1.1.2: Homework
- Page ID
- 197464
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Skills Refresher
The following is a set of review exercises you will need for this section.
For the following exercises, solve the equation.
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\(x-8=19-2 x\)
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\(2 x-9=12-x\)
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\(13 x+5=2 x-28\)
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\(4+9 x=-7+x\)
For the following exercises, solve the system.
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\( \begin{cases}
5x - 2y & = & -13 \\
2x + 3y & = & -9 \\
\end{cases}\) -
\( \begin{cases}
4x + 3y & = & 9 \\
3x + 2y & = & 8 \\
\end{cases}\)
- Answers
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\(9\)
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\(2\)
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\(-3\)
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\(-2\)
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\(x=-3,y=-1\)
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\(x=6,y=-5\)
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Homework
Reading Questions
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What is a "ray"?
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How is the "vertex" of an angle defined?
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What distinguishes a "positive angle" from a "negative angle"?
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How many degrees constitute a full revolution?
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What is an "obtuse angle"?
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What are "supplementary angles"?
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According to the theorem presented, what is true about "vertical angles"?
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What does a "transversal" line do?
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What is the formula for the circumference of a circle in terms of its radius, \( r \)?
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If a circle has a diameter \( d \), how is its radius \( r \) related to \( d \)?
Vocabulary Check
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(MyOpenMath) When two rays share a common initial point they form a(n) ___.
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(MyOpenMath) Trigonometry focuses on oriented angles. An oriented angle starts from an ___ side and ends at a ___ side.
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(MyOpenMath) ___ angles start at an initial side and rotate clockwise to the terminal side.
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(MyOpenMath) A positive angle opens with a ___ rotation.
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We define ___ measure by dividing the full rotation around the circle into 360 segments.
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One-quarter of a complete revolution around a circle forms a ___ angle, which measures ___ degrees.
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___ angles measure between \( 0^{ \circ } \) and \( 90^{ \circ } \), while ___ angles measure between \( 90^{ \circ } \) and \( 180^{ \circ } \).
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The degree system for measuring the size of an angle is also known as the ___ system.
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When two lines intersect, the non-adjacent angles formed from this intersection are called ___ angles.
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A line intersecting two parallel lines is called a ___.
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The set of all points equidistant to a single point on a plane is called a ___.
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The ___ of a circle is simultaneously the largest distance between any two points on a circle and is the name of the line segment that passes through the center of the circle whose endpoints lie on the circle.
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\( \pi \) is a(n) ___ number (unlike \( 3 \), which is a natural number).
Concept Check
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Greek letters like \( \alpha \), \( \beta \), and \( \theta \) are often used to represent angles in Trigonometry. Name one Greek letter that is never used as a label for an angle.
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If a positive angle rotates from its initial to its terminal side, what direction is the rotation?
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If a negative angle rotates from its initial to its terminal side, what direction is the rotation?
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Suppose you have a straight angle and you subtract a right angle. What type of angle is the result?
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Can a right angle be acute? How about obtuse?
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Suppose the vertex of an angle is at the point \( P \), and the points \( R \) and \( S \) lie on the initial and terminal sides of the angle, respectively. Which of the following are correct notations for the angle?
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\( \angle P \)
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\( P \)
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\( m \angle P \)
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\( \triangle P \)
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\( \angle PRS \)
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\( \angle RPS \)
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\( \angle SPR \)
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Quantities measured in degrees use the ___ symbol.
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According to this text, can \( 107^{ \circ } \) have a complement? If so, what is the value of this complement?
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According to this text, can \( 107^{ \circ } \) have a supplement? If so, what is the value of this supplement?
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Can two acute angles be supplementary?
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Choose two of the eight angles formed by a pair of parallel lines cut by a transversal. Those two angles are either equal or ___.
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Write two formulas for the circumference of a circle - one involving the radius of the circle and the other involving the diameter.
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The formula for the circumference of a circle is ___.
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The formula for the area of a circle is ___.
True or False? For the following exercises, determine if the statement is true or false. If true, cite the definition or theorem stated in the text supporting your claim. If false, explain why it is false and, if possible, correct the statement.
