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Find Angles, Area, Perimeter of a Parallelogram


Date: 03/23/2001 at 21:13:31
From: Jamie 
Subject: Finding indicated measures with little information

I can't understand how to find indicated measures when I am given 
little information to begin with. Here is an example of my problem.

           A----------B
           | \       /|
           |  \     / |
           |   \   /  |
           |    \E/   |
           |     \    |
           |    / \   |
           |   /   \  |
           |  /     \ |
           | /       \|
           D----------C

This is a parallelogram.
   Segment AC is perpendicular to BD.
   Angle DEC is equal to 90 degrees.
   Angle EDC is egual to 37 degrees.
   Segment AE and Segment EC are equal to 3.
   Segment BC is equal to 5.

Find:

The measure of EB.
 "    "     "  DB.
               AB.
The perimeter of Triangle AEB.
The measure of angle ACD.
 "    "     "    "   CAB.
The perimeter of parallelogram ABCD.

Please help.
Thank you for your time.


Date: 03/23/2001 at 23:10:31
From: Doctor Peterson
Subject: Re: Finding indicated measures with little information

Hi, Jamie.

Just to keep me from getting confused, I'm going to redraw your 
picture a little more accurately, so the right things look 
perpendicular:

         A             B
          +-----------+
         / \       / /
        /  3\E  /   /
       /     +     /5
      /   /   \3  /
     / / 37    \ /
    +-----------+
   D           C

I see a couple of things immediately. Since AC and BD are 
perpendicular at E, not only DEC but all four angles with vertices at 
E are 90 degrees. Also, since this is a parallelogram, opposite sides 
are equal. 

(I actually didn't need to be told even as much as you were given; 
using trigonometry, I know that angle EDC is really 36.87 degrees; and 
because the diagonals of a parallelogram always bisect one another, I 
didn't have to be told both AE and CE. But you've been given extra 
information so you don't need to do these things.)

Now look at triangles DEC and DEA. Both are right triangles, and both 
legs are the same in each (DE = DE and AE = CE). So these are 
congruent. But since opposite sides of a parallelogram are congruent, 
that means that all four sides are equal, so we have a rhombus. That 
gives me enough to work with.

Now let's look at what you need.

The measure of EB: We have the hypotenuse and leg of a right triangle, 
so Pythagoras will give you the length.

The measure of DB: This will be twice EB, since it is bisected.

The measure of AB: This will be 5, since all four sides are congruent.

The perimeter of Triangle AEB: You now know all three sides.

The measure of angle ACD: You know the other two angles of DEC, so the 
fact that the sum of angles of a triangle gives this to you.

The measure of angle CAB: Alternate interior angles make this the same 
as ACD.

The perimeter of parallelogram ABCD: Since it's a rhombus, you know 
all four sides.

In general, it can take a lot of thinking to work out this sort of 
problem. As you can see, it helps to scout out the territory before 
you try to go anywhere; see what facts you can deduce before you try 
to solve the specific problems they give you. Sometimes, they might 
ask their questions in an order that will lead you through it, as the 
first one in your example could be done without any of my preliminary 
work; but it helps to be aware from the start of basic relations: if 
you see parallel lines and a transversal, you know you can use it to 
equate certain angles, for example, and if you have a parallelogram, 
you know opposite sides are congruent. 

Also, when you read each question, think about the parts you need to 
know to answer it. A perimeter requires knowing all the sides; how 
many of them do you know already, and how can you find the others?

Take it step by step, and you should be able to make it through!

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Geometry
High School Triangles and Other Polygons

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