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7.6: Polygons

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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

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

clipboard_eff2c4c12b2312cd477162aad315c6270.png
Figure 6.2.4

Since a line is 180, we know that angles A1, B, and C1 must add up to 180. A theorem (proven statement) in Geometry states that alternate (opposite sides) interior angles are congruent (equal). Angles A1 and A2 are alternate interior, cut by the transversal (line) connecting angle A2 to the straight line. Angles C1 and C2 follow a similar approach.
Since the measures of angles A1=A2, C1=C2, and A1+B+C1=180, then by substitution, A2+B+C2=180. Therefore, triangle A2BC2 adds up to 180.

Partner Activity 3

The sum of the interior angles of any polygon is represented by: 180(n2).

  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

  2. 180

  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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