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If \( \alpha \) and \( \beta \) are vertical angles, then \( \alpha + \beta = 90^{ \circ } \).
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(MyOpenMath) The complement to \( 47^{ \circ } \) is \( 43^{ \circ } \).
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The supplement to \( 190^{ \circ } \) is \( -10^{ \circ } \).
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Alternate interior angles sum to \( 180^{ \circ } \).
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The distance from the center to the edge of a circle is called the radius of the circle.
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The line connecting the center of a circle to its edge is called a radius.
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The circumference of a square is the sum of the lengths of its sides.
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\( \pi = 3.14 \)
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\( \pi = \frac{22}{7} \)
Basic Skills
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(MyOpenMath) State (if possible) if the angle is acute or obtuse.
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(MyOpenMath) Which is the closest measure to the angle shown below: \( 40^{\circ} \), \( 65^{ \circ } \), \( 95^{ \circ } \), or 120^{ \circ }?
For the following exercises, state (if possible) which angles are acute and which are obtuse. Give the complement and the supplement of each angle, if applicable.
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(MyOpenMath) \( 30^{ \circ } \)
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\( 45^{ \circ } \)
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\( 60^{ \circ } \)
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\( 43^{ \circ } \)
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\( 90^{ \circ } \)
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\( 120^{ \circ } \)
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\( 135^{ \circ } \)
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\( 143^{ \circ } \)
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\( 150^{ \circ } \)
For the following exercises, give the complement of each angle.
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\(60^{\circ}\)
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\(80^{\circ}\)
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\(25^{\circ}\)
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\(18^{\circ}\)
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\(64^{\circ}\)
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\(47^{\circ}\)
For the following exercises, give the supplement of each angle.
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\(30^{\circ}\)
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\(45^{\circ}\)
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\(120^{\circ}\)
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\(25^{\circ}\)
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\(165^{\circ}\)
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\(110^{\circ}\)
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(MyOpenMath) If the measure of \(\angle CAB = 62^{\circ}\), then find the measure of \( \angle CAD \).
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Which angle is supplementary to \( \angle BOC \)?
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Which angle is complementary to \( \angle BOC \)?
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What is the measure of \( \angle EOF \)?
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What is the measure of \( \angle AOE \)?
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What is the measure of \( \angle BOF \)?
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For the following exercises, use the following graph to answer the question. Assume lines \( r \) and \( s \) are parallel.
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(MyOpenMath) \( \angle 3 \) is alternate interior to which other angle(s)?
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(MyOpenMath) \( \angle 5 \) is a corresponding angle to which other angle(s)?
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(MyOpenMath) \( \angle 7 \) and what other angle(s) are vertical angles?
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(MyOpenMath) \( \angle 8 \) is alternate exterior to which other angle(s)?
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(MyOpenMath) Find \( m \angle 5 \) given that \( m \angle 1 = 46^{ \circ } \).
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(MyOpenMath) Find \( m \angle 8 \) given that \( m \angle 6 = 154^{ \circ } \).
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(MyOpenMath) Find \( m \angle 5 \) given that \( m \angle 3 = 82^{ \circ } \).
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(MyOpenMath) Find \( m \angle 3 \) given that \( m \angle 6 = 105^{ \circ } \).
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(MyOpenMath) Find \( m \angle 8 \) given that \( m \angle 3 = 76^{ \circ } \).
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(MyOpenMath) Find the remaining angles given that \( m \angle 1 = 15^{ \circ } \).
For the following exercises, arrows on a pair of lines indicate that they are parallel. Find \(x\) and \(y\).
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(MyOpenMath) The lines \( x \) and \( y \) are parallel, and are cut by transversals \( z \) and \( w \). Given \( \angle E = 57^{ \circ } \), find the measure of all remaining angles.
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(MyOpenMath) The lines \( x \) and \( y \) are parallel, and are cut by transversals \( z \) and \( w \). Given \( \angle H = 36^{ \circ } \) and \( \angle E = 40^{ \circ } \), find the measure of all remaining angles.
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Among the angles labeled 1 through 5 in the figure below, find two pairs of equal angles.
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\(\angle 4+\angle 2+\angle 5= \)_________.
