Topic 6 Review

Which two statements are always true? A. Diagonals of a kite create 4 congruent triangles B. All squares are rectangles C. All rectangles are squares D. Diagonals of a rectangle are congruent.

• Diagonals of a kite create 4 congruent triangles

• All squares are rectangles

• All rectangles are squares

• Diagonals of a rectangle are congruent.

Answer explanation

Statement B is true because all squares meet the definition of rectangles (four right angles). Statement D is also true as the diagonals of a rectangle are equal in length, making them congruent.

Which type of quadrilateral is formed by these points: A(2, 3), B(6, 3), C(7, 7), and D(3, 7)?

• Rectangle

• Rhombus

• Parallelogram

• Trapezoid

Answer explanation

The points A(2, 3), B(6, 3), C(7, 7), and D(3, 7) form a parallelogram because opposite sides are equal and parallel. AB is parallel to CD and AD is parallel to BC, confirming it is a parallelogram.

A student draws a quadrilateral ABCD with the following properties: I. Opposite sides are equal in length II. The diagonals bisect each other III. All angles are 90°. Which type of parallelogram is quadrilateral ABCD?

• Parallelogram

• Rhombus

• Rectangle

• Square

Answer explanation

Quadrilateral ABCD has opposite sides equal, diagonals that bisect each other, and all angles are 90°. These properties define a rectangle, making it the correct choice.

Which statement guarantees that the quadrilateral is a parallelogram?

• One pair of opposite angles are congruent

• One pair of opposite sides are congruent and parallel

• One pair of opposite sides are congruent

• One pair of consecutive angles are congruent

Answer explanation

The statement 'One pair of opposite sides are congruent and parallel' guarantees that the quadrilateral is a parallelogram, as it satisfies the definition of a parallelogram where both pairs of opposite sides are equal and parallel.

Consider the three coordinates of a quadrilateral: A(2, 3), B(6, 3), and C(7, 7). Which of the following could be the fourth coordinate (D) of the parallelogram ABCD? Select three that apply.

• D(3, 7)

• D(1, 5)

• D(11, 7)

• D(1, −1)

Answer explanation

To form a parallelogram, the diagonals must bisect each other. The midpoints of AC and BD must be the same. D(3, 7) and D(11, 7) maintain this property, as does D(1, -1). Thus, these are valid fourth coordinates.

Consider the rectangle ABCD, where the diagonals intersect at point E. If AB = 15 and BE = 8.5, what is the length of AD?

• 21.4

• 10.25

• 13

• 8

Answer explanation

In rectangle ABCD, diagonals bisect each other at E. Since BE = 8.5, then AE = 8.5 as well. Thus, AD = AB - AE = 15 - 8.5 = 6.5. However, since AD is a side, it must equal BE, which is 8.

Consider rectangle QRST, where the diagonals of the rectangle intersect at point U. If QU = 3x + 15 and TU = 7x − 17, what is the length of the diagonal RT?

• 39

• 78

• 8

• 16

Answer explanation

Since QU and TU are segments of the diagonals and they are equal, set them equal: 3x + 15 = 7x - 17. Solving gives x = 8. Thus, QU = 39 and TU = 39. The length of diagonal RT is 2 * QU = 78.

Consider the parallelogram (a)   . If the perimeter of the parallelogram is (b)   , what is the length of (c)   ? Answer: (d)   = (e)  

• 5 units

• 10 units

• 15 units

• 20 units

Answer explanation

In a parallelogram, the perimeter is calculated as 2 times the sum of the length and width. Given the perimeter, we can find either the length or width, making both options correct.

Given a rectangle FGHJ, which of the following statements are NOT necessarily true? Select two.

• A. FH is congruent to GJ

• B. FH is perpendicular to GJ

• C. FG is parallel to HJ

• D. FJ is parallel to GH

• E. Rectangle FGHJ is not a parallelogram.

Answer explanation

Statement B is not necessarily true because while FH and GJ are sides of the rectangle, they are not guaranteed to be perpendicular in all configurations. Statement E is false since all rectangles are parallelograms.

In parallelogram ABCD shown below, m∠DAC = 98° and m∠ACD = 36°. a) Which sides of the parallelogram ABCD are parallel? b) What is the measure of angle B? Explain why.

• AB and CD

• AD and BC

• All sides are equal

• None of the above

Answer explanation

In parallelogram ABCD, sides AB and CD are parallel by definition. To find angle B, we use the fact that opposite angles are equal. Since m∠DAC = 98°, angle B also measures 98°.