Download SD 9-12 Geometry

SD 9-12 Geometry
03/20/2009
Student Name:
___________________________
Class:
___________________________
Date:
___________________________
Instructions:
Read each question carefully and select the correct answer.
1.
Which transformation was performed on the
following figure?
2.
Choose the graph of the inequality below.
y
A.
B.
C.
D.
Rotation
Reflection
Translation
Dilation
9x - 4
A.
B.
C.
D.
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3.
The following graph represents the equation
y = - 4x2 - 4x + 3. Choose the point(s) on the
graph that would solve the equation- 4x2 - 4x
+ 3 = 0.
5.
Choose the correct graph for the equation
below.
y = - |x| - 5
A.
A.
B.
C.
D.
4.
Points R, T, and U
Points R and U
Point T
Point S
Choose the system of inequalities
represented by the following graph.
B.
C.
A.
B.
D.
C.
D.
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6.
Which line represents the equation y = - 7?
7.
What is the measure of any exterior angle of
a regular octagon?
A.
B.
C.
D.
A.
B.
C.
D.
135º
45º
22.5º
157.5º
line d
line c
line b
line a
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8.
Fill in the missing statement in the proof below.
A.
B.
C.
D.
Reflexive Property
Transitive Property
Corresponding Parts of Congruent Triangles are Congruent
Associative Property
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9.
A.
B.
C.
D.
129º
39º
90º
51º
12.
Which picture shows an interior angle?
10.
A.
B.
A.
B.
C.
D.
109º
19º
38º
71º
C.
11.
A.
B.
C.
D.
(27, -9)
(-1, 5)
(27, -7)
(3, 5)
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D.
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13.
Which of the following is NOT true?
16.
About which line are the figures symmetric?
A.
All kites are polygons.
B.
All kites have two sets of congruent
adjacent sides.
C.
All kites have diagonals that are
perpendicular.
D.
All kites are rhombi.
14.
Calculate the sum of the interior angles of
the figure below.
A.
B.
A.
B.
C.
D.
180º
540º
360º
720º
C.
D.
17.
15.
What is the measure of
A.
B.
C.
D.
given.
y=1
y = -1
Calculate the midpoint of line segment RS
given in the diagram.
BAC?
56
34
28
There is not enough information
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A.
B.
C.
D.
(-0.5, 6)
(6, -0.5)
(7.5, 2)
(2, 7.5)
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18.
The mall is 15 miles due south of Jodi's
house. The school is 20 miles due east of the
mall. What is the shortest distance from
Jodi's house to the school?
A.
B.
C.
D.
19.
20.
25 miles
5 miles
35 miles
20 miles
Segment AD is congruent to segment _____.
A.
B.
C.
D.
Fill in the blank.
21.
If m QSN is equal to 90º , then QSP and
PSN are _________________.
A.
B.
C.
D.
complementary angles
supplementary angles
vertical angles
right angles
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Fill in the blank.
Quadrilaterals ABCD and WXYZ are
congruent.
CD
WZ
XW
BC
Point C and point D have symmetry with
respect to the y-axis. What are the
coordinates of point C when point D is the
point (-4, -4)?
A.
B.
C.
D.
(-4, -4)
(-4, 4)
(4, -4)
(4, 4)
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22.
The following are the front and top views of a building. What is the view from the right side of the
building?
A.
B.
C.
D.
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23.
What will the coordinates of point A be if
figure ABC is rotated clockwise around
point D so that point B is at (1, 1)?
A.
B.
C.
D.
24.
(1, 0)
(3, 0)
(4, - 1)
(1, 2)
25.
Choose the graph that represents the
equation.
A.
These two triangles are congruent.
B.
What is the measurement of
A.
B.
C.
D.
QOP?
43º
90º
47º
3º
C.
D.
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26.
Solve the system of equations by graphing.
x + 2y = 7
2x - y = -1
A.
B.
C.
D.
27.
(-1, 7)
(3, 1)
(1, 3)
(7, -1)
A triangle has sides which are 9, 40, and 41 millimeters long. How could you determine if this triangle
is or is not a right triangle?
A.
B.
C.
D.
28.
Find the distance between point R (-1, 0) and point Q (3, 2).
A.
B.
C.
D.
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SD 9-12 Geometry
Answer Key
03/20/2009
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
B
C
B
B
D
A
B
B
B
D
B
A
D
C
C
C
C
A
A
B
C
B
D
C
C
C
D
C
Spatial Relationships - B
Graph Inequalities
Solve Quadratic Equations by Graphing
Graph Systems of Inequalities
Graph Absolute Value
Graph Equations: Constant
Angles and Bisectors
Proofs
Perpendicular Bisector
Angles - D
Translation: Ordered Pair
Interior Angles: Polygons - A
Properties of Kites/Trapezoids
Interior Angles: Polygons - B
Congruence (AAS/ASA/SAS) - B
Symmetry - C
Midpoint
Triangles - B
Angles - B
Congruency - C
Symmetry - D
Spatial Relationships - C
Transformations
Congruence (AAS/ASA/SAS) - A
Graphing Equations - A
Graphing Equations - B
Pythagorean Theorem
Distance Formula
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Study Guide
SD 9-12 Geometry
03/20/2009
Spatial Relationships - B
Spatial relationships include geometric transformations. A transformation is a mapping of a point or shape to a
new location or orientation. Transformations include reflections, rotations, dilations, or translations. All
transformations except for dilations preserve the original size and shape of the image.
First, we'll define each of the transformations mentioned above:
A translation is sliding of a figure from one location to another. (Also called a "slide.")
For example:
A dilation is the image of a figure similar to the original figure. It can be thought of as shrinking or
enlarging a figure.
For example:
A reflection is flipping a figure across a line, just as you would reflect your hand in a mirror.
For example:
A rotation is the movement of a figure in a circular motion around a point. If you drew a figure on a
piece of paper, put the paper on the desk, and turned the paper, you would have a rotation.
