 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

- Compare the shape of the corners, not whether the sides are horizontal or vertical. - Think about what stays the same when a rigid figure is turned. - A square does not stop being a square when it is rotated.

1. Figure b) is a square that has been rotated. 2. Rotating a figure changes the directions in which its sides point, but it does not change its angle measures. 3. A square has four right angles, so figure b) still has four right angles. Therefore Leo is not correct.

No. Rotation changes the square's orientation, not its angle measures. Figure b) still has four 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.

- Trace the whole boundary and inspect each vertex independently. - A horizontal side and a vertical side are perpendicular when they meet. - Do not assume every corner of a concave figure is a square-corner turn.

1. Trace the boundary and check the pair of sides that meets at each vertex. 2. The sides are perpendicular at the four vertices where a horizontal side meets a vertical side. 3. At the two vertices joined to the slanted side, the meeting sides are not perpendicular.

\(4\) vertices

- 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).

- At each vertex, compare the directions of the two sides that meet there. - Perpendicular sides make a square-corner turn. - Determine each figure's count from that figure itself; do not use another figure's answer to force it.

1. In a), perpendicular sides meet at all \(8\) vertices. 2. In b), perpendicular sides meet at \(5\) vertices. 3. In c), perpendicular sides meet at \(3\) vertices.

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

- Do not count every four-sided figure automatically. - Check the angle at each corner of each outline. - Rotation changes orientation, not whether an angle is right.

1. Figure a) has four right angles. 2. Figure b) has two right angles, so it does not qualify. 3. Figure c) has no right angles. 4. Figure d) is rotated, but all four of its angles are right angles. Therefore \(2\) outlines, a) and d), have four right angles.

\(2\): a) and d)

- 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)

- Check each vertex separately for a square-corner turn. - For parallel sides, compare the directions of the two pairs of nonadjacent sides. - The grid can help you distinguish horizontal, vertical, and slanted directions.

1. Only the lower-right vertex has a horizontal side meeting a vertical side, so the quadrilateral has \(1\) right angle. 2. The top side is slanted, so it is not parallel to the horizontal bottom side. 3. The left side is slanted, so it is not parallel to the vertical right side. 4. Therefore the quadrilateral has \(0\) pairs of parallel sides.

a) \(1\) b) \(0\) pairs

- Check both pairs of nonadjacent sides; do not assume both pairs are parallel. - Compare the direction of the top side with the bottom side first. - Then inspect each corner for a square-corner turn.

1. The top and bottom sides have the same direction, so they form one pair of parallel sides. 2. The two slanted side edges have different directions, so they are not parallel. 3. None of the adjacent sides meet at a right angle. 4. Therefore the figure has \(1\) pair of parallel sides and \(0\) right angles.

\(1\) pair of parallel sides and \(0\) right angles

- Perpendicular sides meet at a right angle. - Check adjacent sides, not opposite sides. - A figure can have perpendicular sides without being a rectangle.

1. In a), adjacent sides meet at right angles, so it has perpendicular sides. 2. In b), the sides meet at slanted angles rather than right angles, so it has no perpendicular pair. 3. In c), the left side is perpendicular to both the top and bottom sides, so it has perpendicular sides. The correct choices are a) and c).

a) and c)

- Check each corner separately for a right angle. - Compare opposite sides to find parallel pairs. - One property does not determine the other automatically.

a) The left side meets the top and bottom sides at right angles, so angles \(A\) and \(D\) are right angles. The other two are not. There are \(2\) right angles. b) The top and bottom sides are parallel. The left and right sides are not parallel. There is \(1\) pair of parallel sides.

a) \(2\) b) \(1\) pair

- Compare how each side moves across and up or down on the grid. - Parallel sides have the same direction, even if they point in opposite travel directions around the polygon. - Check all six sides before deciding that you have found every parallel pair.

1. From \(A\) to \(B\), the side moves \(2\) grid cells right and \(1\) cell down. Side \(\overline{DE}\) has the same direction in reverse, so \(\overline{AB} \parallel \overline{DE}\). 2. \(\overline{BC}\) and \(\overline{EF}\) are both horizontal, so \(\overline{BC} \parallel \overline{EF}\). 3. \(\overline{CD}\) and \(\overline{FA}\) have different slants, so they are not parallel.

