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7.6: Polygons

  • Page ID
    51929
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    Below is a table of polygons. There are an infinite amount of polygons, but the following are the shapes taught in elementary school.

    Table 6.2.1: Polygons

    Number of Sides

    Name

    Irregular Polygon

    Regular Polygon

    3 sides

    Triangle

    clipboard_e9815a53801e983b55ed322ef83a27dad.png
    clipboard_ea3e5f4c273546fc93cec49e188606375.png

    4 sides

    Quadrilateral


    clipboard_ef97751f54ca101b86965e14f8e4c15bb.png

    clipboard_ebeeb85777811192d8b6255aedc21f1eb.png

    5 sides

    Pentagon

    clipboard_e14a6c1a69337affb6f579748a00d0ddf.png


    clipboard_e7b434027e06a8170f418c78a0b951a81.png

    6 sides

    Hexagon

    clipboard_eb1e4720be5b0e7e0de346ef74604e55d.png


    clipboard_ec1168c568a4121d01bf108d493bdedfe.png

    8 sides

    Octagon

    clipboard_e2ba5312a8ea77437e7076e6bd84ad5c6.png clipboard_e83c745f3390f704e3f952a32b851e0c8.png

    Definition: Regular Polygon

    A shape whose sides have the same length and whose angles have the same measure.

    Definition: Irregular Polygon

    A shape whose sides differ in length or have angles of different measure.

    Hierarchy of Polygons

    clipboard_ed98984a7968c270d2b9e60f8f724b1f7.png
    Figure 6.2.1: Hierarchy of Polygons

    Polygon Definitions

    Definition: Kite

    A quadrilateral with two consecutive sides having equal lengths and the other two sides also have equal lengths.

    Definition: Trapezoid

    A quadrilateral with at least one pair of opposite sides parallel.

    Definition: Isosceles Trapezoid

    A trapezoid with both angles next to one of the parallel sides having the same size.

    Definition: Parallelogram

    A trapezoid with pairs of opposite sides parallel.

    Definition: Rectangle

    A parallelogram with a right angle.

    Definition: Rhombus

    A quadrilateral with all sides being the same.

    Definition: Square

    A rectangle that has four equal sides.

    Table 6.2.2: Triangles

    Name

    Definition

    Triangle

    SIDES

    Equilateral

    All three sides are equal

    clipboard_e3df189067db6f2f13bdb92f43db4f66a.png

    Isosceles

    Only two sides are equal

    clipboard_e333f30e222860718af08ce594b4bccea.png

    Scalene

    All three sides are different in length


    clipboard_ea6fa1bf332f06943b4af87ea34fea2c5.png

    ANGLES

    Acute

    Each angle is less than 900


    clipboard_e333f30e222860718af08ce594b4bccea.png

    Right

    One angle is 900


    clipboard_e8e1e31e785bcc8132a8048ae9b7615d0.png

    Obtuse

    One angle is more than 900

    clipboard_e54ae0309e1700723b412133434271e2c.png

    Partner Activity 1

    Draw the following triangles

    1. Isosceles right triangle
    2. Scalene obtuse triangle
    3. Equilateral right triangle

    Partner Activity 2

    1. Is a rectangle a square? Is a square a rectangle?
    2. Multiple Choice: Which one is NOT a name for the figure below?
      1. Polygon
      2. Quadrilateral
      3. Parallelogram
      4. Trapezoid
    clipboard_eb1519ddd21524f926d0644bf59598f94.png
    Figure 6.2.2
    1. What is the difference between a regular and irregular polygon?

    Facts about Angles

    clipboard_e03eb67c03d712b118a0b35d7fbfc2c51.png
    Figure 6.2.3
    • Angles in a triangle add up to 1800
    • An angle forming a straight line is also 1800
    • Any quadrilateral (4-sided figure) is 3600
    • Angles which round a point add up to 3600
    • The two base angles of an isosceles triangle are equal

    Why does a triaangle add up to \(180^{\circ }\)

    A full circle is \(360^{\circ}\). Half of a circle, called a semicircle, would then be \(180^{\circ }\). The diameter (a line which passes through the center of the circle) of the semicircle is then also \(180^{\circ }\). Therefore, all straight lines are \(180^{\circ }\). See the figure below. Knowing that all straight lines are \(180^{\circ }\), we look at the figure below of the line and triangle.

    clipboard_eff2c4c12b2312cd477162aad315c6270.png
    Figure 6.2.4

    Since a line is \(180^{\circ }\), we know that angles \(A_1\), B, and \(C_1\) must add up to \(180^{\circ }\). A theorem (proven statement) in Geometry states that alternate (opposite sides) interior angles are congruent (equal). Angles \(A_1\) and \(A_2\) are alternate interior, cut by the transversal (line) connecting angle \(A_2\) to the straight line. Angles \(C_1\) and \(C_2\) follow a similar approach.
    Since the measures of angles \(A_{1}=A_{2}\), \(C_{1}=C_{2}\), and \(A_{1}+B+C_{1}=180\), then by substitution, \(A_{2}+B+C_{2}=180\). Therefore, triangle \(A_{2} B C_{2}\) adds up to \(180^{\circ }\).