For example:
It might be helpful to draw figures on a piece of paper, and rotate them to illustrate rotation of figures.
Example 1: What transformation was performed on the following figure?
The answer is a rotation.
We can use a coordinate plane to show where the parts of a shape are. If we draw the x- and y-axes, we
divide the coordinate plane into four parts, each called a quadrant. The quadrants are numbered as
follows:
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If we reflect a figure (triangle ABC) over the x-axis, what are the coordinates of the reflected figure?
(Use "prime" notation A' to identify the image. A' can be read "A prime.")
A translation moves a figure to another location. Below is triangle ABC moved 3 units left and 5 units
down. We can find the coordinates as follows:
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Graph Inequalities
An inequality is a number sentence that uses is greater than or is less than symbols. For example, 6n <
4 and y 2x - 3 are inequalities.
When graphing an inequality, the student should mentally replace the inequality symbol with an equal
sign in order to graph the inequality as an equation. Then use the table below to decide the type of line
that should be used when drawing the graph.
A dashed line tells the reader that the values on the line ARE NOT included in the inequality. A solid
line tells the reader that the values on the line ARE included in the inequality.
Example 1:
Graph the inequality.
y > - 4x - 6
Step 1: Graph the line that is represented by the inequality. (Remember to mentally replace the > with
=.) This equation is given in y = mx + b form (slope-intercept form), where m is the slope and b is the yintercept. Plot the y-intercept, (0, - 6), then use the slope, - 4, to move up 4 units and to the left 1 unit.
The is greater than symbol (>) is used, refer to the chart above to see that this symbol requires a dashed
line. Connect the two points using a dashed line.
Step 2: Choose a test point to determine which side of the line should be shaded. The most common
test point to use is (0, 0), but it does not matter what point is used. Substitute the test point into the
inequality and simplify.
• If the test point makes the inequality true, shade the side of the line that includes the test point.
• If the test point makes the inequality false, shade the side of the line that does not include the
test point.
In this case, the test point makes the inequality true.
Step 3: Since the test point makes the inequality true, shade the side of the dotted line that includes the
point (0, 0).
Answer:
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Example 2:
Determine the correct inequality for the graph below.
Step 1: Determine the equation of the line. In this case, the y-intercept is at (0, - 6) and the slope
appears to be up 1, over 4 (or ?), as can be seen by points at (8, - 4) and (4, - 5). Therefore, the equation
of the boundary line is y = (?)x - 6.
Step 2: Use the table on page 1 to determine which type of inequality symbol to use (<, >, , or ).
The line on the graph is solid, so the or symbol must be used.
Step 3: Choose a test point from the shaded side of the line and substitute it into each inequality to
determine which of the two inequalities is correct. A good test point to use is (0, - 8), since (0, - 8) is
included in the shaded area of the graph. Since y = (?)x - 6 is true when (0, - 8) is used as the test point,
it is the correct inequality.
To reinforce this skill, create of set of index cards with inequalities written on them and another set with
the graphs of the equations drawn and shaded. Shuffle the cards and lay them face-down on a table in
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columns and rows. Have the student turn two cards over at a time and try to match the inequality to the
correct graph. If the two cards do not match, the next player can try to make a match. If the cards
match, the player keeps the two cards and gets a chance to make another match. The player with the
most matches at the end of the game wins.
Solve Quadratic Equations by Graphing
A quadratic equation is a function that contains polynomial expressions for which the highest power of the
unknown variable is two.
Quadratic functions are written in the form:
y = ax2 + bx + c or f(x) = ax2 + bx + c
f(x) is read "f of x."
Below are a few examples of quadratic functions:
y = x2 + 3x - 4
g(x) = - 2x2 + 6
f(x) = 5x2 - 2x
Graphs of quadratic functions are always in the shape of a parabola. Parabolas can open up or open
down. Examples of each are shown below.
Factoring, using the quadratic formula, and graphing are the three main methods for solving quadratic
equations. This skill focuses on solving quadratic equations by graphing. To solve quadratic equations,
it is necessary to find the values for x in which y equals zero. These values occur at the x-intercept(s), or
the point(s) where the graph crosses the x-axis. The x-intercepts are of the form (x, 0), where the y-value
equals zero. X-intercepts can occur at two points, one point, or no points.
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Example 1:
The following graph represents the equation y = - 3x2 - 3x + 1. Choose the point(s) on the graph that
would solve the equation - 3x2 - 3x + 1 = 0.
Solution:
The x-intercepts are the solutions to the quadratic equation. Therefore, points J and M would solve - 3x2
- 3x + 1 = 0.
Answer: Points J and M.
Graph Systems of Inequalities
An inequality is a number sentence that uses is greater than or is less than symbols. For example, 6n <
4 and y 2x - 3 are inequalities.
When graphing an inequality, the student should mentally replace the inequality symbol with an equal
sign in order to graph the inequality as an equation. Then use the table below to decide the type of line
that should be used when drawing the graph.
A dashed line tells the reader that the values on the line ARE NOT included in the inequality. A solid
line tells the reader that the values on the line ARE included in the inequality.
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Example 1:
Graph the inequality.
y > - 4x - 6
Step 1: Graph the line that is represented by the inequality. (Remember to mentally replace the > with
=.) This equation is given in y = mx + b form (slope-intercept form), where m is the slope and b is the yintercept. Plot the y-intercept, (0, - 6), then use the slope, - 4, to move up 4 units and to the left 1 unit.
The is greater than symbol (>) is used, refer to the chart above to see that this symbol requires a dashed
line. Connect the two points using a dashed line.
Step 2: Choose a test point (that is not on the line) to determine which side of the line should be shaded.
The most common test point to use is (0, 0), but it does not matter what point is used. Substitute the test
point into the inequality and simplify.
• If the test point makes the inequality true, shade the side of the line that includes the test point.