\(\overline{AB} \parallel \overline{DE}\) and \(\overline{BC} \parallel \overline{EF}\)

- The correct drawing must satisfy both conditions, not just one. - Check how many pairs of opposite sides run in the same direction. - Then compare the marked angle with the angle stated on the card.

1. Drawing a) has a marked \(120^\circ\) angle, but it has only one pair of parallel sides. 2. Drawing b) has two pairs of parallel sides, but its marked angle is \(100^\circ\). 3. Drawing c) has two pairs of parallel sides and a marked \(120^\circ\) angle. Therefore drawing c) matches both conditions.

c)

- Start with the property the two figures already share. - Then compare the angles where neighboring sides meet. - Think about what makes a rectangle's corners special.

1. Both figures have two pairs of parallel sides, so that property does not separate them. 2. A rectangle has four right angles, so each pair of neighboring sides is perpendicular. 3. A non-rectangular parallelogram does not have four right angles.

The rectangle has perpendicular neighboring sides and four right angles; the non-rectangular parallelogram does not.

- Check opposite sides in each figure. - One pair is enough to place a figure in the first group. - A figure belongs in the second group only if neither pair of opposite sides is parallel.

1. Figure a) has two pairs of parallel sides. 2. Figure b) has exactly one pair of parallel sides. 3. Figure c) has no parallel sides. 4. Figure d) has two pairs of parallel sides.

At least one pair of parallel sides: a), b), d) No parallel sides: c)

- Check the parallel-side condition first. - Then inspect every corner of the remaining candidates. - Both conditions must be satisfied at the same time.

1. Drawing a) has one pair of parallel sides but also has two right angles. 2. Drawing b) has two pairs of parallel sides. 3. Drawing c) has one pair of parallel sides, and both nonparallel sides are slanted so none of its angles is a right angle. Therefore c) matches both conditions.

c)

- An obtuse angle is wider than a right angle but less than a straight angle. - Check every corner of each figure rather than judging the whole shape at once. - A quadrilateral can contain more than one type of angle.

1. Figure a) is a slanted parallelogram, so it has two obtuse angles. 2. Figure b) has four right angles, so it has no obtuse angles. 3. Figure c) has one wide angle greater than a right angle, so it has an obtuse angle. Therefore a) and c) qualify.

a) and c)

- Test the parallel-side requirement and the angle requirement separately. - A slanted appearance alone does not guarantee the needed properties. - The correct frame must satisfy both conditions.

1. Outline a) has two pairs of parallel sides but has four right angles. 2. Outline b) has two pairs of parallel sides and all four corners are non-right angles. 3. Outline c) has only one pair of parallel sides. Therefore outline b) meets both requirements.

b)

- Check the parallel-side condition and the acute-angle condition separately for every figure. - A figure that satisfies only one condition is not enough. - After checking both properties, look for the one figure that appears in both groups.

1. Figure a) has exactly one pair of parallel sides, but it has no acute angles. 2. Figure b) has exactly one acute angle, but it has no pair of parallel sides. 3. Figure c) has exactly one pair of parallel sides and exactly one acute angle. Therefore c) is the only figure that satisfies both conditions.

c)

- 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) \(\angle DAB\), \(\angle ABC\)

- Think about the direction of a side after two right-angle turns. - Compare the first side with the third side. - Then make the same comparison for the second and fourth sides.

1. Four right angles mean each side turns a quarter-turn from the previous side. 2. The first and third sides therefore have the same direction, and the second and fourth sides have the same direction. 3. Opposite sides form two parallel pairs. The statement is always true.

Always true. A quadrilateral with four right angles has two pairs of parallel opposite sides.

- Rotation does not change angle sizes. - Perpendicular does not mean “one horizontal and one vertical.” - Look at the angle between neighboring sides.

1. Perpendicular describes the angle made by two lines or sides, not whether either one is vertical. 2. Each pair of neighboring sides in the turned square still meets at a right angle. 3. Therefore the neighboring sides are perpendicular even though all four sides are slanted.

Noah is incorrect. Perpendicular sides meet at right angles, and the square's neighboring sides still meet at right angles after the square is turned.

- Use the right angles to describe neighboring sides first. - Then compare opposite sides. - Give one conclusion about perpendicular sides and one about parallel sides.

1. At each vertex, two neighboring sides meet at a right angle, so every pair of neighboring sides is perpendicular. 2. Opposite sides point in the same directions, so there are two pairs of parallel opposite sides.

