Unit 6 Polygons And Properties Of Quadrilaterals Test Version A

The measure of the opposite angles in a parallelogram are equal. Therefore, angle A and angle C must be equal. Setting the two expressions equal to each other, we get x + 20 = 4x - 7. Solving this equation, we find that x = 9. Therefore, x is equal to 9 degrees.

In a parallelogram, opposite sides are congruent. Since AE = 5, CE must also be 5 units long. Therefore, the measure of CE is 5.

In a parallelogram, opposite sides are equal in length. Since BE is equal to 6, and BE is opposite to BD, it means that BD is also equal to 6. Therefore, the length of BD is 12.

In the given rectangle ABCD, AC is equal to 5x and BD is equal to 20. Since opposite sides of a rectangle are congruent, AC and BD must be equal. Therefore, 5x = 20. To solve for x, we divide both sides of the equation by 5, giving us x = 4.

Since AE = 5 and DE is a side of the square, DE must also be equal to 5.

In a rhombus, opposite sides are equal in length. Therefore, MA = AT. Setting the two expressions equal to each other, we get 3x + 10 = 7x - 2. Solving for x, we subtract 3x from both sides and add 2 to both sides, giving us 12 = 4x. Dividing both sides by 4, we find that x = 3.

The measure of angle x in the given rectangle is 120 degrees.

A hexagon has six sides.

In an isosceles trapezoid, the non-parallel sides are congruent. Therefore, the length of AC is equal to the length of BD. Given that AC = 2x + 18 and BD = 5x, we can set up an equation: 2x + 18 = 5x. Solving for x, we find that x = 6. Substituting this value back into the equation for BD, we get BD = 5(6) = 30. Therefore, the length of BD is 30.

The length of AB can be found by setting the expressions for AB and CD equal to each other and solving for x. By equating 4x - 12 to 2x + 10, we can solve for x. Simplifying the equation gives 2x = 22, and solving for x gives x = 11. Plugging this value of x back into the expression for AB, we get AB = 4(11) - 12 = 44 - 12 = 32. Therefore, the length of AB is 32.

Since opposite angles in a parallelogram are congruent, angle A and angle C must be equal. Therefore, the measure of angle C is also 40 degrees.

In an isosceles trapezoid, the legs are congruent. Therefore, if leg AB is equal to 4x, then leg CD must also be equal to 4x. Since it is given that leg CD is equal to 24, we can set up the equation 4x = 24 and solve for x. Dividing both sides by 4, we get x = 6. Therefore, the value of x is 6.

In a parallelogram, opposite angles are equal. Therefore, angle B is also equal to 50 degrees. However, since the sum of the angles in a parallelogram is 360 degrees, angle B can also be calculated by subtracting angle A from 180 degrees. Thus, angle B is 180 - 50 = 130 degrees.

The measure of one exterior angle of a regular octagon is 45 degrees. In a regular octagon, all the exterior angles are congruent. Since the sum of all the exterior angles of any polygon is always 360 degrees, we can divide 360 by the number of sides (8 in this case) to find the measure of each exterior angle. Therefore, each exterior angle of a regular octagon measures 45 degrees.

An isosceles trapezoid and a rectangle have congruent diagonals in common. This means that the diagonals of both shapes have the same length. Diagonals are line segments connecting opposite corners of a shape, and in both an isosceles trapezoid and a rectangle, the diagonals have equal lengths.

A regular hexagon has six equal sides and six equal interior angles. To find the measure of one interior angle, we can use the formula (n-2) * 180 / n, where n is the number of sides of the polygon. Plugging in n=6, we get (6-2) * 180 / 6 = 4 * 180 / 6 = 720 / 6 = 120 degrees. Therefore, the measure of one interior angle of a regular hexagon is 120 degrees.

The interior angle sum of any polygon can be found using the formula (n-2) * 180, where n is the number of sides of the polygon. In the case of an octagon, which has 8 sides, the formula becomes (8-2) * 180 = 6 * 180 = 1080 degrees. Therefore, the correct answer is 1080.

In the given square, all sides are congruent, so statement III is true. The diagonals BD and AC are also congruent because they divide the square into two congruent right triangles, so statement I is true. Additionally, the diagonals BD and AC are perpendicular because they intersect at a right angle at point B, so statement II is also true. Therefore, all three statements are true, making the answer I, II, and III.

A regular octagon is a polygon with 8 sides that are all equal in length and 8 angles that are all equal in measure. This means that it is equilateral (II) as all sides are equal, equiangular (III) as all angles are equal, and convex (I) as all interior angles are less than 180 degrees and the sides do not intersect. Therefore, the correct answer is I, II, and III.

In a rhombus, the diagonals bisect each other at right angles. Therefore, in Rhombus MATH, statement II is true because MA, TA, HT, and MH are all diagonals and they bisect each other at right angles. Statement III is also true because MT is a diagonal and it is perpendicular to AH, which is another diagonal. Therefore, the correct answer is II and III only.

The length of one of the diagonals of a square can be found using the Pythagorean Theorem. In a square, the diagonal forms a right triangle with two sides that are equal to the length of the square's side. By applying the Pythagorean Theorem (a^2 + b^2 = c^2) where a and b are the lengths of the sides of the square and c is the length of the diagonal, we can calculate the length of the diagonal. In this case, since the side of the square is 4 cm, both sides of the right triangle are 4 cm. By substituting these values into the Pythagorean Theorem, we get 4^2 + 4^2 = c^2. Simplifying, we get 16 + 16 = c^2, which gives us c^2 = 32. Taking the square root of both sides, we find that c ≈ 5.7 cm.