{"id":256,"date":"2017-08-23T04:33:24","date_gmt":"2017-08-23T04:33:24","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/elementary-education\/?page_id=256"},"modified":"2017-09-25T20:41:47","modified_gmt":"2017-09-25T20:41:47","slug":"two-and-three-dimensional-figures","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/elementary-education\/two-and-three-dimensional-figures\/","title":{"rendered":"Two- and Three-Dimensional Figures"},"content":{"rendered":"
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Workshop Index<\/a>\u00a0Next Lesson \u27a1<\/a><\/div>\n

<\/p>\n

Two- and Three-Dimensional Figures<\/h1>\n

Objective<\/h4>\n

In this lesson you will classify and describe 2- and 3-dimensional figures, and identify parallel and perpendicular lines. You will use properties of these figures to construct proofs and draw conclusions about relationships among them.<\/p>\n

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Geometry: What\u2019s the Point?<\/h3>\n

The most basic geometric figure is a point<\/abbr>. Because it has neither length nor width, it is 0-dimensional. In fact, every geometric figure is just a collection of points. You may recall that a line segment (usually just called a \u201csegment\u201d) is the set of all points on a line that lie between two points, called the endpoints. A collection of points extending forever in one direction from an endpoint is called a ray<\/abbr>. Extend it in both directions and you get a line<\/strong>. A line segment has one dimension, length. Extend the line perpendicular to its length and you get a 2-dimensional plane<\/abbr>. Extend the plane perpendicular to its surface and you get a 3-dimensional space.<\/p>\n

\"Plane,<\/center>In this section we\u2019ll look at two-dimensional figures. These are often called planar figures because they lie entirely within a plane.<\/p>\n

Angles: Defined and Classified<\/h3>\n

An angle<\/strong> is formed by two rays that share an endpoint. The common endpoint is called the vertex<\/strong> of the angle. This is B. The interior and exterior of the angle are also labeled.<\/p>\n

\"Angles\"<\/center><\/p>\n

Angles are classified by their measures.<\/p>\n\n\n\n\n\n\n
\"Acute<\/td>\nAn acute angle measures less than 90\u00b0.<\/td>\n<\/tr>\n
\"Right<\/td>\nA right angle measures 90\u00b0. The small square at the vertex of this angle indicates that the angle is a right angle.<\/td>\n<\/tr>\n
\"Obtuse<\/td>\nAn obtuse angle measures between 90\u00b0 and 180\u00b0.<\/td>\n<\/tr>\n
\"Straight<\/td>\nA straight angle is formed by two opposite rays. It measures 180\u00b0.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Some Angles Travel in Pairs<\/h3>\n

If the sum of the measures of two angles is 90\u00b0, then the angles are complementary<\/strong>. If their sum is 180\u00b0, then the angles are supplementary<\/strong>.<\/p>\n

\"Compelementary<\/center>\"Angle<\/p>\n
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Question<\/h4>\n

What is the measure of the angle that is supplementary to \u22202 above?<\/p>\n

    \n
  1. 35\u00b0<\/li>\n
  2. 45\u00b0<\/li>\n
  3. 125\u00b0<\/li>\n
  4. 135\u00b0<\/li>\n<\/ol>\n

    Reveal Answer<\/a><\/p>\n

    The correct answer choice is C. Two angles are supplementary if the sum of their measures is 180\u00b0. So the angle that is supplementary to \u22202 measures 180\u00b0- 55\u00b0, or 125\u00b0. Choice A is incorrect. An angle that measures 35\u00b0 is complementary to \u22202.<\/p>\n<\/section>\n

    Recall that two lines are parallel if the distance between them is constant. That is, the distance stays the same. Also recall that perpendicular lines form right angles. There are marks in the figure below that show lines \"Line and \"Line are parallel. The right angle symbol shows lines \"Line and \"Line are perpendicular.<\/p>\n