• If the test point makes the inequality false, shade the side of the line that does not include the
test point.
In this case, the test point makes the inequality true.
Step 3: Since the test point makes the inequality true, shade the side of the dotted line that includes the
point (0, 0).
Answer:
Example 2:
Determine the correct inequality for the graph below.
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Step 1: Determine the equation of the line. In this case, the y-intercept is at (0, - 6) and the slope
appears to be up 1, over 4 (or ?), as can be seen by points at (8, - 4) and (4, - 5). Therefore, the equation
of the boundary line is y = (?)x - 6.
Step 2: Use the table on page 1 to determine which type of inequality symbol to use (<, >, , or ).
The line on the graph is solid, so the or symbol must be used.
Step 3: Choose a test point from the shaded side of the line, and substitute it into each inequality to
determine which of the two inequalities is correct. A good test point to use is (0, - 8), since (0, - 8) is
included in the shaded area of the graph. Since y (?)x - 6 is true when (0, - 8) is used as the test point,
it is the correct inequality.
Systems of Inequalities:
When graphing a system of inequalities, the process is very similar. A system of inequalities is two or
more inequalities. The main difference is that the final solution is the area where the shaded regions
overlap.
Example 3:
Choose the system of inequalities represented by the following graph.
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Step 1: Determine the equations of the boundary lines. The solid line has a y-intercept of - 3 and a
slope of 2/5. The dashed line has a y-intercept of 1 and a slope of - 4. The equations of the boundary
lines are y = (2/5)x - 3 (solid line) and y = - 4x + 1 (dashed line).
Step 2: Use the chart on page 1 to determine which inequality symbols to use.
Step 3: Choose a test point from the shaded region of the graph and substitute it into each of the
inequalities. A good test point to use is (-2, 0). Since (-2, 0) is in the shaded area of the graph, the
inequalities that are true when (-2, 0) is substituted are the correct inequalities.
Answer:
Systems of Inequalities That Are Not Solved For y:
Sometimes systems of equations or inequalities are presented in a form other than y = mx + b. If an
inequality is not in this form, the student should first solve the inequality for y in order to make the
graphing process easier.
Remember, to solve an inequality for y, use inverse operations to isolate the variable and be sure to
follow inequality sign rules when multiplying or dividing.
Rules:
•When multiplying or dividing both sides of an inequality by a positive number, leave the
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inequality sign as is.
•When multiplying or dividing both sides of an inequality by a negative number, reverse the
direction of the inequality sign.
•When adding or subtracting both sides of an inequality by a positive or negative number, leave
the inequality sign as is.
Example 4:
Rewrite the following system of inequalities in order to solve for y. For this example you do not need to
actually graph the system.
8x - 4y > 24
5 6y + x
Step 1: Isolate the y-term by adding or subtracting the x-term.
Step 2: In order to isolate the variable in the first inequality, it is necessary to divide both sides of the
inequality by a - 4. In the second inequality, both sides need to be divided by 6.
Step 3: Simplify each term in the inequality. Remember, since the first inequality was divided by a
negative number, the direction of the inequality sign must be changed. In the second inequality, the sign
remains the same since the division was by a positive number.
nswer:
Example 5:
Graph the solution to the following system of inequalities.
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Step 1: Graph the lines that are represented by the inequalities. The y-intercept of the top line is 4 and
the slope is -1/3. The y-intercept of the second line is -2 and the slope is 1. Use the chart on page 1 to
determine whether the lines should be dashed or solid. The line with the symbol should be solid, and
the line with the > symbol should be dashed.
Step 2: Choose a test point, and substitute it into both inequalities to determine which direction to
shade. It does not matter which point is used as a test point as long as the correct side of the line is
shaded. A good point to use is (0, 0).
Step 3: Since the point (0, 0) makes both inequalities true, shade each inequality on the side of the line
that contains (0, 0).
Step 4: The solution to the two inequalities is the region of the graph where the shading overlaps.
Answer:
Graph Absolute Value
The absolute value of a real number is the distance the real number, x, is from 0 on a number line. The absolute
value of a real number is denoted by placing the real number within two vertical lines: |x|. In other words, |- 5|
(read: the absolute value of - 5) is 5 because - 5 is 5 units from 0 on a number line.
There are two major principles of absolute value:
1. The absolute value of a negative or positive number is always a positive number.
2. The absolute value of 0 is 0.
Graphs of Absolute Value Functions:
The standard form for an absolute value function (y = a|bx + c| + d) is needed to compare graphs of
absolute value functions with the graph of y = |x|.
• a determines whether the graph opens up or opens down
• b determines how wide open the graph is
• c determines whether the graph shifts to the right or the left
• d determines whether the graph shifts upward or downward
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When determining the shifts, it is helpful to use the vertex of the graph as the point of reference. In this
skill, the absolute value equations are given in the form where a is equal to 1 or - 1 and d changes. The
variables b and c do not affect the equations in this skill.
When comparing graphs of absolute value functions, start with the graph of y = |x|, which is shown
below.
Then, determine how the given equation alters this graph. When a is positive, the graph opens up, and
when a is negative, the graph opens down.
When d is positive, the graph has a vertical shift up d units, and when d is negative, the graph has a
vertical shift down d units.
Example 1:
Determine the equation that represents the graph below.
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Solution:
The graph opens down, so the equation must have a negative value of a. It is shifted up five units on the
y-axis, so d must equal 5. Therefore, the equation that represents the graph is y = - |x| + 5.
Answer: y = - |x| + 5
Example 2:
Choose the correct graph for the equation below.
y = |x| - 3
Solution:
The value of a is 1, so the graph opens up, which is true only in graphs C and D. The value of d is -3, so
the graph is shifted down three units on the y-axis. Therefore, the correct answer is D.
Answer: D
A possible activity to reinforce this skill is to have the student create their own graphs on graph paper.
