# WHAT YOU MAY ASSUME IN SOLVING PROBLEMS

## M 360M/396C, Fall 04

You may use the facts, formulas, and theorems below in your solutions unless you are being asked to derive the formula or a special case, explain why the theorem or a special case is true, etc. If you think there is something I have left off this list that is needed to solve a problem, please ask me

1. These are  mathematical facts you may assume  in solving problems in this class. However,  be sure  in writing up  problem solutions to  help the reader know what fact you are using when.
2. If there are any other items you think should be on this list, please contact me.

Geometry

Lines and angles

• There are 180° in a straight angle.
• When two lines intersect, the vertical angles formed are congruent.
• If two parallel lines are cut by a transversal, then the alternate interior angles are congruent and the corresponding angles are congruent.
• If a transversal crosses two lines so as to make the corresponding angles (or alternate interior angles) congruent, then the two lines are parallel.
• The perpendicular bisector of a line segment is the set of all points equidistant from the endpoints of the segment.

Triangles

• The sum of the measures of the angles of a triangle is 180°.
• The Pythagorean Theorem
• The measure of an exterior angle of a triangle equals the sum of the measures of the two nonadjacent interior angles.
• The side-side-side, side-angle-side, and angle-side-angle conditions for congruence of two triangles.
• Two angles of a triangle are congruent if and only if the sides opposite those angles are congruent. (Isosceles triangle conditions)
• Two triangles are similar if and only if their corresponding angles are equal.
• Two triangles are similar if and only if their corresponding sides are all in the same ratio.
• If two triangles have two angles of one congruent, respectively, to two angles of the other, the triangles are similar.
• If two triangles have an angle of one congruent to an angle of the other and the included sides are proportional, then the triangles are similar.
• The altitude to the hypotenuse of a right triangle forms two right triangles similar to each other and to the original triangle.
• If a line is drawn between two sides of a triangle and parallel to the third side, then the new triangle formed is similar to the original triangle.
• The area of a triangle is half the product of the height and the width.
• The lenght of the hypotenuse of a 45°-45°-90° trangle is square root of two times the lenght of the sides. Added 9/9/04
• The lenght of the hypotenuse of a 30°-60°-90° triangle is twice the lenght of the short leg, and the lenght of the long leg is square root of three times the length of the short leg. Added 9/9/04
• The triangle inequality: The length of a side of a triangle is less than the sum of the lengths of the other two sides. Added 11/9/04

Circles

• Two central angles in a circle with equal intercepted arcs have equal measure.
• The ratio of the measure of a central angle of a circle to 360° is equal to the ratio of the length of the intercepted arc to the circumference. (Use 2pi instead of 360° if you are using radian measure.)
• An angle inscribed in a semicircle is a right angle.
• The ratio of the area of a sector of a circle to the area of the entire circle is equal to the ratio of the degrees in the angle of the sector to 360°. (Use 2pi instead of 360° if you are using radian measure.)
• The measure of an angle inscribed in a circle is half the
• measure of the intercepted arc, when the arc is measured in radian or degree measure.
• An angle formed by a tangent and a chord to a circle is measured by half its intercepted arc, when  the arc is measured in radian or degree measure.
• A tangent to a circle is perpendicular to the radius to the point of tangency.
• The perpendicular bisector of a chord of a circle
• lies along a diameter of the circle.
• Formulas for the area and circumference of a circle.

Parallelograms

• Opposite sides of a parallelogram are congruent.
• The diagonals of a parallelogram bisect each other.
• The diagonals of a rhombus are perpendicular.

Trigonometry

• The definitions of the trig functions, both in terms of right triangles and in terms of coordinates of points on circles.
• The relationships between the trig functions -- e.g., tan a = (sin a)/(cos a).
• The trig identities coming from the Pythagorean Theorem: sin2x + cos2x = 1, etc.
• The law of sines.
• The law of cosines.
• Addition and double angle formulas

Analytic Geometry

• The definition of slope in terms of rise and run (change in y over change in x).
• The distance formula (for finding the distance between two points in terms of their coordinates). Added 9/27/04.
• What it means for (a,b) to be on the graph of an equation (i.e., the equation is true when a and b are substituted for x and y, respectively) or on the graph of a function.
• A parabola with vertical axis is the graph of a function of the form

ax2 + bx + c.

• How translating a graph changes the equation for the graph.
• Basics of polar coordinates: what they are, and how to go back and forth between polar and rectangular coordinates.

Algebra

• How to expand a product of expressions and collect terms.
• How to solve equations and inequalities in standard/reasonable cases. (This is necessarily vague; if in doubt, ask.)

Arithmetic Concepts

• Place value concepts (e.g., the digit in the hundreds column stands for that many hundreds)
• Concepts of "divides evenly", "is a factor of", "is a multiple of", "leaves a remainder of"
• Even times even is even; even times odd is even; odd times odd is odd. Added 10/6/04.

Calculus

• Standard formulas for derivatives and integrals
• Differentiation formulas for sum, product, quotient
• Chain rule
• Derivative tests for finding maxima and mimima
• Definition of derivative and integral
• The area under a curve is represented by a certain integral
• Use of derivatives for speed, acceleration.
• Use of derivatives for testing for increasing, decreasing, points of inflection, concavity