Questions and Answers
Are vertical angles congruent?
Are definitions reversible?
Are theorems reversible?
Are opposite rays collinear rays with a common endpoint and extend in the same direction?
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Are supplementary angles congruent?
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If two angles are complementary, are they each acute?
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Is the supplement of an acute angle acute?
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Are the bisectors of vertical angles opposite rays?
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Are vertical angles complementary?
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Does an obtuse triangle have three obtuse angles?
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Is a right triangle isosceles?
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Is the base of an isosceles triangle shorter than either leg?
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Does a median of a triangle bisect the side to which it is drawn?
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If a median of a triangle is also an altitude, is it also an angle bisector?
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Does a triangle have three sides?
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Is any angle congruent to itself?
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If all pairs of corresponding angles of two triangles are congruent, are the two triangles congruent?
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Does a median form a right angle with the opposite side of the triangle?
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Does one point determine a line?
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If ∆SCA is scalene, is SC ≅ CA?
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Is a right triangle equilateral?
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If two angles are both complementary and congruent, are they right angles?
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Is a line segment the shortest path between two points?
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If two points are each equidistant from the endpoints of a segment, do they determine the perpendicular bisector of that segment?
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Are two lines that do not intersect parallel?
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Do alternate interior angles have a common vertex?
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Are the slopes of perpendicular lines opposites?
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If the base of an isosceles triangle is on the x-axis, are the slopes of its legs opposites of each other?
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Are alternate exterior angles acute?
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Is a kite a parallelogram?
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Does a convex polygon have at least one interior angle greater than 180°?
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Is a rectangle a square?
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If a parallelogram is equiangular, is it a rectangle?
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If a parallelogram is equilateral, is it a rhombus?
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Does a quadrilateral have five sides?
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Are the diagonals of a trapezoid congruent?
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Does a polygon have the same number of sides as vertices?
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If two lines are cut by a transversal, are the alternate interior angles congruent?
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Can only one side of a parallelogram have a slope of three?
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Study Notes
Vertical Angles and Definitions
- Vertical angles are always congruent, meaning that they are equal in measure.
- Definitions are always reversible, allowing for flexibility in understanding concepts.
Theorems and Rays
- Theorems are sometimes reversible, depending on the specific theorem.
- Opposite rays are never collinear rays that extend in the same direction; they diverge from their common endpoint.
Angles and Triangles
- Supplementary angles are sometimes congruent, such as in special cases.
- If two angles are complementary, they are always acute, meaning they measure less than 90 degrees.
- The supplement of an acute angle is never acute; it is always obtuse or equal to 90 degrees.
Additional Properties of Angles
- The bisectors of vertical angles are always opposite rays, forming straight angles.
- Vertical angles can sometimes be complementary, depending on their measures.
Triangle Characteristics
- An obtuse triangle never has three obtuse angles; it can only have one obtuse angle.
- A right triangle is sometimes isosceles, but it's not a requirement.
- The base of an isosceles triangle is sometimes shorter than either leg, depending on its configuration.
Median and Triangle Properties
- A median of a triangle always bisects the side to which it is drawn, creating two equal segments.
- If a median of a triangle is also an altitude, it is always an angle bisector, ensuring balanced triangles.
Basic Triangle Facts
- A triangle always has three sides, a fundamental characteristic of its definition.
- Any angle is always congruent to itself, highlighting the reflexive property of equality.
Congruence and Angles
- If all pairs of corresponding angles of two triangles are congruent, then the triangles are sometimes congruent under specific conditions.
- A median sometimes forms a right angle with the opposite side of the triangle, depending on the triangle's type.
Points, Lines, and Segments
- One point never determines a line; at least two points are needed for a line to exist.
- A line segment is always the shortest path between two points, adhering to geometric principles.
Perpendicular Bisectors and Parallel Lines
- If two points are equidistant from the endpoints of a segment, those points always determine the perpendicular bisector of the segment.
- Two lines that do not intersect are sometimes parallel; they can be skew lines in three-dimensional space.
Angles and Slopes
- Alternate interior angles never share a common vertex, as they lie on opposite sides of a transversal.
- The slopes of perpendicular lines are sometimes opposites, indicating a right angle between them.
Properties of Isosceles Triangles
- If the base of an isosceles triangle is positioned on the x-axis, the slopes of its legs are always opposite of each other.
- Alternate exterior angles are sometimes acute, varying with the specific angles involved.
Polygon and Quadrilateral Properties
- A kite is sometimes a parallelogram, depending on its specific properties.
- A convex polygon never has an interior angle greater than 180 degrees, defining its convexity.
- A rectangle is sometimes a square, only when all sides are of equal length.
Parallelograms and Equiangular Properties
- If a parallelogram is equiangular, it is always a rectangle, ensuring right angles at all corners.
- If a parallelogram is equilateral, it is always a rhombus, as all sides are equal in length.
Quadrilateral and Polygon Definitions
- A quadrilateral never has five sides; it is defined specifically as having four sides.
- The diagonals of a trapezoid are sometimes congruent, depending on the nature of the trapezoid.
Polygon Characteristics
- A polygon always has the same number of sides as vertices, which is a fundamental characteristic of polygonal shapes.
- If two lines are cut by a transversal, alternate interior angles are sometimes congruent, based on their positions relative to the transversal.
Slope of Parallelograms
- Only one side of a parallelogram can never have a slope of three, as the slopes of opposite sides are equal in a parallelogram.
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Test your understanding of geometric concepts with these flashcards. Each card presents a statement about geometric properties, asking whether it is always, sometimes, or never true. Perfect for students looking to reinforce their knowledge in geometry.