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Have the student make a T-chart (or x - y chart) of values for different equations. Then have the student
plot the points on the graph paper to graph the absolute value equation. As an enrichment activity,
provide him or her with equations that contain different values of b and c in the equation as discussed
earlier in this tutorial. For example, the student could graph the equations y = |2x| and y = |x - 3| and
determine the effects they have on the graph of y = |x|.
Graph Equations: Constant
Equations of the form y = c and x = c, where c is a constant, are very specific types of linear equations. A
constant is a quantity whose value does not change. For example, 3, - 4, and 56 are all constants. All equations
of the form y = c represent horizontal lines, while those of the x = c form represent vertical lines.
A good method for remembering which equation type is vertical and which is horizontal is to note that
all equations of the form y = c actually cross through point c on the y-axis, and all equations of the form
x = c actually cross through point c on the x-axis. In other words, in the equation y = - 4, the graph
intersects the y-axis at - 4. Therefore, it must be a horizontal line, because no vertical line would pass
through that point except for the y-axis itself.
Example 1:
Graph the equation x = - 2.
Solution:
The equation is in the form x = c. That means that the line must pass through the x-axis at - 2.
Therefore, the graph must be a vertical line that crosses the x-axis at - 2.
Answer:
One way to check to ensure it is the correct graph is to choose points on the line and see if they agree
with the equation. In the above example, the points (- 2, 0), (- 2, 6), and (- 2, - 4) are on the line. These
points all have - 2 as their x-value, so they correspond with the given equation, x = - 2.
Example 2:
Graph the equation y = 8.
Solution:
The equation is in the form y = c. That means that the line must pass through the y-axis at 8. Therefore,
the graph must be a horizontal line that crosses the y-axis at 8.
Answer:
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To check the answer, the points (0, 8), (- 5, 8), and (3, 8) all fall on the line. These points all have 8 as
their y-value, so they correspond with the given equation, y = 8.
An activity to help reinforce this skill could be to start with an equation in either form, x = c or y = c.
For example, x = - 3. Have the student write down six different points that have an x-value of (- 3). For
example, he or she could come up with (- 3, 4), (- 3, - 7), (- 3, 8.9), (-3, - 2.3), etc. Then, have him or
her plot the points on a coordinate graph and connect the points to see that the resulting graph is the line
x = - 3. Repeat this process with an equation in the form not used the first time, for example, y = 1.
Angles and Bisectors
Before working with angles and bisectors, it is important to understand several basic definitions and concepts.
Bisectors:
•An angle bisector is a line, segment, or ray that divides an angle into two congruent adjacent angles. In
other words, it cuts the angle into two equal angles.
•A perpendicular bisector is any line, segment, or ray that forms a 90º angle to a segment at its midpoint.
In other words, it cuts the segment in half and forms right angles at the point of intersection.
Angles:
• A straight line is an angle that contains 180º .
• A circle contains 360º .
• Interior angles are the corner angles or inside angles of a polygon. Examples of interior angles follow:
• When the sides of a polygon are
extended, the Exterior angles are adjacent to the interior angles. Examples of exterior angles follow:
• The sum of the angles in a triangle is
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180º .
• The sum of the interior angles of any polygon can be found by taking the number of sides of the
polygon, subtracting two, and multiplying that result by 180º : 180(n - 2), where n = the number of sides
in the polygon.
• To calculate the measure of the exterior angles of a regular polygon, simply divide 360º by the number
of sides. The formula for finding the measure of each of the exterior angles in a regular polygon is:
Example 1:
What is the measure of any exterior angle of a regular dodecagon (twelve-sided figure)?
Step 1: Use the equation for finding the measure of an exterior angle of a regular polygon.
Step 2: Substitute the value of n into the equation and simplify. The value of n for the dodecagon is
twelve, because it has twelve sides.
Answer: 30º
Example 2:
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Step 1: Recall that the sum of the angles of a straight line is 180º .
Step 2: Substitute the known values of m MRN and m PRQ. Simplify by adding 26 and 9 and
subtracting the sum from both sides of the equation.
Step 3: By definition of an angle bisector, m NRO and m ORP are equal. Therefore, each must be
half of the final angle measure from Step 2.
Answer: m ORP = 72.5º
Example 3:
Step 1: Recall that the sum of the angles in a triangle is 180º . Therefore, m ABC can be found by
subtracting the other two angles from 180º .
Step 2: Because ray BD bisects ABC, m DBA = m DBC, and the measure of each angle is half the
measure of ABC. Divide m ABC by 2 or multiply by 1/2.
Answer: m DBA = 70º
Several of the questions in this skill relate the concepts of bisectors and angles to real world problems.
An example is given below.
Example 4:
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The hour hand on a clock travels the same distance each hour. How many degrees does the hour hand
travel each hour?
Solution:
360º ÷ 12 = 30º
Recall that the total number of degrees in a circle is 360º . Since the hour hand of the clock travels the
entire 360º distance in 12 hours, divide 360º by 12 to determine the number of degrees the hour hand
travels in one hour.
Answer: The hour hand travels 30º each hour.
Proofs
A two-column proof shows the logical progression of numbered statements and the reasons that support them.
Proofs are used to prove theorems. Theorems are statements that are derived from mathematical ideas that were
previously accepted. Below is a list of the theorems and mathematical concepts used in the proofs in this skill.
Definitions:
Perpendicular Lines: Lines or segments that intersect to form right angles (90º ).
Parallel
Lines: Two lines are parallel when, if extended in both directions forever, they will never intersect.
(symbol: ||)
Isosceles Triangle: A triangle with two sides of equal length. The angles opposite those sides are also
congruent (equal). The base of the triangle always represents the side that is not congruent to the other
two sides.
Parallelogram: A quadrilateral with opposite sides parallel. Opposite sides and opposite angles are
congruent.
Midpoint: The point that bisects (divides into two congruent parts) a segment.
Perpendicular Bisector: A segment, line, ray, or plane that is perpendicular (forms 90º angles) to a
segment at its midpoint.
