- Interpret the side markings and angle markings separately. - Decide what the four angle marks guarantee. - 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.

Rectangle. All four angle marks show right angles. The side marks show only opposite sides congruent in pairs, not all four sides congruent, so a square is not guaranteed.

- 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.

- Think about the quadrilateral definitions you know. - Compare the given relationship between opposite sides with those definitions. - Do not assume any side-length or angle facts that are not stated.

1. A parallelogram is a quadrilateral with two pairs of parallel opposite sides. 2. The given quadrilateral matches that definition, so it is a parallelogram.

Parallelogram

- Say the trapezoid definition in your own words. - Check whether the information about \(WXYZ\) meets that definition. - Do not add any side or angle facts that are not given.

1. The definition requires at least one pair of parallel sides. 2. \(WXYZ\) has one pair of parallel sides, so it is a trapezoid.

Trapezoid

- Read the four side labels and the angle marks separately. - What do the side labels tell you about the four sides? - Which quadrilateral definition matches both sets of evidence?

1. All four side labels are \(4\) units, so 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.

Square. All four sides are \(4\) units, and all four angles are marked right.

- Group together sides that have matching tick marks. - For each matching pair, check whether the two sides meet at a vertex. - After finding the two pairs, check whether all four sides are used.

1. Two adjacent sides share one tick mark, so they are congruent. 2. The other two adjacent sides share two tick marks, so they are congruent to each other. 3. The two congruent pairs use all four sides, so the quadrilateral is a kite.

Kite. \(AB\) and \(BC\) are congruent, and \(CD\) and \(DA\) are congruent.

- Think about what a symmetry line through opposite vertices does to the other two vertices. - What side relationships result when those two vertices reflect onto each other? - For part b), what quadrilateral has all four sides congruent?

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 using all four sides, so the quadrilateral belongs to the kite family. 3. A quadrilateral with all four sides congruent is a rhombus.

a) The symmetry makes two pairs of adjacent sides congruent, using all four sides, so it is a kite. b) Rhombus.

- 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 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 this kite is a rhombus. 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 include at least one pair of parallel sides, so the square also meets the inclusive definition of a trapezoid. 5. Its four congruent sides form two pairs of adjacent congruent sides using all four 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, so the parallelogram definition is satisfied. d) Two pairs of parallel opposite sides give at least one parallel pair, satisfying the inclusive trapezoid definition. e) Four congruent sides give two adjacent congruent pairs using all four 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 the stated kite definition needs two adjacent congruent pairs using all four sides.

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. Four congruent sides also give two pairs of adjacent congruent sides using all four 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.

- Read the side marks and angle marks separately. - Combine the marked side and angle properties before choosing a name. - 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. The side marks show four congruent sides, and the angle marks show four right angles. Tilting the drawing does not change side lengths or angle measures.

- 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.

- Would rotating a shape change which sides are parallel? - Which quadrilateral definition matches the parallel-side facts? - Base your answer on the properties, not on how the drawing sits on the page.

1. \(\overline{AB}\) and \(\overline{CD}\) form one pair of parallel opposite sides. 2. \(\overline{BC}\) and \(\overline{AD}\) form a second pair of parallel opposite sides. 3. A quadrilateral with two pairs of parallel opposite sides is a parallelogram. 4. Tilting a figure does not change which sides are parallel.

Parallelogram. It has two pairs of parallel opposite sides, and its orientation does not change that property.

- Classify each figure from its own markings first. - After naming a figure, ask whether that category always belongs to the parallelogram family. - For a figure that is not guaranteed to be a parallelogram, identify which required property is missing from the markings.

1. Figure a) has four right angles, so it is a rectangle. Every rectangle is a parallelogram. 2. Figure b) has four congruent sides, so it is a rhombus. Every rhombus is a parallelogram. 3. Figure c) has two pairs of adjacent congruent sides, so it is a kite. The markings do not show two pairs of parallel opposite sides, so a parallelogram is not guaranteed. 4. Therefore, figures a) and b) are guaranteed to be parallelograms, while figure c) is not.

Figures a) and b) are guaranteed to be parallelograms. a) rectangle b) rhombus c) not guaranteed to be a parallelogram; the markings show two pairs of adjacent congruent sides, not two pairs of parallel opposite sides

- Compare the side properties of a rectangle with the stated kite definition. - If one pair of adjacent sides of a rectangle is congruent, use the rectangle's opposite-side property too. - For part b), compare the square's parallel sides with the stated trapezoid definition.

1. For a rectangle to be a kite, it must have a pair of 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 opposite sides. 4. Because a trapezoid needs at least one pair of parallel sides, every square is also a trapezoid under the inclusive definition.

a) A pair of adjacent sides must be congruent. Then all four sides are congruent, so the rectangle is a square. b) A square has two pairs of parallel sides, so it has at least one pair of parallel sides and is therefore a trapezoid under the inclusive definition.

