Quadrilateral Practice Questions

The interior angles of any quadrilateral always sum to 360°. Set up an equation and solve for the missing angle:

\(85° + 90° + 110° + x = 360°\)

\(285° + x = 360°\)

\(x = 75°\)

This rule applies to all quadrilaterals, regardless of type (parallelograms, trapezoids, irregular shapes, etc.).

In a parallelogram, opposite angles are equal. Since one angle is 65°, the angle directly across from it is also 65°.

Choice B gives 115°, which is the measure of each adjacent (consecutive) angle, not the opposite angle. In a parallelogram, consecutive angles are supplementary (they add up to 180°): \(180° – 65° = 115°\).

In a parallelogram, consecutive angles are supplementary, meaning they add up to 180°:

\(180° – 70° = 110°\)

This is different from opposite angles (which are equal). A parallelogram with a 70° angle has angles of 70°, 110°, 70°, and 110°.

Choice A confuses adjacent angles with opposite angles.

The area of a trapezoid is:

\(A = \dfrac{1}{2}(b_1 + b_2)(h)\)

where \(b_1\) and \(b_2\) are the two parallel bases and \(h\) is the height. Substitute the given values:

\(A = \dfrac{1}{2}(8 + 14)(5) = \dfrac{1}{2}(22)(5) = 55 \text{ cm}^2\)

Choice C forgets to multiply by \(\frac{1}{2}\), and choice D multiplies all three numbers together without adding the bases first.

A rhombus is a quadrilateral with all four sides equal. It does not require right angles.

A square also has four equal sides, but a square must have four right angles as well. Since the question specifies that the angles are not all right angles, it cannot be a square. A rectangle requires right angles but does not require all sides to be equal. A trapezoid only requires one pair of parallel sides.

Perpendicular diagonals are a property of rhombuses (and squares), not all rectangles. A rectangle’s diagonals bisect each other and are equal in length, but they are only perpendicular if the rectangle is also a square.

The other three choices are all true for every rectangle.

The area of a rhombus can be found using its diagonals:

\(A = \dfrac{1}{2}d_1 \cdot d_2\)

Substitute \(d_1 = 12\) and \(d_2 = 16\):

\(A = \dfrac{1}{2}(12)(16) = \dfrac{1}{2}(192) = 96 \text{ cm}^2\)

Choice C forgets the \(\frac{1}{2}\) factor. This formula works because the diagonals of a rhombus are perpendicular and divide it into four right triangles.

The midsegment (or median) of a trapezoid connects the midpoints of the two non-parallel sides. Its length is the average of the two parallel bases:

\(m = \dfrac{b_1 + b_2}{2} = \dfrac{6 + 10}{2} = \dfrac{16}{2} = 8 \text{ cm}\)

The midsegment is always parallel to both bases and its length is always between the lengths of the two bases.

In a parallelogram, opposite sides are equal. So the four sides measure 9, 15, 9, and 15. The perimeter is:

\(P = 2(9) + 2(15) = 18 + 30 = 48 \text{ cm}\)

Choice A adds only two sides (\(9 + 15 = 24\)) without accounting for both pairs. Choice C multiplies the sides instead of adding them.

When diagonals bisect each other, they cut each other into two equal halves at their intersection point. If the full diagonal is 20 cm, then each half is:

\(\dfrac{20}{2} = 10 \text{ cm}\)

This property is true for all parallelograms (and therefore also for rectangles, rhombuses, and squares).

A trapezoid is defined as a quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases, and the non-parallel sides are called legs.

The other three choices all have two pairs of parallel sides. (A rhombus and a rectangle are both special types of parallelograms.)

Find the slopes of all four sides to check for parallel sides:

• AB: \(\frac{0 – 0}{4 – 0} = 0\) (horizontal)
• BC: \(\frac{3 – 0}{5 – 4} = 3\)
• CD: \(\frac{3 – 3}{1 – 5} = 0\) (horizontal)
• DA: \(\frac{0 – 3}{0 – 1} = 3\)

AB is parallel to CD (both have slope 0), and BC is parallel to DA (both have slope 3). Since both pairs of opposite sides are parallel, ABCD is a parallelogram.

A rhombus always has perpendicular diagonals, meaning the diagonals intersect at right angles. This is one of the key properties that distinguishes a rhombus from a general parallelogram.

A rectangle’s diagonals are equal in length but are not necessarily perpendicular (only when the rectangle is a square). A general parallelogram’s diagonals bisect each other but are neither equal nor perpendicular. A trapezoid has no general diagonal properties.

A square is a special rhombus, so its area can be found using the diagonal formula. Since a square’s two diagonals are equal, both diagonals measure 10 centimeters:

\(A = \dfrac{1}{2}d_1 \cdot d_2 = \dfrac{1}{2}(10)(10) = 50 \text{ cm}^2\)

Alternatively, find the side length using the relationship between a square’s side and diagonal: \(d = s\sqrt{2}\), so \(s = \frac{10}{\sqrt{2}} = 5\sqrt{2}\). Then \(A = s^2 = (5\sqrt{2})^2 = 50 \text{ cm}^2\).

Choice C uses 10 as the side length instead of the diagonal.

Let \(w\) represent the width. The length is \(w + 3\). Use the perimeter formula for a rectangle:

\(P = 2l + 2w\)

\(46 = 2(w + 3) + 2w\)

\(46 = 2w + 6 + 2w\)

\(46 = 4w + 6\)

\(40 = 4w \implies w = 10\)

So the width is 10 meters and the length is \(10 + 3 = 13\) meters.

Choice B gives equal dimensions, which would make it a square, not a rectangle that is “3 meters longer than it is wide.”