Math Exam Study Notes PDF

three undefined terms in geometry: point, line, plane point: simplest figure, is a location, no length or width, use capital letter to name point space: the set of all points line: made up of two or more points, any 2 points form a line, a line has no thickness or width, name a line by two points...

three undefined terms in geometry: point, line, plane point: simplest figure, is a location, no length or width, use capital letter to name point space: the set of all points line: made up of two or more points, any 2 points form a line, a line has no thickness or width, name a line by two points on the line or a lowercase script letter Plane; a plane is a 2d figure made up of points, can name by any 3 noncollinear points on the plane or an uppercase script letter , a plane extends indefinitely, non collinear: points that do no lay on the same plane (must be at least 4 points)​ collinear: points that lie on the same plane ray: a figure with one inpoint that continues indefinitely in one direction opp ray: 2 collinear rays with a common endpoint that go in opposite directions angle: a figure formed by two non collinear rays with a common endpoint two rays are called the sides of the angle a common endpoint is the vertex adjacent angles: 2 angles that have a common vertex and a common side, but no common interior points vertical angles; two nonadjacent angles formed by intersecting lines linear pair: two adjacent angles whose non common sides form opposite rays parallel lines: two lines in the same plane that do not intersect intersecting lines: two lines that have one point in common skew lines; two lines not on the same plane that do not intersect transversal: any line or segment that intersects two or more lines at different points. a transversal divides the plane into different pairs of angles alternate interior angles: interior angles, non adjacent, and on opposite sides of the transversal. congruent. alternate exterior angles: exterior angles, non adjacent, and on opposite sides of the transversal. congruent. consecutive interior angles: interior angles that are on the same side of the transversal. supplementary. corresponding angles: one interior angle, one exterior angle on the same side of the transversal. also congruent if 2 lines are cut by and transversal & alt. int. angle are congruent, then the lines are parallel if 2 lines are cut by and transversal & corresp. angle are congruent, then the lines are parallel if 2 lines are cut by and transversal & alt ext angle are congruent, then the lines are parallel if 2 lines are cut by and transversal & consecutive int. angle are congruent, then the lines are parallel if 2 lines are cut by and transversal & perpendicular to two lines, the lines are parallel polygon: a closed figure made up of sides (segments) and angles triangle: a three sides polygon legs: the two sides adjacent to the right angle hypotenuse: the side opposite to the right angle parts of an isos angle- base: the side opposite to the vertex angle legs: the two congruent sides vertex angle: the angle formed by the two legs base angles: the two angles formed by the base and the legs triangle sum theorem: the sum of the interior angles of a triangle is 180 right triangle corollary: in a right triangle, the two acute angles are complementary angle corollary: there can be at most one obtuse or one right angle in a triangle exterior angle: an angle formed by extending the side of the triangle exterior angle theorem: the measure of the exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles isos triangle theorem: in an isos triangle, the two angles opposite to the congruent sides are congruent congruent triangles: triangles with the same size and shape CPCTC: corresponding parts of congruent triangles are congruent SSS: if 3 sides of a triangle are congruent to 3 sides of another triangle, the triangles are congruent SAS: if two sides and the included angle of one triangle is congruent to 2 sides and the included angle of another triangle, then the two triangles are congruent ASA: if 2 angles and the included side of one triangle are congruent to two angles and included side of another triangle, then the triangles are congruent AAS: if two angles and a non included side of one triangle are congruent to two angles and a non included side if another triangle, the the triangles are included hyp leg: if the hyp and a leg of one right triangle is congruent to the hyp and leg of another right triangle, the the triangles are congruent ratio: a comparison of 2 quantities, can represent with fractions, two, and colon :, ratios can be simplified extended ratio: a comparison of more than 2 quantities, written with colon : proportion: an equation that states 2 ratios are equal. written with fraction similar polygons: polygons with the same shape but different sizes polygons are similar if- corresponding angles are congruent( matching angles are congruent) corresponding sides are proportional The ratio of the corresponding side sides is called the scale factor (ratios of corresp. sides in simplest form) if polygons are sim. then their perimeters are also proportional triangle proportionality theorem : if a line is parallel to one side of a triangle and intersects the other two sides, then it divides the sides into segments of proportional lengths midsegment: a segment that joins the midpoints of two sides of a triangle triangle midsegment theorem: if a segment is a midsegment, then it is parallel to the third side and its measure is ½ the length of the third side proportional parts and parallel lines: if three or more parallel lines intersect two transversals, then they cut off the transversals proportionally angle bisector theorem: the angle bisector in a triangle divides the side into parts that are proportional to their corresponding side. special segment theorem: if two triangles are similar, then the corresponding median (mid point), altitude (perpendicular right angle) and angle