Math 7 - GEOMETRY
Two Dimensional Figures will include:
NACS Standard # / Standard7.G.A.2 / Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
7.G.B.4 / Know the formulas for the area and circumference of a circle and use them to solve problems. Give an informal derivation of the relationship between the circumference and area of a circle.
7.G.B.5 / 7.G.5Use Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
7.G.B.6 / Solve real-world and mathematical problems involving area, volume and surface area of two-dimensional and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes and right prisms.
Math 7, Unit 9: Measurement: Two-Dimensional Figures
Notes
Prep for 7.G.B.4 and 7.G.B.5 Angles: Measuring and Classifying
An angle can be seen as a rotation of a line about a fixed point. In other words, if I were to mark a point on a paper, then rotate a pencil around that point, I would be forming angles.
One complete rotation measures 360º. Half a rotation would then measure 180º. A quarter rotation would measure 90º.
Many skateboarders, skiers, ice skaters, etc. practice these kinds of moves. 2 full rotations equals or. One and a half rotations is . Be sure to mention these applications.
Let’s use a more formal definition.
An angle is formed by the union of two rays with a common endpoint, called the vertex. Angles can be named by the vertex.
This angle would be called “angle V”, shown as . However, the best way to describe an angle is with 3 points: one point on each ray and the vertex. When naming an angle, the vertex point is always in the middle.
This angle can now be named three different ways: .
Angles are measured in degrees (°). Protractors are used to measure angles. Here are two interactive websites you might use to show students how to use this measuring tool.
http://www.amblesideprimary.com/ambleweb/mentalmaths/protractor.html
http://www.mathplayground.com/measuringangles.html
You can classify an angle by its measure. Acute angles are greater than 0°, but less than 90º. In other words, not quite a quarter rotation. Right angles are angles whose measure is 90º, exactly a quarter rotation. Obtuse angles are greater than 90º, but less than 180º. That’s more than a quarter rotation, but less than a half turn. And finally, straight angles measure 180º.
Classifying Lines
Two lines are parallel lines if they do not intersect and lie in the same plane. The symbol is used to show two lines are parallel. Triangles ( ) or arrowheads (>) are used in a diagram to indicate lines are parallel.
Two lines are perpendicular lines if they intersect to form a right angle. The symbol is used to state that two lines are perpendicular.
Two lines are skew lines if they do not lie in the same plane and do not intersect.
7.G.B.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
Angle Relationships
Adjacent angles are two angles that have a common vertex, a common side (ray), and no common interior points.
In the examples to the right,
notice and are adjacent.
In this next example, there are many more relationships:
Vertical angles are formed when two lines intersect—they are opposite each other. These angles always have the same measure. We call angles with the same measure congruent ().
In the example to the right, the two yellow
angles are vertical angles. They are formed
by the same two lines, they sit opposite each
other and they are congruent.
Also notice the two green angles are also vertical angles.
Angles will be shown as congruent by using tick marks. If angles are marked with the same number of tick marks, then the angles are congruent.
We call two angles whose sum is 90º complementary angles. For instance, if and , then and are complementary angles. If , then the complement of measures 60°.
Two angles whose sum is 180º are called supplementary angles. If and , then and are supplementary angles.
Example:
Example: Explain why two obtuse angles cannot be supplementary to one another.
The sum of two obtuse angles .
Examples: Determine if each of the following statements is always true, sometimes true or never true. For each answer that you choose ‘sometimes true’ show one example when the statement is true and one example of when the statement is not true.
Your examples should be diagrams with the angle measurements shown.
1. Two adjacent angles are supplementary. sometimes true
2. Two acute angles are supplementary. never true
3. Vertical angles are congruent. always true
4. Vertical angles are adjacent angles. never true
5. If the measure of an angle is called x, the measure of it complement is 180-x. always true
When a pair of parallel lines is cut by a transversal, there is a special relationship between the angles. Discover what this relationship is!
1) Use the lines on a piece of graph paper or a piece of lined paper as a guide to draw a pair of parallel lines. You can also use both edges of your straightedge to create parallel lines on a blank sheet of paper.
2) Draw a transversal intersecting the parallel lines. Label the angles with numbers as in the diagram above.
3) Measure. Calculate the measures of the other three angles that share the same vertex.
4) Measure and calculate the measures of the other three angles. Record your findings into the table.
Angle
/ Measure / Angle / MeasureLook for patterns. Complete the conjecture below:
If two parallel lines are cut by a transversal, then the corresponding angles are ______.
Hopefully, you discovered that the corresponding angles created when a transversal intersects parallel lines are congruent! This will save us time when finding angle measures.
Example: . Find the measure of all other angles.
, since are adjacent and supplementary
, since are vertical angles or are adjacent and supplementary
, since are vertical angles or are adjacent and supplementary
, since are corresponding angles
, , and (using same reasoning as above)
Example: Find the measure of each angle in the diagram below
given .
Example: Find the Explain your thinking for each answer.
Earlier we would have a question like… given the diagram below. Find the measure of x.
Students would have to identify them as complements and then
subtract 46 from 90 to get 44 or
Now we want to begin using monomials and equations. So same type of problem but the question would be… using the diagram below, find the measure of x and the
Knowing the angles are complements we write:
Example: Find the measure of x and the .