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Use parts (a) and (b) to explain why the sum of the angles of a triangle is \(180^{\circ}\).
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\(A B C D\) is a rectangle. The diagonals of a rectangle bisect each other. In the figure, \(\angle A Q D=130^{\circ}\). Find the angles labeled 1 through 5 in order, and give a reason for each answer.
For the following exercises, give an exact answer, and then round your answer to hundredths.
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What is the area of a circle whose radius is 5 inches?
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What is the circumference of a circle whose radius is 5 meters?
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(MyOpenMath) What is the circumference of the following circle?
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(MyOpenMath) What is the circumference of the following circle?
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(MyOpenMath) What is the area of the following circle?
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(MyOpenMath) What is the area of the following circle?
For the following exercises, find the circumference and area for each circle having the given radius \( r \) or diameter \( d \).
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(MyOpenMath) \( r = 7 \text{ feet} \)
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\( r = 8.4 \text{ centimeters} \)
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\( d = 16 \text{ yards} \)
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\( d = 12 \text{ meters} \)
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Find the diameter of a circle having a circumference of \( 4\pi \) kilometers.
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(MyOpenMath) Find the radius of a circle having a circumference of \( 10 \) inches.
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(MyOpenMath) Find the radius of a circle having an area of \( 24\pi \) square meters.
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(MyOpenMath) Find the radius of a circle having an area of 272 square meters.
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(MyOpenMath) Find the diameter of a circle having an area of \( 64 \) square feet.
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Find the circumference of a circle having an area of \( 81\pi \) square miles.
Synthesis Questions
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(MyOpenMath) Find the area of the shaded region. Round your answer to the nearest tenth.
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(MyOpenMath) Find the area of the shaded region. Round your answer to the nearest tenth.
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(MyOpenMath) Suppose the sides of the square are 8 miles in length.
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What is the area of the square?
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What is the area of the circle? Round your answer to the nearest tenth.
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What is the area of the blue shaded region? Round your answer to the nearest tenth.
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State the condition(s) for which \( \alpha \) is an acute angle, and for which \( \alpha \) is an obtuse angle. In each case, give the complement and the supplement of \( \alpha \).
For the following exercises, arrows on a pair of lines indicate that they are parallel. Find \(x\) and \(y\).
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(MyOpenMath) The lines \( x \) and \( y \) are parallel and cut by a transversal \( z \). Given that \( \angle A = k^2 + 22k \) and \( \angle B = 14k + 48 \), find \( k \).
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Climbing Stairs. In each of the following problems, refer to the following image of a staircase and rails. Assume the rails are perpendicular to the ground, they are parallel to each other, and that the top rail (handrail) is parallel to the bottom rail (the black line along the steps).
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What is the relationship between \( \alpha \) and \( \beta \)?
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What is the relationship between \( \beta \) and \( \theta \)?
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What is the relationship between \( \beta \) and \( \phi \)?
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What is the relationship between \( \theta \) and \( \phi \)?
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Find \( \alpha \) if \( \beta = 48^{ \circ } \).
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Find \( \theta \) if \( \alpha = 30^{ \circ } \).
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Applications
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Emergency Light. An ambulance has a rotating light on its roof. The light rotates through one complete revolution every 2 seconds. How long does it take the light to rotate through \( 90^{ \circ } \)?
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Rotation of the Earth (MyOpenMath). The Earth goes through a complete circuit around the sun in approximately 365.25 days. Through how many degrees does the Earth move in one week?
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Earth's Rotation About the Sun. The Earth takes 24 hours to make a complete rotation on its axis. If your math class is 2 hours and 20 minutes, through how many degrees does the Earth turn during your math class?
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Circumference of the Earth. The radius of the Earth is approximately 3,960 miles. Using this approximation, find the circumference of the Earth.
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Earth's Rotation About the Sun. The Earth has a nearly circular orbit about the sun. The radius of this near-circular orbit is approximately 93 million miles. Find the distance the Earth travels when it completes one full orbit about the sun.
Challenge Problems
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(MyOpenMath) A straight piece of wire, \( x \) centimeters long, is bent into a perfect circle. Express the area of the circle, \( A \), in terms of \( x \).