Angle Bisector: A segment, line, or ray that bisects (divides into two congruent adjacent angles) an
angle.
Radii: (plural of radius) The distance from the center of a circle to any point on that circle. All radii of
a given circle are the same length.
Combine Like Terms: When two terms have the exact same variable parts, they can be combined using
their given operation, either addition or subtraction. For example, 5x2 - 2x2 + 11xy + 4xy can be
combined to yield 3x2 + 15xy.
Complementary Angles: Two angles whose measures add up to 90º .
Supplementary Angles: Two angles whose measures add up to 180º .
Congruent: Objects that have exactly the same size and shape. That is, their corresponding angles and
sides are congruent. Synonymous with equal.
Similar: Objects that have exactly the same
shape, but are proportionally a different size. That is, their corresponding angles are congruent and their
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corresponding sides are proportional.
are usually parallel).
Transversal: A line that cuts through a set of lines (that
Alternate Interior Angles: When two lines are cut by a transversal, the inner angles on opposite sides of
the transversal are called alternate interior angles. When parallel lines are cut by a transversal, alternate
interior angles are congruent. In the figure below, a || b (read: "a is parallel to b") and c || d.
Alternate Exterior Angles: When two lines are cut by a transversal, the angles touching the outer side of
each line on opposite sides of the transversal are called alternate exterior angles. When the parallel lines
are cut by a transversal, alternate exterior angles are congruent. In the figure below, a || b and c || d.
Corresponding Angles: When two lines are cut by a transversal, the angles on the same side of the
transversal are corresponding angles. When parallel lines are cut by a transversal, corresponding angles
are congruent. In the figure below, a || b and c || d.
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Vertical Angles: When two lines intersect, the angles that are opposite one another (diagonal) are called
vertical angles. Vertical angles are always congruent.
Properties:
Reflexive Property: For any real number a, a = a.
Symmetric Property: If a = b, then b = a.
Transitive Property: If a = b and b = c, then a = c
Substitution Property: If a = b, then a can be substituted in for b in any equation or expression.
Distributive Property: a(b + c) = ab + ac
Addition, Subtraction, Multiplication, or Division Property of Equality: Applying these operations
correctly will yield a valid statement.
Corresponding Parts of Congruent Triangles are Congruent (CPCTC): When two figures are congruent,
the corresponding angles and corresponding sides are congruent. That is, if triangle ABC is congruent
to triangle DEF, then angle A is congruent to angle D, angle B is congruent to angle E, and angle C is
congruent to angle F. Also, the length of segment AB is equal to the length of segment DE, BC = EF,
and FD = CA.
Angle Sums:
The sum of the angles in a triangle is equal to 180º .
The sum of the angles of a straight line is equal to 180º .
Postulates for Congruency and Similarity:
SSS Postulate: If the lengths of all three corresponding sides of two triangles are congruent, then the
triangles are congruent.
ASA Postulate: If a side on one triangle is congruent to a side on another triangle and the angles on
each end of the side on one triangle are congruent to the angles on each end of the side on the other
triangle, then the triangles are congruent.
SAS Postulate: If an angle on one triangle is congruent to an angle on another triangle and the sides
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containing this angle on one triangle are congruent to those containing the angle on the other triangle,
then the triangles are congruent.
SSS Theorem for similarity: If the lengths of the corresponding sides of two triangles are proportional,
then the triangles are similar.
SAS Theorem for similarity: If an angle on one triangle is congruent to an angle on another triangle and
the sides containing this angle on one triangle are proportional to those containing the angle on the other
triangle, then the triangles are similar.
AA Postulate for similarity: If two of the angles of one triangle are congruent to two angles of another
triangle, then the triangles are similar.
Example 1:
Fill in the missing statement in the proof below.
Solution:
In order to be able to prove that the two lines are parallel in statement 5 of the proof, the appropriate, not
merely correct, information must be chosen in statement 4. Knowing that the reason for statement 4 is
that corresponding parts of congruent triangles are congruent (CPCTC) provides for many correct
options, but only one of them will allow statement 5 to be true in the next step.
Stating that any of the corresponding sides of the triangles are congruent would not provide enough
information to prove that the two lines are parallel in statement 5. The first set of congruent angles that
are named already in Step 4 ( MNP and OPN) are alternate interior angles for the segments MN and
PO, and therefore prove that those two lines are parallel. However, these angles give no information
about the segments MP and NO. PMN is congruent to NOP using CPCTC, but that does not help to
prove that segments MP and NO are parallel either. However, MPN is congruent to ONP, and they
are also alternate interior angles for the segments MP and NO. Therefore, segments MP and NO must be
parallel in statement 5.
Answer:
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Example 2:
Fill in the missing statement in the proof below.
Solution:
Statement 3 states that two triangles are congruent, and statement 4 states that corresponding angles of
those triangles are congruent. Therefore, the reason that statement 4 is true is that corresponding parts
of congruent triangles are congruent.
Answer: Corresponding Parts of Congruent Triangles are Congruent.
An activity that will help reinforce this skill is to create flash cards with each of the definitions,
theorems, and mathematical concepts listed in this study guide. The vocabulary word, theorem, or
concept should be written on the front with an example and/or explanation on the back. Either show the
student the front of the card or the back of the card and have the student state what is on the other side of
the card.
Perpendicular Bisector
A perpendicular bisector is any line, segment, or ray that forms a 90º angle to a segment at its midpoint. In
other words, it cuts the segment in half and forms right angles at the point of intersection.
Example 1:
Step 1: Determine the measure of YXV. Recall that a straight line has a total of 180º .
Step 2: Recall that YVX must be a right angle, by definition of segment YV being a perpendicular
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bisector to segment XZ. The measures of two of the angles of triangle XYV are now known to be 90º
and 37º . The third angle, XYV can be determined by subtracting the sum of these two angles from
180º , the total number of degrees of the interior angles of any triangle.
