Geometry Concepts

Geometric concepts:

Supplementary and complementary angles:

Complementary Angle:  Two angles are complementary when they add up to 90 degrees (a right angle).

Figure A  Complementary angles

Complementary angles DO NOT have to be NEXT to each other their values just need to be 90 degrees when added together.

Other examples are 60 and 30 degrees, 80 and 10 degrees . . . any combination of two values that when added together = 90 DEGREES.

Follow the link by pressing the image (Fig A) for a more through explanation of COMPLEMENTARY angles.

Supplementary angles: Two angles are supplementary if they add up to 180°.

Supplementary angles SO NOT have to be next to each other, their values just need to be 180° when added together.

Examples of Supplementary angles are:  70° + 110° = 180° and 10° + 170° = 180° . . . any combination of two values when added together = 180°

Supplementary angles

1)  Find the supplement AND the complement of a 27 degree angle.
  

In the image provided below (Figure B) Determine the correct answer for each problem.  Be prepare to explain your results.  Know WHY you have chosen the answer that you have.

In problem #2 and #3 you should be aware that when added together that all the angles shown - x, y, z and 105° MUST equal 360°

 

There are two ways to approach this problem:

A)     Each line is STRAIGHT which means that the value of angle x added to 105° MUST equal 180°.  So this gives us the equation 180° = X + 105°; and solving for X yields 75°

B)      Corresponding Angles:  the idea here is that 105° + x + y + z = 360°.  This concept is really a spin-off of the concept discussed in A above.  If X + 105° = 180° AND if Z + 105° = 180° THEN this means that X MUST EQUAL Z!  We can deduce from this that if angle X = angle Z THEN 105° = angle Y.  This is the principle of CORRESPONDING ANGLES.

To better understand the concept of CORRESPONDING ANGLES follow this link and then continue solving problems two and three.

Problem #4 which tells us to find the measure of the missing angle can be solved using the following logic: 

1)      We know that the sum of ALL the interior angles of a triangle MUST equal 180°.

 2)  We are given the value of two of these interior angles

         a.       One is a right angle (per symbol)

         b.      One is 38° 
 3)  Using this information we develop the following equation:  180° = x° + 38° + 90°.

 4)   Solving this equation gives us the answer that x = 52°

     

        To learn more about ANGLES CLICK ON THE IMAGES BELOW

Click on Image

 

Problems #5 and #6 deal with the diameter and the radius of a circle.  To fully understand DIAMETERS we need to know about the circle.  It is not possible to define a diameter without a circle.  A diameter is the linear distance "across a circle".  The radius of a circle can be defined as 1/2 the diameter OR the linear distance from the CENTER of the circle to a line representing the circles circumference.

For a better understanding of this concept, go to the link provided on the image below.  There are problems on this link to give you more practice, work them!

Figure B

Consider the following in regard to these problems:

To find the perimeter of an object is defined for us as finding “the distance” around the object.  We could also say that it is finding the length of “the line” that actually defines the object.

If we are given a square as in problem #7, you will notice that the value of only one side is given.  In order to find the “distance around” the square we will need the value of EACH side not just one.

Think about the definition of a square.  A square is defined for us a four sided polygon WHEN all four sides are EQUAL in length.

If this is true, and if we have been given the length of ONE side, then according to this definition we also have the lengths of the remaining sides which we can use to evaluate the PERIMETER of this square.

The solution is as simple as adding the lengths of each side.  5m + 5m + 5m + 5m = 20m

Do not confuse the terms PERIMETER with AREA or VOLUME.

If were a farmer would you know the difference between how long your fence was from how many cows (standing side by side) could fit inside the fence and whether or not could fit more cows inside your fence if you stacked them on top of one another?

How big “around” is your coffee cup (perimeter).  How big is the flat part of “the bottom” of your cup, how much room is needed to set it down (area)?  How much coffee will your cup hold (volume)?

If you are confused by the three concepts mentioned, think of an example that makes sense to you to help you keep them straight.

Problem #8:  To find the AREA of a triangle you will need to use the following FORMULA ---
   

     A = ½ bh;  This formula is used to determine the “space” that the triangle covers.  If you cut this   

     triangle out and laid it on the floor, it would cover the space that is determined by the equation 

     given.

    The perimeter

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