1.
Through any two points there is exactly one line
Correct Answer
A. True
Explanation
This statement is known as the "two-point form" of a line. It states that for any two distinct points, there exists exactly one line that passes through both of them. This is a fundamental property of lines in Euclidean geometry. It implies that two points uniquely determine a line, and any additional point on that line can be expressed as a linear combination of the two given points. Therefore, the correct answer is true.
2.
In a right-angled triangle, the hypotenuse is always shorter than both of the other two sides.
Correct Answer
A. True
Explanation
The correct answer is False. In a right-angled triangle, the side opposite the right angle, known as the hypotenuse, is always the longest side. This is in accordance with the Pythagorean Theorem, which states that the square of the hypotenuse's length is equal to the sum of the squares of the other two sides. Therefore, the hypotenuse is never shorter than the other two sides; it's always longer.
3.
If two lines intersect, then they intersect in two places
Correct Answer
B. False
Explanation
The statement "If two lines intersect, then they intersect in two places" is false. When two lines intersect, they intersect at a single point. If two lines were to intersect at more than one point, they would no longer be considered lines, but rather overlapping or coinciding lines. Therefore, the correct answer is false.
4.
It is possible for two planes to intersect in exactly one point.
Correct Answer
B. False
Explanation
Two planes can intersect in exactly one point if they are not parallel and if they are not the same plane. However, if the planes are parallel, they will never intersect or if they are the same plane, they will intersect in an infinite number of points. Therefore, it is not always possible for two planes to intersect in exactly one point, making the answer false.
5.
Three points always lie in exactly one plane
Correct Answer
B. False
Explanation
Three points do not always lie in exactly one plane. If the three points are collinear (lying on a straight line), then they do lie in one plane. However, if the three points are not collinear, they can define multiple planes. Therefore, the statement is false.
6.
A plane is defined as an infinite set of points in a flat surface.
Correct Answer
A. True
Explanation
A plane is a two-dimensional surface that extends infinitely in all directions. It is composed of an infinite number of points that lie on a flat surface. This definition aligns with the statement that a plane is defined as an infinite set of points in a flat surface, making the answer true.
7.
Congruent angles are angles that have equal measures.
Correct Answer
A. True
Explanation
Congruent angles are angles that have the same measure. This means that if two angles are congruent, their measures are equal. Therefore, the statement "Congruent angles are angles that have equal measures" is true.
8.
Definitions may be used as reasons in a proof.
Correct Answer
A. True
Explanation
Definitions can be used as reasons in a proof because they provide clear and precise meanings for terms and concepts. By using definitions, we can establish a common understanding of the terms being used and ensure that our reasoning is based on accurate and agreed-upon definitions. This helps to avoid ambiguity and allows for logical and coherent arguments to be made. Therefore, definitions can serve as a solid foundation for constructing a proof.
9.
A statement in the form, If p, then q, is a biconditional.
Correct Answer
B. False
Explanation
A statement in the form "If p, then q" is not a biconditional. A biconditional statement is a statement that is true if both the conditional statement "If p, then q" and its converse "If q, then p" are true. Therefore, the correct answer is False.
10.
The converse of a true conditional is always true.
Correct Answer
B. False
Explanation
The converse of a true conditional is not always true. In a conditional statement, if p implies q, the converse is q implies p. Just because the original statement is true, it doesn't mean that the converse will also be true. The truth value of the converse depends on the specific statements being considered.
11.
The converse of a false conditional is always false.
Correct Answer
B. False
Explanation
The converse of a false conditional is not always false. The converse of a conditional statement is formed by switching the hypothesis and conclusion. If the original conditional is false, it means that the hypothesis is true and the conclusion is false. Therefore, when we switch them in the converse, the hypothesis will be false and the conclusion will be true. Hence, the converse of a false conditional can be true.
12.
Perpendicular lines form congruent adjacent angles
Correct Answer
A. True
Explanation
Perpendicular lines are two lines that intersect at a right angle. When two lines are perpendicular, they form congruent adjacent angles. This means that the angles formed on either side of the intersection point are equal in measure. Therefore, the statement "Perpendicular lines form congruent adjacent angles" is true.
13.
Two angles complementary to the same angle are complementary to each other.
Correct Answer
B. False
Explanation
This statement is false. Two angles that are complementary to the same angle are not necessarily complementary to each other. Complementary angles are two angles that add up to 90 degrees. So, if two angles are each complementary to the same angle, it does not mean that they add up to 90 degrees when combined. Therefore, they may or may not be complementary to each other.
14.
