8.1 Polygons

Similar triangles

Two triangles are called similar if it is possible to proportionally shrink or stretch one of them to a triangle congruent to the other. Congruent triangles are similar triangles, but similar triangles are only congruent if they are the same size to begin with.

Description	Diagram
If all three pairs of corresponding angles of two triangles are equal, then the triangles are similar.
If all pairs of corresponding sides of two triangles are in proportion, then the triangles are similar.	$\frac{x}{p}=\frac{y}{q}=\frac{z}{r}$

The theorem of pythagoras

In the following figure, determine if the two triangles are congruent, then use the result to help you find the unknown letters.

1. Determine congruency

   $D\hat{E}C=B\hat{A}C={55}^{°}$ (angles in a triangle add up to ${180}^{°}$ ).

   $A\hat{B}C=C\hat{D}E={90}^{°}$ (given)

   $\text{DE}=\text{AB}=3$ (given)

   $\therefore \Delta \text{ABC}\equiv \Delta \text{CDE}$
2. Find the unknown variables

   We use Pythagoras to find x:

   $\begin{array}{ccc}\hfill {\text{CE}}^2& =& {\text{DE}}^2+{\text{DC}}^2\hfill \\ \hfill 5^2& =& 3^2+x^2\hfill \\ \hfill x^2& =& 16\hfill \\ \hfill x& =& 4\hfill \end{array}$

   $y={35}^{°}$ (angles in a triangle)

   $z=5$ (congruent triangles, $\text{AC}=\text{CE}$ )

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Triangles

1. Calculate the unknown variables in each of the following figures. All lengths are in mm.

   

2. State whether or not the following pairs of triangles are congruent or not. Give reasons for your answers. If there is not enough information to make adescision, say why.

   

Quadrilaterals

A quadrilateral is a four sided figure. There are some special quadrilaterals (trapezium, parallelogram, kite, rhombus, square, rectangle) which you will learn about in Geometry .

Other polygons

There are many other polygons, some of which are given in the table below.

Angles of regular polygons

Polygons need not have all sides the same. When they do, they are called regular polygons. You can calculate the size of the interior angle of a regular polygon by using:

where $n$ is the number of sides and $\widehat{A}$ is any angle.

Summary

• Make sure you know what the following terms mean: quadrilaterals, vertices, sides, angles, parallel lines, perpendicular lines,diagonals, bisectors and transversals.
• The properties of triangles has been covered.
• Congruency and similarity of triangles
• Angles can be classified as acute, right, obtuse, straight, reflex or revolution
• Theorem of Pythagoras which is used to calculate the lengths of sides of a right-angled triangle
• Angles:
  • Acute angle: An angle $0^{\circ }$ and ${90}^{\circ }$
  • Right angle: An angle measuring ${90}^{\circ }$
  • Obtuse angle: An angle ${90}^{\circ }$ and ${180}^{\circ }$
  • Straight angle: An angle measuring ${180}^{\circ }$
  • Reflex angle: An angle ${180}^{\circ }$ and ${360}^{\circ }$
  • Revolution: An angle measuring ${360}^{\circ }$
• There are several properties of angles and some special names for these
• There are four types of triangles: Equilateral, isoceles, right-angled and scalene
• The angles in a triangle add up to ${180}^{\circ }$

Exercises

1. Find all the pairs of parallel lines in the following figures, giving reasons in each case.

   1. 

   2. 

   3. 

2. Find angles $a$ , $b$ , $c$ and $d$ in each case, giving reasons.

   1. 

   2. 

   3. 

3. Say which of the following pairs of triangles are congruent with reasons.

   1. 

   2. 

   3. 

   4. 

4. Identify the types of angles shown below (e.g. acute/obtuse etc):

   

5. Calculate the size of the third angle (x) in each of the diagrams below:

   

6. Name each of the shapes/polygons, state how many sides each has and whether it is regular (equiangular and equilateral) or not:

   

7. Assess whether the following statements are true or false. If the statement is false, explain why:
   1. An angle is formed when two straight lines meet at a point.
   2. The smallest angle that can be drawn is 5°.
   3. An angle of 90° is called a square angle.
   4. Two angles whose sum is 180° are called supplementary angles.
   5. Two parallel lines will never intersect.
   6. A regular polygon has equal angles but not equal sides.
   7. An isoceles triangle has three equal sides.
   8. If three sides of a triangle are equal in length to the same sides of another triangle, then the two triangles are incongruent.
   9. If three pairs of corresponding angles in two triangles are equal, then the triangles are similar.
8. Name the type of angle (e.g. acute/obtuse etc) based on it's size:
   1. 30°
   2. 47°
   3. 90°
   4. 91°
   5. 191°
   6. 360°
   7. 180°
9. Using Pythagoras' theorem for right-angled triangles, calculate the length of x:

   

Challenge problem

1. Using the figure below, show that the sum of the three angles in a triangle is 180 ${}^{\circ }$ . Line $DE$ is parallel to $BC$ .