BASIC GEOMETRY GLOSSARY

Transcription

1 BASIC GEOMETRY GLOSSARY Acute angle An angle that measures between 0 and 90. Examples: Acute triangle A triangle in which each angle is an acute angle. Adjacent angles Two angles next to each other that share a common vertex and a common side. ABC is adjacent to CBD.

2 Alternate exterior angles - A pair of angles that are exterior to the lines and on alternate sides of the transversal. 1 and 8 and 2 and 7 are alternate exterior angles. Alternate interior angles - A pair of angles that are interior to the lines but on alternate sides of the transversal. 3 and 6 and 4 and 5 are alternate interior angles.

3 Altitude (1) Height. (2) The perpendicular segment from the vertex of a triangle to the line that contains the opposite side. Angle Two rays that have a common endpoint. Examples:

4 Angle bisector A ray that divides an angle into two equal adjacent angles. Angle of depression The angle formed by the horizontal and the line of sight to an object below the horizontal.

5 Angle of elevation - The angle formed by the horizontal and the line of sight to an object above the horizontal. Angle of incidence - The angle formed by a ray incident on a surface and a perpendicular to the surface at the point of incidence. The angle of incidence = the angle of reflection. Where i is the angle of incidence and r is the angle of reflection.

6 Angle of reflection - The angle formed by a reflected ray and a perpendicular to the surface at the point of reflection. The angle of reflection = the angle of incidence. Where i is the angle of incidence and r is the angle of reflection. Area - The amount of square units that covers a given surface. ASA (1) Suppose that we have two triangles ABC and DEF. If a side of ABC and two angles that have this side of ABC as one of their sides are equal to the corresponding side and two angles of the triangle DEF, then triangles ABC and DEF are equal. The proof for the above is: Given: AC CD, B D. Prove: VABC VEDC Proof: Statement Reason AC CD Given B D Given ACB ECD Vertical Angles VABC VEDC ASA

7 (2) If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar, known as AA similarity. Base The face or lower side of a geometric shape. Base angle of an isosceles triangle The angle that is opposite of one of the equilateral sides of an isosceles triangle. There are two base angles in an isosceles triangle.

8 Betweenness of points - In the figure, Point B is between A and C while Point Y is not between A and B. For B to be between A and C, all three points must be collinear and B must lie on segment AC. Centroid Where the three medians of a triangle intersect. Also called the center of gravity.

9 Circle Set of all points in a plane a given distance from a given point. Circumcenter - The intersection of the three perpendicular bisectors of a triangle. Circumference The distance around a circle. The formula for circumference is: c = 2π r or c = π d, where r = radius and d = diameter.

10 Circumscribed A polygon is circumscribed around a circle if all the sides of the polygon are tangent to the circle. Collinear - In the same line. Points A, B, C, D and E are collinear while F is not. Complementary angles Two angles whose sum measures 90. Where EBD and DBC are complementary angles.

11 Concave polygon A polygon that has at least one interior angle greater than 180 and has some of its sides bent inward. Examples: Concentric circles Circles that have different radii but share the same center.

12 Cone A solid figure that has a circle for its base and tapers to a point. Congruent - Having the same size and shape. Where V ABC is congruent to V DEF. Congruent angles - Two angles that have the same measure. Written as A B.

13 Congruent polygons - Two polygons that are equal in shape and size. Written as A B. Congruent segments - Two segments that are equal in length. Written as A B. Converse A reversed condition. For example: if a b, then its converse is b a. Converse of the Pythagorean theorem This states that if the sum of the squares of the two shorter sides of a triangle equals the square of the longest side of the triangle, then the triangle is a right triangle. This is written as: c 2 = a 2 + b 2. Convex polygon A polygon in which each interior angle is less than or equal to 180. Examples: Coplanar Within the same plane.

14 Corresponding angles Two non-adjacent angles on the same side of the transversal, with one angle interior and one angle exterior to the lines. Where 2 and 6 are corresponding angles. Cosine - The ratio of the length of the side that is adjacent to an angle to the length of the hypotenuse in a right triangle. cosb (cosine of B ) = AB, where AB is the length of the adjacent sides and BC is the BC length of the hypotenuse.

15 CPCTC When you have two congruent triangles, then all six pairs of corresponding parts (sides and angles) are congruent. This statement is usually known as corresponding parts of congruent triangles are congruent, or CPCTC for short. For the above, if VABC V XYZ then AB ZY, BC YX, AC XZ, A Z, B Y and C X. Cube - A square prism that has six equal square sides. Examples: Cylinder - A solid with circular ends and straight sides.

16 Diagonal A line segment joining two non-adjacent vertices of a polygon. Diameter The distance across a circle through its center. The diameter is AB. Dilation A transformation that enlarges or reduces a figure.

