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Correct Answer
B. All rectangles are quadrilaterals
Explanation The statement "All rectangles are quadrilaterals" is true because a rectangle is a type of quadrilateral that has four sides and four angles. A quadrilateral is a polygon with four sides. Since a rectangle satisfies the definition of a quadrilateral, it can be concluded that all rectangles are quadrilaterals.
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2.
Identify the quadrilateral which has one pair of parallel sides.
A.
Kite
B.
Trapezoid
C.
Rectangle
D.
Rhombus
Correct Answer
B. Trapezoid
Explanation A trapezoid is a quadrilateral that has one pair of parallel sides. This means that two sides of the trapezoid are parallel while the other two sides are not. Therefore, the correct answer is trapezoid. A kite has two pairs of consecutive sides that are equal in length, a rectangle has four right angles and opposite sides that are equal in length, and a rhombus has four sides of equal length.
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3.
If a quadrilateral is a parallelogram, then its consecutive angles are ?
A.
Complementary
B.
Congruent
C.
Supplementary
D.
Equal
Correct Answer
C. Supplementary
Explanation If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. This is because in a parallelogram, opposite sides are parallel and opposite angles are congruent. Since consecutive angles are the ones that share a side, they are also opposite angles. Therefore, they are congruent and their measures add up to 180 degrees, making them supplementary.
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4.
Which of these descriptions would NOT guarantee that the figure was a square?
A.
A parallelogram with perpendicular diagonals
B.
A quadrilateral with all right angles and all sides congruent
C.
Both a rectangle and a rhombus
D.
A quadrilateral with all sides and all angles congruent
Correct Answer
A. A parallelogram with perpendicular diagonals
Explanation A parallelogram with perpendicular diagonals does not guarantee that the figure is a square because a rectangle can also have perpendicular diagonals. Therefore, this description can apply to both squares and rectangles, making it not a guarantee of a square.
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5.
Which of these descriptions would NOT guarantee that a figure was a parallelogram?
A.
A quadrilateral with consecutive angles supplementary
B.
A quadrilateral with the diagonals bisecting each other
C.
A quadrilateral with two opposite sides congruent
D.
A quadrilateral with one pair of opposite sides both parallel and congruent
Correct Answer
C. A quadrilateral with two opposite sides congruent
Explanation A quadrilateral with two opposite sides congruent does not guarantee that the figure is a parallelogram because a rhombus also has two opposite sides congruent, but it is not a parallelogram. A parallelogram requires that both pairs of opposite sides are congruent.
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6.
Isosceles trapezoid ABCD has legs AB and CD and base BC. If AB = 3y - 9, BC =8y - 2 and CD = 7y - 10, find the value of y.
Correct Answer .25 1/4
Explanation The given trapezoid is isosceles, which means that the lengths of the legs AB and CD are equal. Therefore, we can set up an equation: 3y - 9 = 7y - 10. Solving this equation, we get y = 1/4 or 0.25. Therefore, the value of y is 0.25 or 1/4.
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7.
Parallelogram JKLM (pt O is diagonal intersection), angle KLO = 80, angle MLO = 40, find the measure of angle KJM
Correct Answer 120
Explanation In a parallelogram, opposite angles are equal. Since angle KLO is 80 degrees and angle MLO is 40 degrees, angle KJM must also be 80 degrees in order for the opposite angles to be equal. However, since the sum of the angles in a triangle is 180 degrees, angle KJM must be 180 - 80 = 100 degrees. Therefore, the measure of angle KJM is 100 degrees, not 120 degrees.
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8.
The four properties of the diagonals of squares are
A.
Congruent to each other, congruent to the sides, bisect both sets of angles, bisect each other
B.
Bisect each other, bisect one set of angles, are congruent to each other, are perpendicular to each other
C.
Are congruent to each other, bisect both sets of angles, are perpendicular to each other, are twice the length of the sides
D.
Are perpendicular to each other, bisect each other, bisect both sets of angles, are congruent to each other
Correct Answer
D. Are perpendicular to each other, bisect each other, bisect both sets of angles, are congruent to each other
Explanation The answer states that the properties of the diagonals of squares are that they are perpendicular to each other, bisect each other, bisect both sets of angles, and are congruent to each other. This means that the diagonals of a square intersect at a right angle, divide each other into equal segments, divide the angles of the square into equal parts, and have the same length. These properties are unique to squares and help define their geometric characteristics.
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9.
Rhombus MATH, angle HMA = 100, find the measures of angles MTH and MAT
Correct Answer 50 and 80
Explanation In a rhombus, opposite angles are equal. Since angle HMA is given as 100 degrees, angle MTH is also 100 degrees. Additionally, the sum of the angles in a rhombus is always 360 degrees. Since angle HMA is 100 degrees, the sum of angles MAT and MTH must be 360 - 100 = 260 degrees. Therefore, angle MAT is 260 - 100 = 160 degrees. Thus, the measures of angles MTH and MAT are 100 degrees and 160 degrees respectively.
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10.
LMNP is a rectangle. Find the value of x and the length of each diagonal.
LN = 5x - 8 and MP = 2x + 1
Correct Answer 3, 9.9
Explanation In a rectangle, the diagonals are equal in length. So, we have:
LN = MP
Substituting the given expressions for LN and MP, we get:
5x - 8 = 2x + 1
Solving this equation for x, we get:
5x - 2x = 8 + 1
3x = 9
x = 9 / 3 = 3
Now, substituting x = 3 into the expressions for LN and MP, we get:
LN = 5(3) - 8 = 7
MP = 2(3) + 1 = 7
So, the length of each side of the rectangle is 7 units.
The length of the diagonal (D) of a rectangle can be found using the Pythagorean theorem:
D = sqrt(LN^2 + MP^2)
Since LN = MP = 7, we get:
D = sqrt(7^2 + 7^2) = sqrt(98) ≈ 9.9 units
So, the length of each diagonal of the rectangle is approximately 9.9 units.
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