bisector are proportional to the corresponding sides parallelogram: a quadrilateral with two pairs of opposite sides parallel. arrows sign parallel rectangle: a parallelogram with four right angles diagonals divide a rectangle into four isos. triangles rhombus: a parallelogram that has four congruent sides square: a parallelogram with 4 congruent sides and 4 right angles properties: since a square has four congruent sides, it must be a rhombus. Since a square has four right angles, it must be a rectangle. a square has the properties of a parallelogram, rectangle, and rhombus. the diagonals form four cong. isos, right triangles trapezoid: a quadrilateral with 1 pair of parallel sides isos trapezoid: a trap. with 2 cong. legs properties of isos trap: non ll legs are cong. diagonals are cong. The base angle are cong. opp angles are supp. parallel: two lines are parallel if and only if their slopes are the same perpendicular: two lines are perp iaoi their slopes have a product of -1, aka opp reciprocals trig: trig. is the study of triangle measurements. a trigonometric ratio: is the ratio of the lengths if the sides of a right triangle. the tree most common trigonometric ratios are sine, cosine, and tangent law of sines and cosines: when a triangle does not have a right angle, we can use the law of sines or law of cosine to solve. sines:​ an angle and opposite side pair any 2 angles and ant 1 side cosine: all sides no angles any 2 sides with an included angle( sas) circle: the set of all points in a plane that are equidistant from a given point. given point is at the center of the circle radius: a seg. that has one endpoint at the center of the circle and one endpoint on the circle chord: a segment that has both endpoints on the circle diameter: a chord that passes through the center of a circle arc of the chord: the arc whose endpoints are also endpoints of the chord arc of the chord theorem: in a circle, if the diameter is perp to a chord, then it bisect the chord and it bisects the arc in a circle, or in congruent circles, two chords are cong. iaoi they are the same distance from the center of the circle tangent: a tang. line intersects a circle at exactly one point called the point of tangency if 2 seg from the same ext point are tangent to a circle, then they are cong a polygon is circumscribed around a circle, then all sides are tangent a tangent line intersects a circle at exactly one point a secant line intersects a circle at exactly 2 points in a figure, the height must always be per to the base of the figure the area of the shaded region is the whole minus the parts(unshaded) the area of the irregular region is the sum of the pieces when we combine two or more figures together we call this a composite figure The height of a triangle is also called an altitude. look for a right angle in a trap. The bases have different lengths so we must use their average. we call this the median The perimeter of a circle is called its circumference. regular polygo: polyg that has all sides congruent and all angles congruent a 3 sided regular polygon is an equilateral triangle a 4 sides regular polygon is a square apothem: a seg from the center of the polygon that is perpendicular to a side of the polygon. it also bisect that side into 2 cong, parts perimeter: the total distance around the outside of a two dimensional figure, found by doing the sum of all sides Area: the amount of space in the inside of a two dimensional figure, found by using the product surface area: the amount of space on the outside of a three dimensional figure, found by doing the sum of the area of each face of the figure volume: the amount of space on the inside of a three dimensional figure, found by multiplying the area of the base times the height of the figure right prism: a three dimensional figure that has two opposite faces that are the same size and shape. these two faces are called the bases and are parallel to each other the faces connecting the bases are all rectangular in shape and are perpendicular to the two bases we name the right prism byt the shape of its circles regular pyramid: a three dimensional figure composed of 1 base that is a regular polygon (congruent sides) anf triangular faces that intersect at the same point the segment that represents the height of the pyramid is perpendicular to the base and intersect at the center of the polygon (used for volume) the segment that represents the height of the triangular faces is called the slant height (used for surface area) cone: a pyramid with a circular base sphere: a round three dimensional figure where every point on the surface is an equal distance to the center hemisphere: a three dimensional figure that is ½ of a sphere. it is made up of one flat side and a curved side midsegment: a segment that joins the midpoints of two sides of a triangle median: s segment that connects a vertex of a triangle to the midpoint of the opposite sides(every triangle has three medians) centroid: the point at which the three medians intersect in a triangle centroid theorem: the distance from the centroid to any of the vertices of a triangle is ⅔ the length of the median altitude: a segment that connects to a vertex of a triangle to the opposite sides so that its perpendicular to that side orthocenter: the point at which the three altitudes intersect in a triangle perpendicular bisector: a line, segment, or ray that divides a segment into two congruent segments and is perpendicular to the segment circumcenter: the point at which the three perpendicular bisectors intersect at a triangle perpendicular: any point on the perpendicular bisector of a triangle is equidistant to the vertices of the triangle incenter: the point at which the three angle bisectors intersect in a triangle angle bisector theorem: any point on the angle bisector of a triangle is equidistant to the sides of the triangle triangle inequality theorem: the sum of the length of any twp sides is greater than the length of the third side