chapter 1
What is midpoint?
A point that divides the segment into two congruent segments.
what is a theorem? How is it different from a postulate?
A theorem is a statement the can be proven; a postulate is a rule that is accepted without proof.
Complete the statement with sometimes, always,or never.
If two lines are not coplanar, then they ? intersect.
never
What is the triangle sum theory?
The sum of the measure of the interior angles of a triangle is 180 degrees.
Complete the statement with always, sometimes, and never.
If the perpendicular bisector of a triangle goes through the opposite vertex, then the triangle is ? isosceles.
Always; congruent sides are created.
<a and <b are complementary. Find the m<a and the m<b.
m<a=(2x-20)
m<b=(3x+5)
m<a=58,m<b=122
Explain why writing a proof is an example of deductive reasoning, not inductive reasoning.
A proof is deductive reasoning because it uses facts, definitions, accepted properties and laws of logic.
Write an equation of the line that passes through the given point P and has the given slope m.
P(5,4), m=4
y=4x-16
If (0,3) translates to (1,2), then (2,5) translates to ?
(3,4)
What is the angle bisector theorem?
If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle.
Find the coordinates of the midpoint of the segment with the given endpoints.
C(3,5) and D(7,5)
(5,5)
rewrite the conditional statement in if-then form.
Two angles are supplementary if they are a linear pair.
if two angles are a linear pair then they are supplementary.
Tell which line through the given points is steeper.
Line 1:(-4,2),(-3,6)
Line 2:(1,6),(3,8)
line 1
the angle between two sides of a triangle is called the ? angle
included
Is it possible to build a triangle using the given side lengths? If so, list the angles of the triangle in order from least to greatest measure.
PQ=√58, QR=2√13, PR=5√2
yes;<Q,<P,<R
The expressions 3x-9 and 23-5x represent the lengths(in feet) of two sides of an equilateral triangle. Find the length of a side.
3ft
Consider the pattern 5,5r,5r2,5r3,....For what values of r will the values of the numbers increase? For what values of r will the values of the numbers be decreasing?Explain.
r>1;0<r1; raising numbers greater than one by successive natural number increases the result while raising a number between 0 and 1 by a successive natural numbers decrease the result.
Two lines are cut by a transversal. Suppose the measure of a pair of alternate interior angles is 90 degrees. Explain why the measure of all four interior angles must be 90 degrees.
The adjacent interior angles are supplementary thus the measure of the other two angles are must be 90 degrees.
Given that △ABC⩭ △DEF, m<A=70 degrees, m<B=60 degrees, m<C=50 degrees, m<D=(3x+10),m<E=(y/3+20), and m<f=(z2+14), find the values of x, y, z.
x=20, y=120, z=6
What is the hinge theorem?
If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second.
let x be an angle measure. let y1 be the measure of a complement of the angle and let y2 be the measure of a supplement of the angle.
write equations for y1 as a function of x, and for y2 as a function of x. what is the domain of each function? Explain.
y1=90-x,0<x<90; y2=180-x, 0<x<180; the measure of a complement must be less than 90 degrees and the measure of the supplement must be less than 180 degrees.
Use inductive reasoning to form a conjecture about whether the sum of an even integer is even or odd.Then use deductive reasoning to show that the conjecture is true. Let the integer be 2m and the odd integer be 2n+1.
Conjecture:The sum of an even integer and an odd integer is odd.
Reasoning: 2m+(2n+1)=(2m+2n)+1=2(m+n)+1, which is odd.
Slope can be used to describe an average rate of change. to write an average rate of change, rewrite the slope fraction so the denominator is one.
In 2000, a business made a profit of $8500. In 2006, the business made a profit of 15,400. Find the average rate of change in dollars per year from 2000 to 2006.
$1150 per year
Suppose you have the equations y=ax+b, y=cx+d, and y=ex+f. A. When will these three lines form a triangle? B. Draw the triangle formed when a= 4/3, b=1/3,c=-4/3, d=41/3,e=0, and f=-1. Classify the type of triangle it is.
A. a, c, and e would have to have different slopes. this is because the lines are parallel.
B. isosceles
Suppose you are given a triangle and are asked to draw all of its perpendicular bisectors and angle bisectors.
A. For what type of triangle would you need the fewest segments? What is the minimum number of segments you would need? Explain.
B. For what type of triangle would you need the most segments? What is the maximum number of segments you would need? Explain.
A. Equilateral; 3; the angle bisector would also bee the perpendicular bisector.
B. Scalene; 6; each angle bisector would be different than the corresponding perpendicular bisector.