    What can you conclude about lines \"Line and \"Line? If two lines are parallel, and one of these lines is perpendicular to a third line, then the other parallel line is also perpendicular to the third line. That means that \"Line and \"Line are perpendicular.<\/p>\n

    \"Parallel<\/center>In the figure above, \u2220ABC and \u2220DBF are vertical<\/abbr> angles. Pairs of vertical angles are formed where two lines intersect. \u2220GFH and \u2220BFE are also vertical angles. Can you find two other pairs of vertical angles in the figure?<\/p>\n

    Vertical angles are congruent.<\/em> That means that they have the same measure. We know that \u2220ABC and \u2220DBF are vertical angles, and \u2220ABC is a right angle, so we can conclude that \u2220DBF = 90\u00b0.<\/p>\n

    Practice, Practice, Practice:<\/h3>\n
    \n

    Question<\/h4>\n

    If m\u2220ABC = 40\u00b0, what is the measure of \u2220BED?<\/p>\n

    \"Angle<\/center><\/p>\n
      \n
    1. 40\u00b0<\/li>\n
    2. 45\u00b0<\/li>\n
    3. 50\u00b0<\/li>\n
    4. 60\u00b0<\/li>\n<\/ol>\n

      Reveal Answer<\/a><\/p>\n

      The correct answer is C. Because m\u2220ABC = 40\u00b0 and \u2220ABC and \u2220DBE are vertical angles, then m\u2220DBE = 40\u00b0. Because the sum of the measures of the angles in a triangle is 180\u00b0, then the measure of \u2220BED is 180\u00b0 – (40\u00b0 + 90\u00b0), or 50\u00b0.<\/p>\n<\/section>\n

      Constructions<\/h3>\n

      \"Compass<\/center>These tools are a typical compass<\/abbr> and straight edge. They are the tools that you and your students will use to construct simple geometric figures by hand. A compass is used for drawing arcs and circles, and for copying lengths. The straight edge is used only to draw straight lines. While a ruler is typically used as a straight edge, it will not be used to measure lengths during constructions.<\/p>\n

      Given a line segment, how would you construct the midpoint<\/abbr> and perpendicular bisector<\/abbr> of that segment using a compass and a straight edge?<\/p>\n