- For each part, compare the requested shape with the defining properties of the larger and smaller categories. - For the kite, think about two equal adjacent pairs that do not all have the same length. - For the rectangle, ask what side property a rhombus needs that a rectangle does not always have.

1. A parallelogram works for part a). It has two pairs of parallel sides, so it has at least one pair and is a trapezoid under the inclusive definition. 2. For part b), a kite can have consecutive side lengths \(3\), \(3\), \(5\), and \(5\) units. It has two pairs of adjacent congruent sides using all four sides, but not all four sides are congruent, so it is not a rhombus. 3. For part c), a \(3\)-unit by \(5\)-unit rectangle works. It has four right angles, but its four sides are not all congruent, so it is not a rhombus.

a) A parallelogram; it has two pairs of parallel sides, so it also meets the inclusive trapezoid definition. b) A kite with consecutive side lengths \(3\), \(3\), \(5\), and \(5\) units; it has two adjacent congruent pairs but not four congruent sides. c) A \(3\)-unit by \(5\)-unit rectangle; it has four right angles but not 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.

- Separate side-length conditions from angle conditions. - Look for figures that belong to more than one quadrilateral category. - For the kite statement, use only the side condition in the definition. - 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 using all four sides, not right angles. A non-square rhombus is a counterexample because it 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

- First classify each card from only the facts written on it. - Do not assume a right angle, equal side, or parallel side unless the facts guarantee it. - After naming the shape, use the quadrilateral hierarchy to decide whether parallelogram must also apply.

1. Card A has four congruent sides, so rhombus is the most specific guaranteed name. Every rhombus is a parallelogram, so Card A is guaranteed to be a parallelogram. 2. Card B has two pairs of adjacent congruent sides using all four sides, so it is a kite. The two pairs have different lengths, so it is not a rhombus. Those side lengths do not guarantee opposite sides are parallel, so a parallelogram is not guaranteed. 3. Card C has four right angles, so rectangle is the most specific guaranteed name. Every rectangle is a parallelogram, so Card C is guaranteed to be a parallelogram.

Card A: rhombus; yes, it is guaranteed to be a parallelogram because every rhombus is a parallelogram. Card B: kite; no, a parallelogram is not guaranteed because the given side lengths do not guarantee two pairs of parallel opposite sides. Card C: rectangle; yes, it is guaranteed to be a parallelogram because every rectangle is a parallelogram.

- 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 \(\overline{AB}\) is parallel to \(\overline{CD}\), \(\overline{AE}\) is also parallel to \(\overline{CD}\). 2. It is given that \(\overline{CE}\) is parallel to \(\overline{AD}\). 3. Quadrilateral \(AECD\) therefore has two pairs of parallel opposite sides, so it is a parallelogram.

Parallelogram. \(\overline{AE}\) lies on \(\overline{AB}\), so \(\overline{AE}\) is parallel to \(\overline{CD}\). Also, \(\overline{CE}\) is parallel to \(\overline{AD}\). Thus both pairs of opposite sides are parallel.

- 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 whether each pair of congruent sides is adjacent or opposite. - Recall that a parallelogram is defined by relationships between opposite sides. - Look for a quadrilateral that satisfies the congruent-side condition but does not force both pairs of opposite sides to be parallel.

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.

- Consider the parallel-side information and the side-length information separately. - For each name in the list, ask whether its definition must be satisfied. - If you are unsure about a name, imagine two shapes that satisfy the givens and ask whether that name must fit both.

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

- Start by deciding which opposite sides are parallel. - Then use side congruence and right-angle evidence to decide whether a more specific name applies. - List only names that are guaranteed by the grid or markings, not by appearance.

1. In figure a), the top and bottom sides are horizontal, so they are parallel. The left side is vertical, while the right side moves \(3\) units horizontally and \(4\) units vertically, so those sides are not parallel. The side lengths are \(8,5,5,4\), so no more specific listed category is guaranteed. Only trapezoid applies. 2. Figure b) has four right angles. Its horizontal side length is \(5\) units and its vertical side length is \(3\) units, so it is not a rhombus or square. Rectangle, parallelogram, and trapezoid apply. 3. In figure c), each slanted side moves \(3\) units horizontally and \(4\) units vertically, so 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. The slanted side is not perpendicular to the horizontal base, so a rectangle or square is not guaranteed.

a) trapezoid. Evidence: the top and bottom are parallel, but the other two sides are not parallel. b) trapezoid, parallelogram, rectangle. Evidence: all four angles are marked right, and adjacent side lengths are \(5\) units and \(3\) units. c) trapezoid, parallelogram, rhombus, kite. Evidence: opposite sides are parallel and all four sides share the same congruence mark.