Step 3: If Steps 1 and 2 are repeated on triangle VYZ, it's easy to see the same angle measure will be the
result. Therefore, ZYV is also equal to 53º .
Step 4: XYZ is equal to XYV + ZYV.
Answer: m XYZ = 106º
Example 2:
Step 1: Since segment NH is the perpendicular bisector of segment MK, it cuts the segment into two
parts of equal length. Therefore, ML = LK. Since MK = 1.9, divide by two to determine ML and LK.
Step 2: Segment ML is given to be the same length as segment IJ and in Step 1 it is known that segment
ML is also the same length as segment LK. Therefore, ML = IJ = LK.
Step 3: Since segment KI is the perpendicular bisector of segment HJ, it cuts the segment into two parts
of equal length. Therefore, HI = IJ.
Step 4: To determine the length of segment HJ, we must add the lengths of segments HI and IJ.
Step 5: Because segment NH is the perpendicular bisector of segment GJ, it also cuts the segment into
two parts of equal length. Therefore, GH = HJ.
Step 6: To determine the length of segment GI, we must add the lengths of segments GH and HI.
Answer: GI = 2.85 m
Angles - D
This study guide shows the student how to define and apply the characteristics of alternate interior angles,
alternate exterior angles, corresponding angles, and vertical angles. Learning to recognize these types of angles
is helpful when trying to prove that two angles are congruent or that two lines are parallel.
Important Definitions:
Parallel lines: Two lines in the same plane are parallel when, if extended in both directions forever, they
will never intersect. Some examples of parallel lines are railroad tracks, columns in a building, and rows
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in a field. The symbol for parallel is ||.
Transversal: A line that cuts through a set of lines (that are usually parallel).
Notice the transversal intersects each line in exactly one point.
Congruent: Objects that have exactly the same size and shape. Synonymous with equal.
Alternate interior angles: When two lines are cut by a transversal, the inner angles on opposite sides of
the transversal are called alternate interior angles. When parallel lines are cut by a transversal, alternate
interior angles are congruent. In the figure below, line a is parallel to line b (a || b) and line c is parallel
to line d (c || d).
Alternate exterior angles: When two lines are cut by a transversal, the angles touching the outer side of
each line on opposite sides of the transversal are called alternate exterior angles. When the parallel lines
are cut by a transversal, alternate exterior angles are congruent. In the figure below, a || b and c || d.
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Corresponding angles: When two lines are cut by a transversal, the angles on the same side of the
transversal are corresponding angles. When parallel lines are cut by a transversal, corresponding angles
are congruent. In the figure below, a || b and c || d.
Vertical angles: When two lines intersect, the angles that are opposite one another (diagonal) are called
vertical angles. Vertical angles are always congruent.
Example 1:
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Step 1: Determine if 2 and 6 fit any of the special types of angles. Since r || s, 2 and 6 are
corresponding angles.
Step 2: Since corresponding angles are congruent when parallel lines are cut by a transversal, m 6 =
m 2. Therefore, m 6 = 121º .
Answer: m
6 = 121 º
Example 2:
Step 1: Determine if the given angles fit any of the special types of angles and pick out the information
that is needed (example the m 8 is not needed).
Step 2: Since g and h are parallel, 5 and 11 are alternate exterior angles.
Step 3: Since alternate exterior angles are congruent when parallel lines are cut by a transversal, m 11
= m 5. Therefore, m 11 = 112 º .
Answer: m 11 = 112 º
Example 3: Madison plays soccer on a rectangular field. During practice she saved the ball from going
out of bounds by kicking it across the field. The ball traveled in a diagonal path from sideline to sideline
forming a 63º angle. If no one is able to retrieve the ball on the other side, what would be the angle
measure where it would exit the field (m s)?
Step 1: Recognize that rectangles have parallel sides, and the diagonal created a transversal. This
makes the angles in question alternate interior angles.
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Step 2: Since alternate interior angles are congruent when parallel lines are cut by a transversal, the
angles in question are equal. Therefore m s = 63º .
Answer: m s = 63º .
Example 4:
Step 1: Recognize that 1 and 14 do not fit any of the special angles.
Step 2: Recall that supplementary angles have a sum of 180º . 1 and 2 are supplementary angles
because together they make a straight angle, which measures 180º . To find the measure of 2, subtract:
180º - m 1 = m 2. m 2 = 180º - 59º = 121º
Step 3: Since g and h are parallel, 2 and 14 are corresponding angles. Corresponding angles are
congruent when parallel lines are cut by a transversal, so m 2 = m 14 = 121º .
Answer: m 14 = 121º
An active way to explore the angles created by parallel lines that are cut by a transversal is laid out as
follows. First, draw several sets of parallel lines on a piece of paper and include a transversal with each
set. Next, trace some of the angles formed by this transversal onto a piece of tracing or wax paper.
Have the student slide the tracing/wax paper around the diagram and compare the sets of angles by
placing a traced angle over another angle in the diagram. Are they congruent? Have the student identify
the types of angles he or she sees.
Translation: Ordered Pair
This study guide will show the student how to perform a translation on an ordered pair to get the image. A
translation of an ordered pair is the slide of a coordinate point to a new point given a rule.
The diagram below represents the translation of A under the translation (x + 6, y + 4).
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.
The rule is given in the form of an expression. The values of x and y are substituted into the expression
to find the new point. This new point is the image.
Important Definitions:
Ordered pair: A point in the coordinate plane denoted by (x, y) where x moves in the horizontal
direction and y moves in the vertical direction. Other names include coordinate point and coordinate
pair.
Translation: The rule (or expression) that produces the slide.
Image: The resulting coordinate point after the translation expression has been evaluated.