As diagrams become more complex we may find examples like the following.
What we can easily see is there are three angles that total . So we could sum all three and set them equal to 360.
Example: Find the measure of m.
Solution:
90 + 165 + m = 360
255 + m = 360
255 – 255 + m = 360 – 225
m = 135
In this diagram notice that p is vertically opposite from and equal to the sum of angles with measurements 22 and 26, or a sum of 48.
Here three lines meet at a point. Notice that the two aangles and the angle 144are angles on a line and sum to 180.We could write the equation a + a + 144 = 180.
Here again three lines meet at a point. Notice that angles x and y are complementary. Angle x is a vertical angle to the angle marked 21, so x = 21.
Here three lines meet at a point. Notice that part of angle z is a right angle and the additional portion is vertical to the 48 angle. So angle z = 48 + 90 = 138.
Three lines meet at a point; =154. What is the measure of ?
Some students may just start writing equations and figuring out that must be 64
(154 – 90 = 64). Then = 64 since it is a vertical angle to . Finally .
OR Students they may want to use some colored pencils to lightly shade portions of the diagram. This will aid with their visualization skills
Example: In a pair of complementary angles, the measurement of the larger angle is four times the measure of the smaller angle. Find the measures of the two angles.
Angle 1 = 18
Angle 2 = 4(18) = 72
Example: The measures of two supplementary angles are in the ratio of 4:5. Find the two angles.
Example: The measure of a complement of an angle is more than three times the angle. Find the measurements of the two angles.
x + (3x+8) = 90
Angle 1 = 20.5
Angle 2 = 3(20.5) + 8 = 61.5 + 8 = 69.5
Example: Find the measurement of the two missing angles.
Example: Two lines meet at the common vertex of two rays. Set up and solve an appropriate
equation for x and y.
Example: Two lines meet at the common vertex of a ray. Set up and solve the appropriate
equations to determine x and y.
**Example: The measurement of the complement of an angle exceeds the measure of the angle by 25%. Find the angle and its complement.
x + (x + x) = 90
x + x = 90
x = 90
x = 90
x = 40
90 – x = 90 – 40 = 50
The angle is 40 and its complement is 50.
Prep for 7.G.B.6 Area of Rectangles, Squares and Rhombi
One way to describe the size of a room is by naming its dimensions. A room that measures 12 ft. by 10 ft. would be described by saying it’s a 12 by 10 foot room. That’s easy enough.
There is nothing wrong with that description. In geometry, rather than talking about a room, we might talk about the size of a rectangular region.
For instance, let’s say I have a closet with dimensions 2 feet by 6 feet. That’s the size of the closet.
Someone else might choose to describe the closet by determining how many one foot by one foot tiles it would take to cover the floor. To demonstrate, let me divide that closet into one foot squares.
By simply counting the number of squares that fit inside that region, we find there are 12 squares.
If I continue making rectangles of different dimensions, I would be able to describe their size by those dimensions, or I could mark off units and determine how many equally sized squares can be made.
Rather than describing the rectangle by its dimensions or counting the number of squares to determine its size, we could multiply its dimensions together.
Putting this into perspective, we see the number of squares that fits inside a rectangular region is referred to as the area. A shortcut to determine that number of squares is to multiply the base by the height. More formally area is defined as the space inside a figure or the amount of surface a figure covers. The Area of a rectangle is equal to the product of the length of the base and the length of a height to that base. That is.
Most books refer to the longer side of a rectangle as the length (l), the shorter side as the width (w). That results in the formula . So now we have 2 formulas for the areas of a rectangle that can be used interchangeably. The answer in an area problem is always given in square units because we are determining how many squares fit inside the region. Of course you will show a variety of rectangles to your students and practice identifying the base and height of those various rectangles.
Example: Find the area of a rectangle with the dimensions 3 m by 2 m.
Example: The area of a rectangle is 16 square inches. If the height is 8 inches, find the base.
Solution: A = bh
16 = b 8
2 = b
Example: The area of the rectangle is 24 square centimeters. Find all possible whole number dimensions for the length and width.
Solution:
Example: Find the area of the rectangle.
Be careful! Area of a rectangle is easy to find, and students may quickly multiply to get an answer of 18. This is wrong because the measurements are in different units. We must first convert feet into yards, or yards into feet.
Since 1 yard = 3 feet we start with
We now have a rectangle with dimensions 3 yd. by 2 yd.
The area of our rectangle is 6 square yards.
If we were to have a square whose sides measure 5 inches, we could find the area of the square by putting it on a grid and counting the squares as shown below.
Next, if we begin with a 6 x 6 square and cut from one corner to the other side (as show in light blue) and translate that triangle to the right, it forms a new quadrilateral called a rhombus (plural they are called rhombi). Since the area of the original square was 6 x 6 or 36 square units then the rhombus has the same area, since the parts were just rearranged. So we know the rhombus has an area of 36 square units. Again we find that the Area of the rhombus = bh.
Be sure to practice with a number of different rhombi so your students are comfortable with identifying the height versus the width of the figures.