Free math worksheets

- Focus on the angle marks before judging the figure's appearance. - Ask which quadrilateral category is guaranteed by four right angles. - Check whether the side marks justify a more specific category.

1. All four angles are marked as right angles. 2. A quadrilateral with four right angles is a rectangle. 3. The side markings show opposite sides congruent in pairs, but they do not show all four sides congruent. Therefore, a square is not guaranteed.

\(ABCD\) is a rectangle.

- Compare the tick marks on all four sides. - Identify the quadrilateral category determined by those side markings. - Separate what the diagram marks from what its shape may merely appear to show.

1. The four matching side marks show that all four sides are congruent. 2. A quadrilateral with four congruent sides is a rhombus. 3. A square would also need four right angles. No right-angle information is given, so a square is not guaranteed.

A rhombus is guaranteed. A square is not guaranteed because no right-angle information is given.

- Which quadrilaterals can be folded across a diagonal so that the two halves match? - What side-length relationships result when two vertices reflect onto each other? - Imagine folding a rectangle so that its vertices match. Where are the fold lines?

1. If a line of symmetry passes through two opposite vertices, the other two vertices are reflections of each other. 2. This creates two pairs of adjacent congruent sides, so the quadrilateral belongs to the kite family. 3. A rectangle that is not a square has exactly two lines of symmetry. 4. Each of those lines passes through the midpoints of a pair of opposite sides.

a) The kite family. b) Two lines of symmetry, each through the midpoints of a pair of opposite sides.

- 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

- Separate what the four congruent sides guarantee from what might require extra information. - Name the quadrilateral category defined by four congruent sides. - For part b), compare a rhombus with the defining angle property of a square.

1. A quadrilateral with four congruent sides is a rhombus, so rhombus must describe the kite. 2. A square has four congruent sides and four right angles. Therefore, if the rhombus also has four right angles, it is a square.

a) Rhombus b) Four right angles

- 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

- Use the parallel-side information to identify a guaranteed broad category. - Then use the four congruent sides to decide whether a more specific category must apply. - Ask which remaining names would require right-angle information.

1. Two pairs of parallel opposite sides make \(J\) a parallelogram. 2. Because all four sides are congruent, the parallelogram is also a rhombus. 3. No right angles are given, so rectangle and square are not guaranteed.

Parallelogram and rhombus must apply. Rectangle and square are not guaranteed.

- For each category, recall its defining side or angle property. - Match that definition to a property every square has. - For the final explanation, compare what “square” specifies with what the broader names require.

1. A square has four right angles, which meets the defining angle property of a rectangle. 2. A square has four congruent sides, which meets the defining side property of a rhombus. 3. A square has two pairs of parallel opposite sides, so it is a parallelogram. 4. Those two pairs of parallel opposite sides include at least one parallel pair, so the square also meets the inclusive definition of a trapezoid. 5. Its four congruent sides form at least two pairs of adjacent congruent sides, so it meets the inclusive definition of a kite. 6. “Square” is the most specific name because it combines four right angles with four congruent sides.

a) Four right angles, so the rectangle definition is satisfied. b) Four congruent sides, so the rhombus definition is satisfied. c) Two pairs of parallel opposite sides. d) Two pairs of parallel opposite sides give at least one parallel pair, satisfying the inclusive trapezoid definition. e) Four congruent sides give at least two pairs of adjacent congruent sides, satisfying the inclusive kite definition. “Square” is the most specific name because it combines four right angles with four congruent sides.

- Compare side markings and angle markings separately in each figure. - A square needs both four congruent sides and four right angles. - For every choice, state the markings that make the classification guaranteed rather than relying on appearance.

1. Figure Z has four congruent sides but no right-angle information, so a rhombus is guaranteed but a square is not. 2. Figure X has four congruent sides and four right angles, so it is a square. 3. Figure Y has four right angles and opposite sides marked congruent in pairs, so it is a rectangle. The markings do not guarantee that all four sides are congruent, so a square is not guaranteed.

a) Figure Z. All four sides are marked congruent, but no right-angle information is given. b) Figure X. All four sides are marked congruent and all four angles are marked right. c) Figure Y. All four angles are marked right, but the side markings do not guarantee that all four sides are congruent.

- Start with the angle information. - Then ask whether the side lengths justify a more specific category. - Use the most specific name that must be true.

1. Four right angles guarantee that the quadrilateral is a rectangle. 2. The side lengths are not all congruent, so it is not a square or a rhombus. 3. Therefore, rectangle is the most specific guaranteed classification.

Rectangle

- Start with the defining angle property of a rectangle. - Think about how many pairs of opposite sides are parallel in every rectangle. - Separate properties every rectangle must have from properties only squares have.