Determining the Image After a Translation:
In order to perform a translation of a point, one needs the rule. In mathematics, a translation rule looks
like this:
If a number is added to x: move right
If a number is subtracted from x: move left
If a number is added to y: move up
If a number is subtracted from y: move down
In the translation above, x + 2 means that the x-coordinate is moved to the right 2 units (in the positive x
direction) and y - 3 means that the y-coordinate is moved down 3 units (in the negative y direction.)
Example 1:
Step1: Take the x-coordinate, 4, and move it right two units. In other words, add 4 + 2 = 6. The new xcoordinate is 6.
Step 2: Take the y-coordinate, 5, and move it down 3 units. In other words, subtract 5 - 3 = 2. The new
y-coordinate is 2.
Step 3: The image appears as follows: (new x-coordinate, new y-coordinate).
Answer: (6, 2)
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Interior Angles: Polygons - A
This study guide will show students how to identify the interior angles of polygons. It will also show students
how to find the measure of an interior angle of a regular polygon.
A polygon is a closed figure made up of at least three line segments. The number of sides of a polygon
is equal to its number of interior angles.
These are polygons:
These are not polygons:
Distinguishing Between Interior Angles and Other Angles:
Interior angles are the angles inside or the corners of the polygon.
These are interior angles:
These are not interior angles:
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Example 1:Name all possible interior angles in the figure below.
Step 1: Recall that interior angles are inside the polygon. These angles include 3, 8, 9, and 15.
Answer:
3,
8,
9, and
15
Example 2:Which figure shows an interior angle?
Step 1: Look for the figure that shows an angle inside the polygon (this eliminates figures b and c).
Step 2: Interior angles are in the corners of the polygon so that eliminates figure d.
Step 3: By process of elimination, we are left with figure a. Verify that figure a fits the definition of an
interior angle.
Answer: figure a
Determining the Measure of an Interior Angle of a Regular Polygon:
A formula is needed to find the measure of an interior angle.
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Simply substitute the number of sides of the polygon into the formula and evaluate.
A regular polygon has sides of the same length and has the same interior angle measures. Students
should be familiar with the following regular polygons:
Triangle: a polygon with 3 sides
Square: a polygon with 4 sides
Pentagon: a polygon with 5 sides
Hexagon: a polygon with 6 sides
Heptagon: a polygon with 7 sides
Octagon: a polygon with 8 sides
Nonagon: a polygon with 9 sides
Decagon: a polygon with 10 sides
Hendecagon (also an undecagon): a polygon with 11 sides
Dodecagon: a polygon with 12 sides
Example 3: Find the interior angle measure of a regular hendecagon. Round your answer to the nearest
degree.
Step 1: Recognize that a hendecagon has 11 sides and replace n with 11 in the formula.
Step 2: Simplify using the order of operations, so parentheses are first, (11 - 2 = 9).
Step 3: Multiply 180º by 9. Divide 1,620 by 11 to get 147.2727273º . Finally, round 147.2727273 to
the nearest degree to get 147º
Answer: 147º
A fun activity to reinforce these concepts is laid out as follows. Have the student draw or trace various
regular polygons. Have the student measure the interior angles of these polygons with a protractor and
record their measures. Have the student verify his or her data by using the formula given above. If data
and the measurements match exactly, the student has calculated correctly.
Properties of Kites/Trapezoids
This tutorial will focus on the properties that are specific to trapezoids and kites. Students will be asked to
identify these polygons given their properties or vice versa.
It is important to define some of the terms that will be used.
•Parallel lines: Two lines in the same plane that extend in opposite directions and never intersect. Some
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examples of parallel lines are straight railroad tracks, columns in a building, rows in a field.
•Perpendicular: Two intersecting lines that create a 90º angle at the point of intersection.
•Congruent: Figures that are exactly the same size and shape. Congruent is synonymous with equal.
•Diagonal: A line segment that connects two vertices (corners) of a polygon that are not adjacent (next
to each other).
Properties of Kites and Trapezoids
A trapezoid is a quadrilateral (a four-sided figure) that has exactly one pair of parallel sides. These
parallel sides are called bases and are used to find the area. The other two sides of a trapezoid are called
legs.
These are examples of trapezoids.
An isosceles trapezoid is a trapezoid that has congruent legs. They also have congruent diagonals and
two pairs of congruent base angles.
A kite is a quadrilateral that has two distinct sets of adjacent sides (they have a common vertex) with the
same length, but opposite sides are NOT congruent. The diagonals in a kite are perpendicular.
This is an example of a kite:
A rhombus is a special quadrilateral where both pairs of opposite sides are parallel and congruent. The
diagonals of a rhombus are perpendicular bisectors of each other.
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Example 1: In which of the following categories do trapezoids belong?
a) quadrilaterals
b) circles
c) parallelograms
d) hexagons
Answer a is the correct answer. A trapezoid is a four-sided figure, which is the definition of a
quadrilateral.
Answer b is not correct. A circle does not consist of line segments, so it is not a trapezoid.
Answer c is not correct. A parallelogram has two pairs of opposite parallel sides; a trapezoid has only
one.
Answer d is not correct. A hexagon has six sides; a trapezoid has four.
Example 2: A kite has exactly _____ pair(s) of opposite parallel sides.
a) 3
b) 2
c) 1
d) 0
Answer a is not correct. A kite does not have a pair of opposite parallel sides.
Answer b is not correct. A kite does not have a pair of opposite parallel sides.
Answer c is not correct. A kite does not have a pair of opposite parallel sides.
Answer d is correct. A kite does not have a pair of opposite parallel sides.
An activity that would serve useful for reinforcement of this concept is to have the student take index
cards and write one property of a trapezoid on one side and the word trapezoid on the other. Do this
with each property. Do the same for kites. Use these as flash cards. Shuffle all the cards. Draw one
card and show or read the student the property. Have the student determine whether it is a property of a
trapezoid or a kite. For a more challenging game, add some cards that have properties of other
polygons.
Interior Angles: Polygons - B
In order to find the sum of the interior angles of various polygons, it is important to define some of the terms
that will be used.