    Partner Activity 3

    The sum of the interior angles of any polygon is represented by: \(180(n-2)\).

    1. Find the sum of the interior angles of a triangle, using the formula.
    2. Find the sum of the interior angles of a pentagon, using the formula.
    3. Find the sum of the interior angles of a 15-sided polygon, using the formula.
    4. What is the sum of the EXTERIOR angles of a pentagon?

    Complementary and Supplementary Angles

    Definition: Complementary Angles

    Complementary angles are any two angles with a sum of 900. See angles C and D below.

    clipboard_efbff43d5836f64a5eb4d2fec3accc75c.png
    Figure 6.2.5: Complementary angles

    Definition: Supplementary Angles

    Supplementary angles are any two angles with a sum of 1800. See angles A and B below.

    clipboard_ea5953eda36c76b93f6d3d9cef9479da6.png
    Figure 6.2.6: Supplementary angles

    Partner Activity 4

    1. You have two supplementary angles. One angle is 300. What is the measure of the other angle?
    2. One angle is complementary to another angle. The first one is 490. What is the measure of the second angle?

    Practice Problems

    (Problems 1 – 4) Find the measure of angle b.

    1. clipboard_ec14be2401d090cf5fe600ddea5f3d385.png

    2. clipboard_eb02cffea674b32fcc53befa014ffbabb.png

    3. clipboard_efcebe6c65c84e280fbf120593a905a6a.png

    4. clipboard_e7fe2f7e7378980512c9cec61de7e3267.png

    (Problems 5 – 6) Find the measure of each angle indicated.

    1. clipboard_e94e47cb092f4bf935a2a0d9093732ec5.png

    2. clipboard_e5b9ba5ec6e872c039ed5aea2f49dfa8b.png

    (Problems 7 – 10) Classify each angle as acute, obtuse, right or straight.

    1. \(121^{\circ}\)

    2. \(180^{\circ}\)

    3. clipboard_e73f4b6f0daef268c0f4073e351ef5d91.png

    4. clipboard_ee38498a22df05476181af08b22e22d7c.png

    (Problems 11 – 12) Classify each triangle by its angles.

    1. clipboard_ed5f8ed5c05769bcd5e4e6ce626426bfa.png

    2. clipboard_ee5524cd6e2bbc494229b8cffc558b28e.png

    (Problems 13 – 14) Classify each triangle by its angles and sides.

    1. clipboard_e18aefe08d1db488f24446aa9a9e2965d.png

    2. clipboard_e6cd65ce277cea3b3a752dd07ff741a13.png

    (Problems 15 – 16) Sketch an example of the type of triangle described.

    1. Acute Isosceles

    2. Right Obtuse

    (Problems 17 – 18) Write the name of each polygon.

    1. clipboard_e139f4267d32957b14b79b17e0f24a13b.png

    2. clipboard_e4425ebd7370a86950d1a40c6a21d50b8.png

    (Problems 19 – 22) Find the interior angle sum for each polygon. Round your answer to the nearest tenth, if necessary.

    1. clipboard_e3c2b9aa29fc38024f44c7ebac05ab0c9.png

    2. clipboard_ec5d901f522f9332a65222443215fec22.png

    3. clipboard_e1010d9a2ca8537fa094715092fb3c33f.png

    4. clipboard_e1c7488d83f5fc13712bcf52bbe303e76.png

    (Problems 23 – 26) State if the polygon is regular or irregular.

    1. clipboard_e557180c8a9450649b62906d96ae453a0.png

    2. clipboard_e14f9d3e7eb79d8d4e8cdfe4b99a9864c.png

    3. clipboard_e471663a194bf9fea286ad1a1326b30d1.png

    4. clipboard_ef92cb43985e8e356e668fac55b27d008.png


    This page titled 7.6: Polygons is shared under a not declared license and was authored, remixed, and/or curated by Amy Lagusker.

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