Math 460 Cheat Sheet, Study notes of Algebra

Download Math 460 Cheat Sheet and more Study notes Algebra in PDF only on Docsity! Math 460 “Cheat Sheet” Basic Facts (BF1) SSS: Three sides determine a triangle up to congruence. (BF2) SAS: Two sides and an included angle determine a triangle up to congruence. (BF3)ASA: Two angles and an included side determine a triangle up to congruence. BF4: Ratios of corresponding sides for two similar triangles are the same. (The definition of similar is that the angles are the same.) BF5: If two lines are crossed by a transversal, then: if the lines are parallel the corresponding angles are the same; if two corresponding angles are the same, lines are parallel. BF6: Lengths, angles and areas add up. BF7: Through two points there is exactly one line. BF8: On a ray there is exactly one point at a given distance from the endpoint. BF9: A line segment extends to a line. (Line segments finite, lines are infinite in both directions.) BF10: Line segments have midpoints. BF11: Angles have bisectors. BF12: It is possible to find line perpendicular to a given line through a given point. BF13: It is possible to fine a line parallel to a given line through a point not on the line. BF14: Two lines parallel to a third line are parallel to each other. BF15: The area of a rectangle is base times height. Some Major Theorems Theorem 1: When two lines cross, adjacent angles add up to 180 degrees. Vertical angles are equal. Theorem 2: Suppose that two lines l and m are crossed by a transversal. a) l and m are parallel if and only if alternate interior angles are equal. b l and m are parallel if and only if each pair of interior angles add up to 180 degrees. Theorem 3: Sum of angles of a triangle are 180 degrees. Theorem 5: Opposite sides of triangle are equal if and only if opposites angles are equal. (Such a triangle is isoceles.) Theorem 7: The area of triangle is one half base times height. Theorem 8: If ∆ABC ∼ ∆DEF and the ratio AB/DF = r, then the area of the first triangle is r2 times the area of the second. Theorem 9: (Pythagorean theorem). The square of the hypotenuse of a right angle triangle is the sum of the squares of the sides. Theorem 10: If two right triangles have the hypotenuse and leg matching, then they are congruent. Theorem 11,12: Given a parallelogram (which means opposite sides are parallel), the opposites sides (thm 11) and angles (thm 12) are equal. Theorem 13: If a pair of sides are equal and parallel, then it’s a parallelogram. Theorem 14: A quadralateral is a parallelogram if and only if diagonals bisect each other. Theorem 17, 18: (Thm 17) In a triangle ABC, let D be a midpoint of AC and suppose E is a point of BC with DE parallel to AB. Then E is a midpoint of BC and DE = AB/2. (Thm 18) Conversely, if E is a midpoint of BC, then DE is parallel to AB and DE = AB/2. Theorem 19, 20: In triangles ABC abd DEF, (Thm 19) if 6 C = 6 F and AC/DF = BC/EF or (Thm 20) if AB/DE = AC/DF = BC/EF , then they are similar. Theorem 22: The area of a triangle ABC is 1 2AB · AC · sin 6 A. (If 6 A is part of right triangle, then sin 6 A is given by opposite/hypotenuse.) Theorem 23: In a triangle ABC, sin 6 A BC = sin 6 B AC = sin 6 C AB Theorem 24: The perpendicular bisectors of a triangle are concurrent. (The point where they meet is the circumcenter.) Theorem 25. Given a triangle with circumcenter O, suppose that a circle with center O that goes through one of the vertices of the triangle. Then it also goes through the other two vertices. Theorem 26: The angle bisectors of a triangle are concurrent. (The point where they meet is called the incenter.) Theorem 27: The altitudes of a triangle are concurrent. (The point where they meet is called the orthocenter.) 1