1. Every rectangle has four right angles, so a) is guaranteed. 2. Every rectangle has two pairs of parallel sides, so b) is guaranteed. Having two pairs also guarantees at least one pair, so d) is guaranteed. 3. A rectangle does not have to have four congruent sides, so c) is not guaranteed.

a), b), and d)

- For figure a), compare the four numerical side labels and group equal adjacent sides. - For figure b), interpret the matching side marks. - A rhombus needs all four sides congruent, while an inclusive kite needs at least two adjacent congruent pairs.

1. In figure a), two adjacent sides are \(5\) units each and the other two adjacent sides are \(4\) units each. It is a kite, but its four sides are not all congruent, so it is not a rhombus. 2. In figure b), all four sides have the same congruence mark, so all four sides are congruent. Therefore, figure b) is a rhombus. 3. Under the inclusive definition, four congruent sides also provide at least two pairs of adjacent congruent sides. Therefore, both figures are kites.

a) Figure b) b) Figures a) and b) c) Figure a) has adjacent congruent pairs of \(5\) units and \(4\) units; figure b) has all four sides congruent.

- Interpret the side markings and the angle markings separately. - Combine the marked side and angle properties to identify the most specific category. - Decide whether changing a figure's orientation changes any of those properties.

1. The matching side marks show that all four sides are congruent. 2. The angle marks show that all four angles are right angles. 3. A quadrilateral with four congruent sides and four right angles is a square. 4. Rotating or tilting a figure does not change its side lengths or angle measures.

Square

- Compare the defining properties of a rectangle and a square. - Focus only on the side property that the square definition adds. - State the property for all four sides, not just for one selected pair.

1. A rectangle already has four right angles. 2. A square has four right angles and four congruent sides. 3. Therefore, the additional defining side property needed is that all four sides are congruent.

All four sides must be congruent.

- What side-length property defines a kite? - What happens to a rectangle when adjacent sides are congruent? - What condition defines a trapezoid under the inclusive definition? - What makes a trapezoid isosceles?

1. For a rectangle to be a kite, it must have congruent adjacent sides. 2. Opposite sides of a rectangle are already congruent, so congruent adjacent sides make all four sides congruent. The rectangle is then a square. 3. A square has two pairs of parallel sides, so it meets the inclusive definition of a trapezoid. 4. For either pair chosen as bases, the other two sides are congruent and there is a line of symmetry perpendicular to the bases through their midpoints. Therefore, a square is a special isosceles trapezoid under the stated inclusive definition.

a) Adjacent sides must be congruent, making the rectangle a square. b) A square has at least one pair of parallel sides, congruent legs, and a line of symmetry perpendicular to the chosen bases. Under the inclusive definition, it is a special isosceles trapezoid.

- 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 defining 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 four 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 four right angles. 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 information about opposite sides. - Then use the information that all four sides are congruent. - Compare the angle information with the defining angle property of a square.

1. Two pairs of parallel opposite sides make the quadrilateral a parallelogram. 2. A parallelogram with four congruent sides is a rhombus. 3. A square must have four right angles. Because this quadrilateral has no right angles, it is not a square.

Rhombus. It is not a square because none of its angles is a right angle.

- Imagine folding each kind of quadrilateral along possible mirror lines. - Compare the horizontal, vertical, and diagonal folds that work for a square and a non-square rectangle. - For a statement about every trapezoid, think about whether a lopsided trapezoid can match across any fold.

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 lines of symmetry. 3. Statement c) is false. A trapezoid does not have to have a line of symmetry; a trapezoid with unequal nonparallel sides can have none.

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

- Use the peg grid to compare the directions of opposite sides. - A side perpendicular to a horizontal side would run vertically along a peg column. - For part c), count horizontal and vertical peg spaces rather than judging side length by appearance.

1. Quadrilateral B has horizontal top and bottom sides. Its two slanted sides make the same horizontal and vertical move between pegs, so both pairs of opposite sides are parallel. A side meeting the horizontal base is slanted rather than vertical, so the angles are not right angles. Thus, B is a parallelogram but not a rectangle. 2. Quadrilateral A has horizontal top and bottom sides, but its other two sides have different directions. It has one pair of parallel opposite sides, so it is a trapezoid but not a parallelogram. 3. Quadrilateral C has horizontal top and bottom sides and vertical left and right sides, so it has four right angles and is a rectangle. Its horizontal sides span \(4\) peg spaces while its vertical sides span \(2\) peg spaces, so not all four sides are congruent. Therefore, it is not a square.

a) Quadrilateral B b) Quadrilateral A c) Quadrilateral C

- Identify the two pairs of opposite sides in quadrilateral \(AECD\). - Use the fact that \(E\) lies on \(\overline{AB}\) together with the given parallel bases. - Compare the two resulting pairs of parallel sides with the definition of a parallelogram.

1. Segment \(\overline{AE}\) lies on \(\overline{AB}\). Because \(\overline{AB} \parallel \overline{CD}\), it follows that \(\overline{AE} \parallel \overline{CD}\). 2. It is given that \(\overline{CE} \parallel \overline{AD}\). 3. Quadrilateral \(AECD\) therefore has two pairs of parallel opposite sides, so it is a parallelogram.