17 Distance The distance between two points A and B is written as AB. Dodecahedron A solid figure with 12 regular pentagon faces. Examples: Equilateral triangle A triangle with all sides congruent.

18 Equiangular triangle A triangle that has angles with the same measurement. Exterior angles The angles that are on the outer sides of two lines cut by a transversal. Where 1, 2, 7 and 8 are exterior angles. Heptagon - A seven-sided polygon. The sum of the angles is 900.

19 Hexagon - A six-sided polygon. The sum of the angles is 720. Hexagonal prism A prism composed of two hexagonal faces and six parallelograms. Hypotenuse of a right triangle - The side of a right triangle that is opposite the right angle.

20 Icosahedron A solid figure with 20 equilateral triangle faces. Examples: Incenter - The intersection of the bisectors of the three angles in the triangle ABC.

21 Included angle The angle formed by two sides of a polygon. Where A is an included angle formed by sides a and d, B is an included angle formed by sides a and b, C is an included angle formed by sides b and c, and D is an included angle formed by sides c and d. Included side The side that is between two angles in a polygon. Where a is an included side formed by A and B, b is an included side formed by B and C, c is an included side formed by C and D, and d is an included side formed by D and A. Inscribed - A polygon is inscribed in a circle if all its vertices are on the circle.

22 Interior angles Angles that are on the inner sides of two lines cut by a transversal. Where 3, 4, 5 and 6 are interior angles.

23 Intersecting lines Two lines that cross at only one point. Inverse Reciprocal The opposite of the reciprocal of x is Examples: 2 is the inverse reciprocal of is the inverse reciprocal of 2 1. x Isosceles trapezoid A quadrilateral with one pair of sides parallel with at least two sides the same length. Isosceles triangle A triangle with at least two congruent sides.

24 Kite A quadrilateral that has two distinct pairs of consecutive equilateral sides. Legs of a right triangle - Either of the two sides that form a right angle of a right triangle. Legs of an isosceles triangle - One of the two congruent sides in an isosceles triangle.

25 Line A set of points that are perfectly straight and extend forever. Can be named by a lowercase letter or by two points on the line. If naming suur by two points on the line, a double-headed arrow is used over the two letters, ex. AB. suur Written as line XY or XY. Linear pair Two supplementary adjacent angles that form a line with their noncommon sides. Line of symmetry A line that separates a figure into two congruent, or identical, parts.

26 Examples: The dashed lines are the lines of symmetry. Median (1) The median of a trapezoid is parallel to the bases, and its measure is onehalf the sum of the measures of the bases. The median of a trapezoid is the segment that joins the midpoints of its legs, as shown below. (2) The segment from the vertex in a triangle to the midpoint of the opposite side. Median = ( ) 2 = 7.5 Midpoint - The point that divides a line into two equal parts. Where M is the midpoint of AB. Non-collinear Not in the same line.

27 Where points A, B and D are non-collinear. Noncoplanar Not within the same plane. Obtuse angle An angle that has a measure of greater than 90 but less than 180. Obtuse triangle A triangle that has one obtuse angle. Octagon - An eight-sided polygon. The sum of the angles is 1080.

28 Octahedron A solid figure with eight equilateral triangle faces. Examples: Orthocenter - The intersection point of the three altitudes of a triangle. Parallel lines Lines in the same plane that do not intersect.

29 suur suur Written as AB P XY. Parallelogram A quadrilateral with both pairs of opposite sides parallel. Pentagon - A five-sided polygon. The sum of the angles is 540. Perimeter - The distance around a figure.

30 The perimeter of this figure is: 2cm + 2cm + 3cm = 7cm. Perpendicular bisector A line, or line segment, that intersects a given line segment at its midpoint and forms right angles. suur Line XY is the perpendicular bisector of segment AB. Written as XY is the perpendicular bisector of AB. Perpendicular lines Lines that intersect to form right angles.

31 suur suur Line XY is perpendicular to line AB. Written as XY AB. Plane - A set of points that form a flat surface that extends without end in all directions. Pi - Written π ; it is the ratio of the circumference to the diameter of a circle. Circumference/Diameter. Also rounded to Point Generally represented by a dot, but have no size. Use capitol letters to name them. Polyhedron - A three-dimensional solid that consists of polygons, usually joined at their sides.