Two angles congruent to the same angle are congruent to each other
Correct Answer
A. True
Explanation
This statement is true because congruent angles have the same measure. If two angles are congruent to the same angle, it means that they have the same measure as that angle. Therefore, the two angles must also have the same measure as each other, making them congruent to each other.
15.
Angles formed by a transversal and two parallel lines are either complementary or congruent.
Correct Answer
B. False
Explanation
This statement is false. Angles formed by a transversal and two parallel lines can have various relationships, including being supplementary, congruent, or neither. The angles can also be alternate interior angles, alternate exterior angles, corresponding angles, or vertical angles. Therefore, the statement that the angles are either complementary or congruent is incorrect.
16.
Parallel and skew lines are coplanar.
Correct Answer
B. False
Explanation
Parallel lines are coplanar, meaning they lie on the same plane. However, skew lines do not lie on the same plane, they are in different planes and do not intersect. Therefore, the statement that parallel and skew lines are coplanar is false.
17.
If vertical angles are acute, the adjacent angle to them must be obtuse.
Correct Answer
A. True
Explanation
Vertical angles are formed when two lines intersect. By definition, vertical angles are congruent, meaning they have the same measure. If vertical angles are acute, it means they have a measure less than 90 degrees. Since vertical angles are congruent, both angles must have a measure less than 90 degrees. If one of the vertical angles is acute, the other angle must also be acute. Therefore, the adjacent angle to the vertical angles must be obtuse, as the sum of the measures of adjacent angles is always 180 degrees. Hence, the statement is true.
18.
The sum of exterior angles of a polygon always add up to 180 degrees.
Correct Answer
B. False
Explanation
The sum of exterior angles of a polygon always add up to 360 degrees, not 180 degrees. Each exterior angle of a polygon is formed by extending one side of the polygon and the sum of all exterior angles is always equal to 360 degrees.
19.
Regular, in terms of polygons, means that all sides are equal.
Correct Answer
B. False
Explanation
Regular polygons do have all sides equal, but the statement is incorrect because it states that regular, in terms of polygons, means all sides are equal. In reality, regular refers to polygons that have all sides and angles equal. So, while having equal sides is a characteristic of regular polygons, it is not the only requirement. Therefore, the correct answer is False.
20.
Opposite sides of a rectangle must be parallel
Correct Answer
A. True
Explanation
In a rectangle, opposite sides are always equal in length and parallel to each other. This is one of the defining properties of a rectangle. Therefore, it is true that opposite sides of a rectangle must be parallel.
21.
The diagonals of a rhombus must be perpendicular.
Correct Answer
A. True
Explanation
A rhombus is a quadrilateral with all sides of equal length. Since the opposite sides of a rhombus are parallel, the diagonals are also perpendicular to each other. This can be proven using the properties of a rhombus, such as the fact that opposite angles are equal and the diagonals bisect each other. Therefore, the statement that the diagonals of a rhombus must be perpendicular is true.
22.
Consecutive angles of a rhombus are always complementary.
Correct Answer
B. False
Explanation
The statement is false because consecutive angles of a rhombus are not always complementary. In a rhombus, opposite angles are equal, but they are not necessarily complementary. Complementary angles are two angles that add up to 90 degrees, while in a rhombus, the sum of two consecutive angles is always 180 degrees. Therefore, the statement is incorrect.
23.
The diagonals of a rectangle are always perpendicular
Correct Answer
B. False
Explanation
The statement that the diagonals of a rectangle are always perpendicular is false. While it is true that the diagonals of a square, which is a special type of rectangle, are always perpendicular, this is not the case for all rectangles. In a general rectangle, the diagonals can have any angle between them, depending on the dimensions of the rectangle. Therefore, the statement is not universally true for all rectangles.
24.
Opposite sides of a parallelogram must be congruent
Correct Answer
A. True
Explanation
In a parallelogram, opposite sides are parallel and congruent. This is because a parallelogram is a quadrilateral with both pairs of opposite sides parallel. Since parallel lines have the same length, the opposite sides of a parallelogram must be congruent. Therefore, the given statement is true.
25.
Each diagonal of a rectangle always bisects a pair of opposite angles
Correct Answer
A. True
Explanation
In a rectangle, the diagonals are lines that connect opposite corners. Since a rectangle has four right angles, each diagonal divides the rectangle into two congruent right triangles. In each of these triangles, the diagonal bisects the pair of opposite angles, meaning it divides each angle into two equal parts. Therefore, it can be concluded that each diagonal of a rectangle always bisects a pair of opposite angles, making the answer true.