Free Math Worksheets

- Use the definition of a rectangle. - Decide which names are guaranteed without knowing the side lengths. - An inclusive trapezoid has at least one pair of parallel sides.

1. A parallelogram with four right angles is a rectangle. 2. It is also a trapezoid because it has at least one pair of parallel sides. 3. The side lengths are not given, so it is not necessarily a kite, rhombus, or square.

Rectangle and trapezoid

- Name quadrilaterals with four congruent sides. - Consider the special case that also has four right angles.

1. A quadrilateral with four congruent sides is a rhombus. 2. If it also has four right angles, it is a square. 3. Therefore, the two possible more specific names are rhombus and square.

Rhombus and square

- Recall the definition of a parallelogram. - A quadrilateral with four congruent sides and four right angles has a special name.

1. A quadrilateral with two pairs of parallel sides is a parallelogram. 2. A parallelogram with four congruent sides and four right angles is a square.

a) Parallelogram b) Square

- Name the broad category with two pairs of parallel opposite sides. - Decide whether parallel sides determine the angle measures. - Recall the defining angle property of a rectangle.

1. A quadrilateral with both pairs of opposite sides parallel is a parallelogram. 2. A rectangle is a special parallelogram with four right angles. 3. A parallelogram can have angles that are not right angles, so it does not have to be a rectangle.

A parallelogram. Unlike a rectangle, a general parallelogram does not have to have four right angles.

- Identify which choices have two pairs of parallel sides. - Eliminate figures that must have four congruent sides. - Pay close attention to the condition that is ruled out.

1. Squares, rectangles, rhombuses, and parallelograms all have two pairs of parallel sides. 2. Squares and rhombuses always have four congruent sides, so they do not meet the second condition. 3. A non-square rectangle and a general parallelogram can have two different side lengths.

(2) Rectangle and (4) parallelogram

- State a property both figures share. - Compare how many sides must be congruent in each figure. - Think of a rhombus as a more specific type of parallelogram.

1. Both a parallelogram and a rhombus have two pairs of parallel opposite sides. 2. In a general parallelogram, only opposite sides must be congruent. 3. In a rhombus, all four sides must be congruent. Therefore, every rhombus is a special type of parallelogram.

In a general parallelogram, each pair of opposite sides is congruent. In a rhombus, all four sides are congruent.

- Use the information about the pairs of parallel sides. - What is true about opposite sides of that quadrilateral?

1. A quadrilateral with both pairs of opposite sides parallel is a parallelogram. 2. Opposite sides of a parallelogram have equal lengths. 3. Therefore, the side opposite the \(14\,\text{cm}\) side is also \(14\,\text{cm}\) long.

a) The quadrilateral is a parallelogram. b) The opposite side is \(14\,\text{cm}\) long.

- Recall how the diagonals of a parallelogram divide each other. - Relate one half of a diagonal to its full length. - Match each given length to a half-diagonal or a full diagonal.

1. The diagonals of a parallelogram bisect each other at their intersection. 2. Therefore, \(AC = 2 \times AS = 2 \times 6\,\text{cm} = 12\,\text{cm}\). 3. Also, \(DS = BD \div 2 = 10\,\text{cm} \div 2 = 5\,\text{cm}\).

\(AC = 12\,\text{cm}\) and \(DS = 5\,\text{cm}\)

- Use the minimum requirements in each definition. - Think of examples that belong to more than one quadrilateral category. - Remember that “at least one pair” allows two pairs.

1. Yes. A parallelogram has two pairs of parallel sides, so it satisfies the requirement of at least one pair. 2. Yes. A kite can have two different pairs of congruent adjacent sides without having all four sides congruent. 3. No. A rectangle must have four right angles, but its four sides do not have to be congruent. 4. Yes. Under the inclusive definition, a parallelogram is a trapezoid with two pairs of parallel sides.

1. Yes. 2. Yes. 3. No. 4. Yes.

- Compare the side-length requirements for parallelograms and rhombuses. - Compare the angle requirements for rhombuses and squares. - Think of a rectangle that is not a square.

