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
- 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.
- If there are any other items you think should be on this list,
please contact me.
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
- 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.
- 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
- 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
- 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
- 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
- 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
- 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.
- Opposite sides of a parallelogram are congruent.
- The diagonals of a parallelogram bisect each
- The diagonals of a rhombus are perpendicular.
- 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
- 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.
- How to expand a product of expressions and
- The quadratic formula.
- How to solve equations and inequalities in
standard/reasonable cases. (This is necessarily vague; if in doubt, ask.)
- 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.
- Standard formulas for derivatives and integrals
- Differentiation formulas for sum, product,
- Chain rule
- Derivative tests for finding maxima and mimima
- Definition of derivative and integral
- The area under a curve is represented by a
- Use of derivatives for speed, acceleration.
- Use of derivatives for testing for increasing, decreasing, points
of inflection, concavity