Weight Loss Made a Bit Easier: Realistic and Practical Advice for Healthy Eating and Exercise
Available on Amazon.com in paperback/Kindle formats for $6.25/$2.99. Please click here for details.
Home | My Math and Education Books | Math Lessons | Ask a Math Question
Site Info | Contact Info | Tutoring Info | LarryZafran.com | Tweet
Lesson 127: More About Triangles and Quadrilaterals
In this lesson we'll learn more about the properties and triangle and quadrilaterals.
Before starting this lesson, make sure that you review all of the previous lessons that we've had about angles, triangles, and quadrilaterals, and make sure that you understand them.
This lesson has a lot of definitions and concepts that need to be memorized. Very often a test question is designed to simply see if you understand what a term means.
First,
there are four main types of triangles. The first is an
isosceles triangle. It has two sides that are
equal in measure, and one side that isn't. Notice in the
picture that the two equal sides have tick marks. That shows
us they are equal. Notice at the angles opposite those sides
have rounded angle marks. That shows us that those two angles
are equal, which makes sense, since they are opposite equal sides.
The two equal angles at the bottom are called base angles,
and the angle at the top is called the vertex angle.
Next
is a scalene triangle. That is a triangle
with three unequal sides, and therefore three unequal angles.
Next is an equilateral triangle. It has three
equal sides and three equal angles. Each angle is 60 degrees,
since the sum of the angles in a triangle is 180 degrees.
Next
is a right triangle, which we've seen quite a lot
of. It has a right angle, and two other angles. The
other two angles must add up to 90 degrees, since there are 180
degrees in a triangle, and we've already used 90 for the right
angle. Recall that the side opposite the right angle is called
the hypotenuse, and we use the Pythagorean Theorem for problems
involving right triangles.
Knowing that all triangles have a sum of angles equal to 180 degrees, we can easily solve associated algebra problems. For example, if we are told that the three angles in a triangle are 7x + 5, 3x, and 4x - 7, and we are asked to find x, all we need to do is add those up, and set them equal to 180. Then we can use basic algebra to solve for x. Once we know x, we can easily determine the measure of each angle.
Now let's look at some quadrilaterals. First we'll look at
parallelogram. Angles that share a common side, such as A and
B are called consecutive. B and C are also
consecutive. Consecutive angles are always supplementary (sum
to 180 degrees). Notice that the angles which are opposite
each other, such as A and C, are always equal in measure.
Notice that the diagonals are not equal in length, but they do
bisect each other. That means that each
diagonal is cut in half by the other one.
Now let's look at a rectangle. First, notice that if we
draw a diagonal line through the rectangle, we form two right
triangles. If we draw a diagonal line the other way, notice
that the two diagonals are equal in length. They always will
be. The two diagonals also bisect each other.
We can use all of these facts in solving algebra problems that are
tied in to the geometry of shapes.
Let's look at a rhombus. A rhombus is a special type of
parallelogram with all four sides equal. The diagonals bisect
each other, but are not equal in length unless the rhombus happens
to be a square.
Next we have a trapezoid. Recall that a trapezoid has two
parallel sides, and two non-parallel sides. We call the
parallel sides bases, and the non-parallel sides legs. If the
legs are equal, we say that the trapezoid is isosceles, and the
angles that are opposite the legs will be equal. Consecutive
angles in a trapezoid are supplementary.
Remember that you can ask a math question if you have additional questions about a topic, or you can contact me if you have any comments or suggestions for this site.