1. A parallelogram is a rhombus when all four sides are congruent. 2. A rhombus is a square when it has a right angle. Then all four angles are right angles. 3. A square has four right angles, so it meets the definition of a rectangle. A rectangle does not have to have four congruent sides, so it does not have to be a square.

a) All four sides must be congruent. b) It must have a right angle. c) A square has all the properties of a rectangle, but a rectangle does not have to have four congruent sides.

- Use the defining properties of each quadrilateral. - A counterexample must satisfy the condition but not the conclusion. - Remember that one shape can belong to several categories.

1. Statement a is true because a square has four right angles. 2. Statement b is false. A rhombus with angles \(60^\circ\) and \(120^\circ\) is not a square. 3. Statement c is true because a parallelogram has two pairs of parallel sides and therefore at least one pair. 4. Statement d is false. A non-square rhombus has four congruent sides but is not a rectangle.

a) True b) False; a non-square rhombus is a counterexample. c) True d) False; a non-square rhombus is a counterexample.

- Start with the parallel-side property. - Then use the equal-side property. - Think about which diagonals divide the rhombus into mirror-image halves.

1. Two pairs of parallel opposite sides make the quadrilateral a parallelogram. 2. A parallelogram with four congruent sides is a rhombus. 3. Because it has no right angles, it is not a square. 4. A non-square rhombus has two lines of symmetry, along its diagonals.

It is a rhombus. Its two diagonals are lines of symmetry.

- Think of each diagonal as a segment joining opposite vertices. - Decide which properties must hold for every figure in the category. - Compare the required diagonal properties of parallelograms, kites, and rhombuses.

1. Statement a) is true. The diagonals of every parallelogram bisect each other. 2. Statement b) is false. A kite can have diagonals of different lengths. For example, a kite can have one diagonal of length \(5\,\text{cm}\) and the other of length \(6\,\text{cm}\). 3. Statement c) is true. The diagonals of every rhombus intersect at a right angle.

a) True b) False. A kite can have diagonals of different lengths. c) True

- Compare the mirror lines of a non-square rectangle and a square. - Use the quadrilateral hierarchy to compare squares and rhombuses. - Think of a trapezoid whose nonparallel sides have different lengths.

1. Statement a) is false. A rectangle that is not a square has exactly two lines of symmetry. A square has four. 2. Statement b) is true. A square has four congruent sides, so it meets the definition of a rhombus. 3. Statement c) is false. A trapezoid does not have to have a line of symmetry.

a) False. A non-square rectangle has exactly two lines of symmetry; a square has four. b) True c) False. A trapezoid does not have to have a line of symmetry.

- Separate side-length conditions from angle conditions. - Look for figures that belong to more than one quadrilateral category. - Recall what the definition of a kite requires. - Use the defining angle property of a rectangle.

1. Statement a) is false. Four congruent sides make a rhombus, but a square must also have four right angles. A rhombus with angles of \(60^\circ\) and \(120^\circ\) is a counterexample. 2. Statement b) is true. A square has four congruent sides, so it is a rhombus. 3. Statement c) is false. The definition of a kite requires two pairs of adjacent congruent sides, not right angles. A non-square rhombus is a counterexample because it also meets the kite side condition and can have no right angles. 4. Statement d) is true. Every rectangle has four right angles, so all four interior angles are congruent.

a) False. A rhombus with angles of \(60^\circ\) and \(120^\circ\) has four congruent sides but is not a square. b) True c) False. A non-square rhombus can meet the kite side condition and have no right angles. d) True

- Recall the parallel-side property of a parallelogram. - Decide whether the two congruent side pairs of a kite must have the same length. - Recall the defining angle property of a rectangle.

1. Statement a) is true. A rhombus has two pairs of parallel opposite sides, so it is a parallelogram. 2. Statement b) is false. A kite needs two pairs of adjacent congruent sides, but the two pairs may have different lengths. For example, its consecutive side lengths can be \(3\,\text{cm}\), \(3\,\text{cm}\), \(5\,\text{cm}\), and \(5\,\text{cm}\). 3. Statement c) is true. A quadrilateral with four right angles is a rectangle.

a) True b) False. A kite can have consecutive side lengths of \(3\,\text{cm}\), \(3\,\text{cm}\), \(5\,\text{cm}\), and \(5\,\text{cm}\). c) True

- Pay attention to the words “exactly” and “at least.” - Compare adjacent side lengths with opposite side lengths. - Recall the definition of a rhombus.