32 Polygon - A closed plane figure that is formed by joining three or more line segments at their endpoints. Examples: Prism A solid figure that has two bases that are parallel, congruent polygons and with all other faces that are parallelograms. Examples: Pyramid A solid figure with a polygon base and with all other faces that are triangles that share a common vertex. Examples: Pythagorean Theorem - For a right triangle a 2 + b 2 = c 2, where a and b are the lengths of the triangle s legs and c is the length of the triangle s hypotenuse. Pythagorean Triples - A set of three whole numbers that can be side lengths of a right triangle. 3, 4 and 5, where c is the greatest number. Quadrilateral - A polygon with four sides. The sum of the angles is 360. Examples:

33 Radius A line segment that is drawn from one point on the circle to the center of the circle. Radius is AB. Ray A line segment that has one endpoint and goes on forever in only one direction. uuur Ray AB, written as AB. Reciprocal One of two numbers that have a product of 1. The reciprocal of x is 1 x. Examples: 2 is the reciprocal of is the reciprocal of 2 Rectangle - A quadrilateral with four right angles. The sum of the angles is 360. Examples:

34 Rectangular prism (1) A solid figure that with two bases that are rectangles and with all other faces that are parallelograms. (2) A prism in the shape of a rectangle. Examples: Reflection - The figure formed by flipping a geometric figure about a line to obtain a mirror image.

35 Regular polygon A polygon whose sides are equal and whose angles are equal. Examples: Rhombus A polygon with four congruent sides. The sum of the angles is 360. Examples: Right angle An angle whose measure is 90.

36 Right Prism A prism that has two special characteristics: all lateral edges are perpendicular to the bases and all lateral faces are rectangular. Right triangle - A triangle that has a right angle. Rotation Turning a geometric figure about a fixed point.

37 Same-side interior angles Interior angles on the same side of a transversal. Where 3 and 5, 4 and 6 are same-side interior angles. SAS (1) Suppose that we have two triangles ABC and DEF. If two sides and an angle of ABC are equal to two sides and an angle of DEF, then the triangle ABC is equal to the triangle DEF. The proof for the above is: Given: AC CD, BC CE. Prove: VABC VEDC Proof: Statement Reason AC CD Given BC CE Given ACB ECD Vertical Angles VABC VEDC SAS

38 (2) If the measures of two sides of a triangle are proportional to the measures of two corresponding sides of another triangle and the included angles are congruent, then the triangles are similar. 4 5 = 8 10 Scalene triangle A triangle with no congruent sides. Examples: Segment - Line segment AB is a part of line x between points A and B including these points.

39 Segment bisector Any line, segment or ray that intersects a segment at its midpoint. Line X is a segment bisector. Similar polygons Polygons that have the same shape, but not necessarily the same size.

40 Sine The ratio of the length of the side opposite an angle to the length of the hypotenuse in a right triangle. sinb (sine of B ) = AC, where AC is the length of the opposite side and BC is the BC length of the hypotenuse. Skew lines - Lines are skew if they do not lie in the same plane. Slope of a line - The steepness of a line. Sphere - A solid figure that has all points on the surface the same distance from the center.

41 Square A quadrilateral with four right angles and four congruent sides. The sum of the angles is 360. Square prism - A solid figure that with bases that are squares and with all other faces that are parallelograms. SSS (1) If three sides of the triangle ABC are equal to three sides of the triangle DEF, then triangles ABC and DEF are equal. The proof for the above is: Given: AC CD, AB BD. Prove: VABC VDBC Proof: Statement Reason AC CD Given AB BD Given

42 BC CB Common Side VABC VDBC SSS (2) If the measures of the corresponding sides of two triangles are proportional, then the triangles are similar = = Supplementary Two angles are said to be supplementary if the sum of their measures is 180. Supplementary angles Two angles whose measure equals 180. m ACD + m DCB = = 180 Surface area The sum of all the areas of the surfaces of a solid figure.

43 Tangent (1) The ratio of the length of the side opposite an angle to the length of the side adjacent to the angle in a right triangle. tanb (tangent of B) = AC, where AC is the length of the opposite side and AB is the AB length of the adjacent side. (2) A line is tangent to a circle if it intersects the circle in exactly one point. suur AB is tangent to the circle at point C.

44 Tetrahedron A solid with four equilateral triangles as faces. Transformation - A change in size, shape, or position of a geometric figure. Translation When you move a geometric figure to a new position without turning or flipping it. Transversal A line that intersects two or more lines. Trapezoid A quadrilateral with exactly one pair of opposite sides parallel. The sum of the angles is 360. Examples:

45 Trigonometry Mathematics dealing with triangular measurement. Triangle - A three-sided polygon. The sum of the angles is 180. Examples: Triangular prism A prism composed of two triangular faces and three parallelograms. Vertex The common endpoint of two rays that form an angle. Examples:

46 Vertex angle of an isosceles triangle The angle that is formed in the isosceles triangle where the two congruent sides (legs) meet. Also known as the angle opposite the base of the isosceles triangle. Vertical angles Angles of the same measure that form two intersecting lines. ACB and DCE are vertical angles. Vertices The points in a figure where the lines meet. Volume - The number of cubic units needed to occupy a given space