Parallelogram

- Use the opposite-side properties of each given parallelogram. - Compare \(\overline{AB}\) and \(\overline{EF}\) through their relationships with \(\overline{CD}\). - Use the collinearity of \(A\), \(D\), and \(F\) to determine the long side of \(ABEF\). - After the proof in part a), use the side lengths of the resulting parallelogram for its perimeter.

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 \(2 \cdot (AB + AF) = 2 \cdot (5\,\text{cm} + 9\,\text{cm}) = 28\,\text{cm}\).

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

- Separate what is guaranteed from what is left unknown about the second pair of opposite sides. - Try one example with exactly one pair of parallel sides and another with two pairs. - Compare the most-specific names of those two possible figures.

1. The information guarantees at least one pair of parallel opposite sides, but it does not tell whether the second pair is parallel. 2. One possible figure is a non-parallelogram trapezoid with exactly one pair of parallel sides and two adjacent right angles. 3. Another possible figure is a rectangle, which has two pairs of parallel sides and four right angles. 4. Because these possibilities have different most-specific names, the given information is not sufficient to determine one most-specific classification.

No. For example, a non-parallelogram trapezoid with two adjacent right angles and a rectangle can both satisfy the information.

- Pay attention to where the congruent sides are located, not only how many pairs there are. - Compare the side condition for a kite with the side relationships in a parallelogram. - A counterexample must satisfy the claim's condition but fail its conclusion.

1. The claim is false because it does not say the congruent sides are opposite sides. 2. A non-rhombus kite has two pairs of adjacent congruent sides. 3. Such a kite does not need to have two pairs of parallel sides, so it does not have to be a parallelogram.

False. A non-rhombus kite is a counterexample because its congruent sides can be adjacent rather than opposite.

- Use the parallel-side condition to identify every broad category it guarantees. - Use the four-congruent-side condition separately to identify additional guaranteed categories. - Keep names requiring right angles in the “could apply” group unless right angles are given.

1. Two pairs of parallel opposite sides make \(Q\) a parallelogram and therefore also a trapezoid under the inclusive definition. 2. All four sides are congruent, so \(Q\) is a rhombus and also a kite under the inclusive definition. 3. No right-angle information is given. The figure could be a rectangle or square, but neither is guaranteed.

Must apply: trapezoid, parallelogram, kite, rhombus Could apply but are not guaranteed: rectangle, square

- Treat the requested move counts, side lengths, and markings as evidence you must report, not as decorative details. - Establish parallel-side information before using the quadrilateral hierarchy. - Then use side congruence and right-angle evidence to decide which more specific names are justified and which are ruled out.

1. Figure a) has horizontal top and bottom sides. One other side is vertical, while the other moves \(3\) units horizontally and \(4\) units vertically, so those two sides are not parallel. The labeled side lengths are \(8,5,5,4\), giving only one adjacent congruent pair. Thus, only trapezoid applies. 2. In figure b), each slanted side moves \(3\) units horizontally and \(4\) units vertically when followed from the lower base to the upper base, so the slanted sides are parallel; the horizontal bases are also parallel. Thus, it is a parallelogram and trapezoid. Adjacent side lengths are \(6\) and \(5\), so rhombus and kite are not guaranteed. A slanted side is not vertical to a horizontal base, so rectangle and square do not apply. 3. Figure c) has horizontal side length \(5\) and vertical side length \(3\). The four angle marks are right angles. Thus, it is a rectangle, parallelogram, and trapezoid. Because adjacent side lengths differ, it is not a rhombus, kite, or square. 4. In figure d), each slanted side moves \(3\) units horizontally and \(4\) units vertically, while the top and bottom are horizontal, so both pairs of opposite sides are parallel. All four sides have matching congruence ticks, so it is a rhombus and kite as well as a parallelogram and trapezoid. A slanted side has a nonzero horizontal move, so it is not perpendicular to the horizontal base; rectangle and square do not apply. 5. Figure e) has matching congruence ticks on all four sides and right-angle marks at all four vertices. Thus, square, rectangle, rhombus, kite, parallelogram, and trapezoid all apply.

a) trapezoid. Evidence: the top and bottom are horizontal; one other side is vertical and the other moves \(3\) horizontally and \(4\) vertically; the side lengths are \(8,5,5,4\). b) trapezoid, parallelogram. Evidence: each slanted side moves \(3\) horizontally and \(4\) vertically; adjacent side lengths are \(6\) and \(5\). c) trapezoid, parallelogram, rectangle. Evidence: a horizontal side is \(5\) units, a vertical side is \(3\) units, and all four angles are marked right. d) trapezoid, parallelogram, rhombus, kite. Evidence: each slanted side moves \(3\) horizontally and \(4\) vertically, all four sides share a congruence mark, and a slanted side is not vertical to the horizontal base. e) trapezoid, parallelogram, rectangle, rhombus, kite, square. Evidence: all four sides share a congruence mark and all four angles are marked right.