Every pair of neighboring sides is perpendicular, and the quadrilateral has two pairs of parallel opposite sides.

- Check the parallel-side condition and the angle condition separately for each figure. - A figure that satisfies only one condition is not enough. - After narrowing by one property, use the other property to distinguish the remaining choices.

1. Figure a) has two pairs of parallel sides, but it has four right angles and no acute angles. 2. Figure b) has exactly two acute angles, but only one pair of parallel sides. 3. Figure c) has two pairs of parallel sides and exactly two acute angles. 4. Therefore c) is the only figure with both properties.

c)

- Determine each figure's properties independently. - Count pairs of parallel sides separately from right angles. - Check every corner; do not infer one figure's result from another figure's result.

1. Figure a) has \(2\) pairs of parallel sides and \(4\) right angles. 2. Figure b) has \(2\) pairs of parallel sides and \(0\) right angles. 3. Figure c) has \(1\) pair of parallel sides and \(2\) right angles. 4. Figure d) has \(1\) pair of parallel sides and \(0\) right angles.

a) \(2\) pairs of parallel sides; \(4\) right angles b) \(2\) pairs of parallel sides; \(0\) right angles c) \(1\) pair of parallel sides; \(2\) right angles d) \(1\) pair of parallel sides; \(0\) right angles

- Treat the side relationships and angle types as two separate checks for each figure. - Parallel sides point in the same direction even when they are different lengths. - Compare each corner with a right angle to decide whether it is acute or obtuse.

1. The triangle has no parallel side pair. Two of its angles are acute. 2. The pentagon has one pair of parallel sides, the top and bottom sides. It has one acute angle. 3. The hexagon has three pairs of parallel opposite sides. All six angles are obtuse, so it has no acute angles.

a) \(0\) pairs of parallel sides; \(2\) acute angles b) \(1\) pair of parallel sides; \(1\) acute angle c) \(3\) pairs of parallel sides; \(0\) acute angles

- Count the sides of the polygon. - Use the angle mark as given information. - A single example that satisfies the condition but not the claimed conclusion disproves an “any” statement.

1. The shown figure has five sides, so it is a pentagon. 2. The marked angle is a right angle. 3. A five-sided polygon with a right angle is a counterexample to Eli's claim, so the claim is false.

Yes. The figure is a pentagon, so it has five sides, and it has a right angle. Therefore a polygon with a right angle does not have to have four sides.

- Check parallel-side pairs separately from angle types. - Use the marked corners as exact right-angle information. - Compare each unmarked corner with a right angle before deciding whether statement III or IV is true.

1. The bottom and top sides are parallel, and the left and right vertical sides are parallel. The other two sides are not parallel, so there are exactly two parallel pairs. 2. The two marked bottom corners are right angles, and no other corner is right, so there are exactly two right angles. 3. Each of the four remaining angles is wider than a right angle, so they are obtuse. There are no acute angles. Therefore statements I, II, and IV are true.

I, II, and IV

- Picture a slanted quadrilateral whose opposite sides are parallel. - Separate what “parallel” tells you about side directions from what is known about the corner angles. - Ask for the smallest extra angle fact that would force the perpendicular relationships around the whole quadrilateral.

1. Two pairs of parallel sides do not force right angles; a slanted parallelogram can have acute and obtuse angles. 2. Therefore Leo's conclusion is not guaranteed from parallel sides alone. 3. It is enough to know that one angle is a right angle. Because each opposite side is parallel to its partner, the perpendicular relationship carries to the other corners, so all four angles are right angles.

No. Two pairs of parallel sides are not enough. Knowing that any one angle is a right angle would guarantee that the quadrilateral has four right angles.

- Separate what a right angle tells you from what parallel sides require. - A right angle fixes the relationship between only two neighboring sides. - Ask whether the directions of the remaining sides are forced.

1. One right angle tells us that one pair of neighboring sides is perpendicular. 2. It does not determine the directions of the other three sides. 3. Those other sides can be arranged so that a pentagon has a pair of parallel sides, or so that it has no parallel sides. Therefore the two pentagons do not have to have the same number of pairs of parallel sides.

No. Knowing that each pentagon has one right angle does not determine its parallel-side relationships, so the two pentagons can have different numbers of parallel-side pairs.