1. Statement a) is false. A parallelogram is a trapezoid under the inclusive definition, and it has two pairs of parallel sides. 2. Statement b) is false. A general parallelogram has congruent opposite sides, not necessarily two pairs of congruent adjacent sides. A non-square rectangle is a counterexample. 3. Statement c) is true. A quadrilateral with four congruent sides is a rhombus.

a) False. A parallelogram is a trapezoid with two pairs of parallel sides. b) False. A non-square rectangle is a parallelogram but not a kite. c) True

- Recall the side and angle properties that define each quadrilateral. - Count the pairs of parallel sides in each figure. - Use the grid directions to decide whether any angles are right angles. - Compare the horizontal and vertical changes along adjacent sides of the possible kite. - A rhombus must have four congruent sides.

1. For part a), quadrilateral B is a parallelogram because both pairs of opposite sides are parallel. It is not a rectangle because it has no right angles. 2. For part b), quadrilateral A is a trapezoid because its top and bottom sides are parallel. Its other pair of opposite sides is not parallel, so it is not a parallelogram. 3. For part c), quadrilateral C is a kite. Its two lower adjacent sides each move \(2\) grid units horizontally and \(2\) grid units vertically. Its two upper adjacent sides each move \(2\) grid units horizontally and \(3\) grid units vertically. The sides within each pair are congruent, but the upper pair is longer than the lower pair, so not all four sides are congruent. Therefore, C is not a rhombus.

a) Quadrilateral B; both pairs of opposite sides are parallel, but its angles are not right angles. b) Quadrilateral A; it has one pair of parallel sides, so it is a trapezoid but not a parallelogram. c) Quadrilateral C; it has two pairs of adjacent congruent sides, but the two pairs have different lengths.

- Identify quadrilateral \(AECD\). - Recall the relationship between opposite sides of a parallelogram. - Relate \(AB\), \(AE\), and \(EB\).

1. Since \(\overline{AD} \parallel \overline{CE}\) and \(\overline{AE} \parallel \overline{CD}\), quadrilateral \(AECD\) is a parallelogram. 2. Opposite sides of a parallelogram are congruent, so \(AE = CD = 5\,\text{cm}\). 3. Because \(AB = AE + EB\), \(EB = 12\,\text{cm} - 5\,\text{cm} = 7\,\text{cm}\).

\(EB = 7\,\text{cm}\)

- Compare the lengths of the two parallel bases. - In an isosceles trapezoid, the extra length of the longer base is split equally between the two ends. - Divide the difference between the base lengths by \(2\).

1. In an isosceles trapezoid, the shorter base is centered over the longer base, so the extra length of the longer base is divided equally between its two ends. 2. The difference between the base lengths is \(20\,\text{cm} - 12\,\text{cm} = 8\,\text{cm}\). 3. Therefore, \(AE = 8\,\text{cm} \div 2 = 4\,\text{cm}\).

\(AE = 4\,\text{cm}\)

- Use the opposite-side properties of each parallelogram. - Lines parallel to the same line are parallel to each other. - Use the collinearity of \(A\), \(D\), and \(F\) to find \(AF\). - Use the perimeter formula for a parallelogram.

1. In parallelogram \(ABCD\), \(\overline{AB} \parallel \overline{CD}\) and \(AB = CD = 5\,\text{cm}\). 2. In parallelogram \(DCEF\), \(\overline{CD} \parallel \overline{EF}\) and \(CD = EF = 5\,\text{cm}\). 3. Therefore, \(\overline{AB}\) and \(\overline{EF}\) are parallel and congruent. A quadrilateral with one pair of opposite sides both parallel and congruent is a parallelogram, so \(ABEF\) is a parallelogram. 4. Since \(A\), \(D\), and \(F\) are collinear, \(AF = AD + DF = 5\,\text{cm} + 4\,\text{cm} = 9\,\text{cm}\). 5. The perimeter is \(2 \times (AB + AF) = 2 \times (5\,\text{cm} + 9\,\text{cm}) = 28\,\text{cm}\).

a) \(\overline{AB}\) and \(\overline{EF}\) are parallel and congruent, so \(ABEF\) is a parallelogram. b) \(28\,\text{cm}\)