C972 College Geometry
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Free C972 College Geometry Questions
What is the relationship between vertical angles?
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Vertical angles form a linear pair
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Vertical angles are supplementary
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Vertical angles are congruent
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Vertical angles are complementary
Explanation
Vertical angles are formed when two lines intersect, creating opposite pairs of angles. These opposite angles always have the same measure because they are created by the same intersecting lines. This equality in measure means the angles are congruent. Their relationship is based on position and symmetry rather than forming a straight line or adding to a specific total.
What does the Angle Addition Postulate state?
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The distance between two points can only be found using the Pythagorean Theorem
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Vertical angles are congruent
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If two angles are supplementary, they form a linear pair
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If G is in the interior of an angle, then the measures of the smaller angles add up to the measure of the larger angle
Explanation
The Angle Addition Postulate states that if a point lies in the interior of an angle, then the measure of the whole angle is equal to the sum of the measures of the two smaller angles created by that point. This postulate allows for the calculation of unknown angles when an angle is divided by a ray or line segment within it. It is a fundamental principle in geometry used to break down and analyze angles in various constructions and proofs.
What are collinear points?
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Points on different lines
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Points on the same plane
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Points on the same line
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Points that intersect
Explanation
Collinear points are points that lie on the same straight line. This geometric relationship shows that a single line can pass through all of the points without deviation. Collinearity is important when studying lines, segments, and coordinate geometry because it describes alignment and helps determine if points share the same linear path. Points can lie on the same plane without being collinear, so the plane alone does not guarantee collinearity.
In teaching coordinate proofs, to prove a quadrilateral is a parallelogram, what vector condition must hold?
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AB = DC and AD = BC
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Diagonals bisect
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Midpoints of diagonals coincide
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All of the above
Explanation
A quadrilateral is a parallelogram if any of several equivalent conditions are satisfied. Vectors provide a convenient method: opposite sides must be equal (AB = DC and AD = BC). Additionally, the diagonals of a parallelogram bisect each other, meaning the midpoints of the diagonals coincide. Since all these conditions are equivalent and confirm that the quadrilateral is a parallelogram, "all of the above" is the correct choice.
In a trig identities lesson without calculations, students use unit circle diagrams. sin²θ + cos²θ =1 represents what property?
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Pythagorean identity on unit circle
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Angle addition
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Double angle
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Law of sines
Explanation
On the unit circle, any point corresponding to angle θ has coordinates (cosθ, sinθ). By the Pythagorean theorem applied to the radius of the unit circle (r = 1), the relationship sin2 θ + cos2 θ= 1 holds. This is the Pythagorean identity, a fundamental property of trigonometric functions derived geometrically from the unit circle.
Surface area of a cylinder with r =3, h =5 (exclude bases)?
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2π35 = 30π
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2π9 + 2π35
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π35
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2π3*(3+5)
Explanation
The lateral surface area of a cylinder (excluding bases) is calculated using the formula A=2πrh. Substituting r = 3 and h = 5: A = 2π(3)(5) = 30π. This formula represents the area of the curved surface of the cylinder only, without including the top and bottom circular bases.
A teacher introduces 3D geometry with everyday objects. The dihedral angle is best described as the angle between what?
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Two intersecting planes
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Edge and face
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Vertex and edge
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Volume measures
Explanation
A dihedral angle is the angle formed between two intersecting planes in three-dimensional space. It is measured along the line where the planes meet, which is their common edge. Using everyday objects helps students visualize this concept by observing angles formed by, for example, book covers or boxes.
For a circle with radius 5 cm, what is the length of a 72° arc?
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(72/360)2π5 = 2π cm
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72π/5 cm
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5π cm
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(72/360)π25 = (1/5)π*25 = 5π cm
Explanation
How many units is the radius of the circle in Circle Sector C1?
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4
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120º
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18 units
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3
Explanation
The radius of a circle is the distance from the center of the circle to any point on its circumference. It is measured in linear units, not degrees, because degrees measure angles, not lengths. Based on the provided options, the correct choice identifies the linear measurement corresponding to the circle’s radius. This length defines the size of the circle and is essential for calculations involving circumference, area, and sectors.
What is the relationship between two angles that are both adjacent and whose noncommon sides are opposite rays?
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They form a linear pair
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They are congruent
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They are vertical angles
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They are complementary
Explanation
Two angles that are adjacent and whose noncommon sides form opposite rays create a straight line together, which defines a linear pair. According to the Linear Pair Postulate, the measures of these two angles add up to 180 degrees, making them supplementary. The defining characteristic is their position along a straight line rather than their individual measures being equal or forming a right angle.
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