GEOMETRIC CONCEPTS IN PROBLEM SOLVING

ANGLE – WHEN 2 RAYS HAVE A COMMON ENDPOINT WHICH FORMS A VERTEX AND IS MEASURED IN DEGREES

As the Angle Increases, the Name Changes

Type of Angle / Description
Acute Angle / an angle that is less than 90°
Right Angle / an angle that is 90° exactly
Obtuse Angle / an angle that is greater than 90° but less than 180°
Straight Angle / an angle that is 180° exactly
Reflex Angle / an angle that is greater than 180°

Circle Graphs

Circle graphs, also called pie charts, are a type of graph used to represent a part to whole relationship. Properties of Circle Graphs:

  • They are circular shaped graphs with the entire circle representing the whole.
  • The circle is then split into parts, or sectors. ( DECIMAL X 360 FOR DEGREES)
  • Each sector represents a part of the whole.
  • Each sector is proportional in size to the amount each sector represents; therefore it is easy to make generalizations and comparisons. (THE PERCENT DECIMAL TIMES 100)

In this unit we will review how to read a circle graph. An example of a circle graph is given below.

Revenue Sources for the Federal Government

MAKING A CIRCLE GRAPH

Percent of Hours of a Day Spent on Activities

ACTIVITY / HOURS / PERCENT OF DAY
Sleep / 6 / 25 % OR .25 X 360 = 90 DEGREES
School / 6 / 25 % OR .25 X 360 = 90 DEGREES
Job / 4 / 17% OR .17 X 360 = 61.2 DEGREES
Entertainment / 4 / 17% OR .17 X 360 = 61.2 DEGREES
Meals / 2 / 8% OR .08 X 360 = 28.8 DEGREES
Homework / 2 / 8% OR .08 X 360 = 28.8 DEGREES

TYPES OF ANGLES

Complementary Angles

Two Angles are Complementary if they add up to 90 degrees (a Right Angle).

These two angles (40° and 50°) are Complementary Angles, because they add up to 90°. They do not have to be together to =90
. /

Supplementary Angles

Two Angles are Supplementary if they add up to 180 degrees.
These two angles (140° and 40°) are Supplementary Angles, because they add up to 180°.
Notice that together they make a straight angle. /
But the angles don't have to be together.
These two are supplementary because 60° + 120° = 180° /

Vertical Angles are the angles opposite each other when two lines cross

They are called "Vertical" because they share the same Vertex (or corner point)

In this example, a° and b° are vertical angles.
The interesting thing here is that vertical angles are equal:

a° = b°

(in fact they are congruent angles) /

Congruent Angles have the same angle in degrees. Congruent figures same angles same size similar figures same angles proportional size.


These angles are congruent.

They don't have to point in the same direction.

Similar Figures

When two figures are similar, the ratios of the lengths of their corresponding sides are equal.

To determine if the triangles below are similar, compare their corresponding sides.

Are these ratios equal?

10/ 6 = 5/3 = 30 =30 so yes

If you know that two objects are similar, you can use proportions and cross products to find the length of an unknown side. Let's find the length of side DF, labeled x.

We can write a proportion, like this:

INDIRECT MEASURE

THE USE OF SIMILAR FIGURES IN THE FORM OF TRIANGLES TO SOLVE FOR A MEASURE OF A TALL OBJECT THAT YOU ARE UNABLE TO MEASURE SUCH AS A TREE OR BUILDING USING A KNOWN HEIGHT AND SHADOW TO THE SHADOW OF AN UNKNOWN HEIGHT TO CREATE 2 SIMILAR TRIANGLES TO SOLVE BY RATIO PROPRORTION

If a building casts a 103-foot shadow at the same time that a 32-foot flagpole casts as 34.5-foot shadow. How tall is the building? Since the triangles are similar, set up a proportion and solve:



34.5 × h = 103 × 32
34.5h = 3296
h = 95.5362318841... /

The building is about 95.5 feet tall.

Missing Measure of a Triangle- all triangles have 180 degrees subtract known measures to solve for unknown

Types of Triangle

There are seven types of triangle, listed below. Note that a given triangle can be more than one type at the same time. For example, a scalene triangle (no sides the same length) can have one interior angle 90°, making it also a right triangle. This would be called a "right scalene triangle".

Isosceles / / Two sides equal
Equilateral / / All sides equal
Scalene / / No sides equal
Right Triangle / / One angle 90°.
Obtuse / / One angle greater than 90°
Acute / / All angles less than 90°
Equiangular / / All interior angles equal

To solve for a missing measure if one angle is 90 and the other is 30 then 180-90-30= 60 for the missing angle

A transformation is a general term for four specific ways to manipulate the shape of a point, a line, or shape. The original shape of the object is called the pre-image and the final shape and position of the object is the image under the transformation.

Types of transformations in math

  • Translation
  • Reflection
  • Rotation
  • Dilation

A translation is the same as sliding/shifting an object. The notation for translate is T(+a,+b)--where a and b represent how much you slide in the x and the y directions, respectively. For instance, look at the picture below.

Rotations the turning of an object around a point by degrees

The three main Transformations are:

Rotation / / Turn!
Reflection / / Flip!
Translation /