      Follow these steps:<\/p>\n\n\n\n\n\n\n
      1.<\/td>\nBegin with line segment XY.<\/td>\n\"Table<\/td>\n<\/tr>\n
      2.<\/td>\nPlace the compass at point X. Adjust the compass radius so that it is more than half of XY. Draw two arcs as shown here.<\/td>\n\"Table<\/td>\n<\/tr>\n
      3.<\/td>\nWithout changing the compass radius, place the compass on point Y. Draw two arcs intersecting the previously drawn arcs. Label the intersection points A and B.<\/td>\n\"Table<\/td>\n<\/tr>\n
      4.<\/td>\nUsing the straight edge, draw line AB. Label the intersection point M. Point M is the midpoint of line segment XY, and line AB is perpendicular to line segment XY.<\/td>\n\"Table<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n\n\n\n\n
      Given a point on a line, how would you construct a line perpendicular to the given line that passes through the given point?<\/th>\n<\/tr>\n
      1.<\/td>\nBegin with a given line k that contains point P.<\/td>\n\"Table<\/td>\n<\/tr>\n
      2.<\/td>\nPlace the compass on point P. Using any radius, draw arcs intersecting line k at two points. Label the intersection points X and Y.<\/td>\n\"Table<\/td>\n<\/tr>\n
      3.<\/td>\nPlace the compass at point X. Adjust the compass radius so that it is more than half of XY. Draw an arc as shown here.<\/td>\n\"Table<\/td>\n<\/tr>\n
      4.<\/td>\nWithout changing the compass radius, place the compass on point Y. Draw an arc intersecting the previously drawn arc. Label the intersection point A.<\/td>\n\"Table<\/td>\n<\/tr>\n
      5.<\/td>\nUse the straight edge to draw line AP. Line AP is perpendicular to line k.<\/td>\n\"Table<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n\n\n\n
      Given an angle, how would you construct the bisector of that angle?<\/th>\n<\/tr>\n
      1.<\/td>\nLet point P be the vertex of the given angle. Place the compass on point P and draw an arc across both sides of the angle. Label the intersection points Q and R.<\/td>\n\"Table<\/td>\n<\/tr>\n
      2.<\/td>\nPlace compass on point Q, measure distance QR, and from the point Q using that distance draw an arc on the interior of the angle QPR.<\/td>\n\"Table<\/td>\n<\/tr>\n
      3.<\/td>\nWithout changing the radius of the compass, place it on point R and draw an arc intersecting the one drawn in the previous step. Label the intersection point W.<\/td>\n\"Table<\/td>\n<\/tr>\n
      4.<\/td>\nUsing the straight edge, draw ray PW. This is the bisector of \u2220QPR.<\/td>\n\"Table<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n\n\n\n
      Given a line segment, how would you construct an equilateral triangle with sides congruent to the given line segment?<\/th>\n<\/tr>\n
      1.<\/td>\nBegin with line segment TU.<\/td>\n\"Table<\/td>\n<\/tr>\n
      2.<\/td>\nCenter the compass at point T, and set the compass radius to TU. Draw an arc as shown.<\/td>\n\"Table<\/td>\n<\/tr>\n
      3.<\/td>\nKeeping the same radius, center the compass at point U and draw another arc intersecting the first one. Let point V be the point of intersection of the two arcs.<\/td>\n\"Table<\/td>\n<\/tr>\n
      4.<\/td>\nDraw line segments TV and UV. Triangle TUV is an equilateral triangle.<\/td>\n\"Table<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n\n\n\n\n
      Given a line and a point not on the line, how would you construct a line parallel to the given line through the given point?<\/th>\n<\/tr>\n
      1.<\/td>\nBegin with the givens, point P and line k.<\/td>\n\"Table<\/td>\n<\/tr>\n
      2.<\/td>\nDraw any line through point P, intersecting line k. Call the intersection point Q. Now we\u2019ll construct an angle with vertex P, congruent to the angle of intersection.<\/td>\n\"Table<\/td>\n<\/tr>\n
      3.<\/td>\nCenter the compass at point Q and draw an arc intersecting both lines. Without changing the radius of the compass, center it at point P and draw another arc.<\/td>\n\"Table<\/td>\n<\/tr>\n
      4.<\/td>\nSet the compass radius to the distance between the two intersection points of the first arc. Now center the compass at the point where the second arc intersects line PQ. Mark the arc intersection point R.<\/td>\n\"Table<\/td>\n<\/tr>\n
      5.<\/td>\nLine PR is parallel to line k.<\/td>\n\"Table<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n\n\n\n\n\n
      How would you construct a triangle from three given line segments?<\/th>\n<\/tr>\n
      1.<\/td>\nBegin with the givens, segments BC, AC, and AB.<\/td>\n\"Table<\/td>\n<\/tr>\n
      2.<\/td>\nDraw a segment AX longer than any of the given segments.<\/td>\n\"Table<\/td>\n<\/tr>\n
      3.<\/td>\nSet the compass radius to the length of segment AB, the longest segment. Center the compass at point A and draw an arc that intersects segment AX. Mark the intersection point B.<\/td>\n\"Table<\/td>\n<\/tr>\n
      4.<\/td>\nSet the compass radius to the length of segment AC. Center the compass at point A and draw an arc that intersects segment AX.<\/td>\n\"Table<\/td>\n<\/tr>\n
      5.<\/td>\nSet the compass radius to the length of segment BC. Center the compass at point B and draw an arc that intersects segment AX twice. Mark the intersection of this arc and the arc that was constructed in step 4 point C.<\/td>\n\"Table<\/td>\n<\/tr>\n
      6.<\/td>\nConstruct triangle ABC.<\/td>\n\"Table<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Review<\/h3>\n