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A polygon is a closed figure made up of at least three line segments. The number of sides of a polygon
is equal to its number of interior angles.
Interior angles are the angles formed by two sides of the polygon.
These are polygons:
These are not polygons:
Students should be familiar with the following polygons:
•Triangle: a polygon with 3 sides
•Rectangle: a polygon with 4 sides
•Square: a polygon with 4 equal sides
•Pentagon: a polygon with 5 sides
•Hexagon: a polygon with 6 sides
•Heptagon: a polygon with 7 sides
•Octagon: a polygon with 8 sides
•Nonagon: a polygon with 9 sides
•Decagon: a polygon with 10 sides
•Hendecagon (Undecagon): a polygon with 11 sides
•Dodecagon: a polygon with 12 sides
Regular Polygon: All of the sides and angles are congruent (they have the same measure). The list
above consists of regular polygons if all of the sides are congruent.
Convex Polygon: A polygon where no line that contains a side of the polygon also contains a point on
the inside of the polygon. Regular polygons are convex.
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Concave Polygon: A polygon that is not convex. They contain an interior angle greater than 180º . If
the line segments (sides) are extended in both directions, the side will consist of points which lie inside
the polygon. An example is seen below.
Determining the Sum of the Interior Angles of a Polygon:
The same formula is needed to find the sum of the interior angles of regular, convex, and concave
polygons:
Example 1: Find the sum of the interior angles of a regular nonagon.
Step 1: Recognize that a nonagon has 9 sides. Replace n with 9 in the formula.
Step 2: Simplify using the order of operations, starting with parentheses, 9 - 2 = 7.
Step 3: Multiply 180º by 7 to get 1,260º .
Answer: 1,260º
Example 2: Calculate the sum of the interior angles of the figure below.
Step 1: Determine that the figure has 16 sides.
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Step 2: Substitute 16 for n in the formula.
Step 3: Subtract 2 from 16 to get 14.
Step 4: Multiply 14 by 180º to get 2,520º
Answer: 2,520º
Another way to find the sum of the interior angles is by choosing a vertex point and drawing a line to
each of the other nonadjacent vertex points to create triangles. Count the number of triangles inside the
figure. Since a triangle has an interior angle sum of 180º , multiply the number of triangles found by
180º . This gives the sum of the interior angles of the polygon. A diagram is given below.
A fun activity to reinforce these concepts could be to have the student draw or trace various polygons, or
look for some around the house. Then, have the student measure the interior angles of these polygons
with a protractor, add the angle measures together, and record the result. Have the student verify his or
her data using the formula given above.
Congruence (AAS/ASA/SAS) - B
Engineers and people in the construction field use triangle congruency on a daily basis to make sure that things
like rafters are congruent so the roof of a house does not sink in. This study guide will introduce students to
postulates, which are used to determine triangle congruency. A postulate is a statement which is taken to be
true without proof.
Review of Triangle Congruency Symbols
Tick marks are used to show that sides are congruent. All sides in a diagram that are marked with the
same number of tick marks are congruent.
Arcs are used to show that angles are congruent. All angles in a diagram that are marked with the same
number of arcs are congruent.
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Angles in a triangle are named by their vertex, the point at which two line segments meet. They are also
named with the vertex in the middle.
Postulates for Congruency:
ASA (Angle, Side, Angle) Postulate: If two angles and the included side (the side between the two
angles) of one triangle are congruent to two angles and the included side of another triangle, then the
triangles are congruent.
SAS (Side, Angle, Side) Postulate: If two sides and an included angle of one triangle are congruent to
two sides and an included angle of another triangle, then the triangles are congruent.
AAS (Angle, Angle, Side) Postulate: If two angles and a non-included side of one triangle are
congruent to two angles and a non-included sides of another triangle, then the triangles are congruent.
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Determining that Triangles are Congruent:
Example 1: What is the measure of
YXZ?
Step 1: Write out all the information given in the diagram.
Step 2: Determine if the triangles are congruent and, if congruent, by what postulate? These triangles
are congruent by the SAS postulate.
Step 3: Since the triangles are congruent, m TUV equals m XYZ. m XYZ = 34º .
Step 4: Remember that the sum of the 3 angles of a triangle is 180º . To determine m YXZ, subtract
180 - 57 - 34 = 89.
Answer: 89º
Example 2: Using only the information presented in the diagram, determine if the following triangles
are congruent and state which congruence theorem was used.
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Step 1: State the given information as shown in the diagram.
Step 2: The triangles are congruent by the AAS postulate.
Answer: The triangles are congruent by the AAS postulate.
Example 3: Using only the information below, and your knowledge of triangle congruence postulates,
which pair of triangles are congruent?
A is not correct. The angles of one triangle are congruent to the angles of another. This proves
similarity, not congruence.
B is correct. Two angles and a non-included side of one triangle are congruent to two angles and a nonincluded side of another triangle. The triangles are congruent by the AAS postulate.
C is not correct. SSA is not true for every triangle, that is why it is not a postulate.
D is not correct. There is not enough information to prove congruency.
Example 4:
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A.
B.
C.
D.
2 and 4
1 only
1, 3, and 4
4 only
A is not correct. Although SAS does prove that the triangles are congruent, SSS cannot be proven using
only what is presented in the drawing.
B is not correct. While AAS does prove that the triangles are congruent, SAS does too, and it is not part
of option B.
C is correct. AAS, ASA, and SAS will all prove that the two triangles are congruent using only what is
presented in the drawing.
D is not correct. SAS is not the only postulate that will prove these triangles congruent using only what
is presented in the drawing.
Answer: C
Example 5: Jacob is constructing rafters for a house. The middle piece of wood stands 4 feet tall and is
perpendicular to the base. He attaches two pieces of wood at the top of the middle piece which create a
68º angle with the base on either side. Using only the information presented, which postulate would be
used to prove that the sides of the rafters are equal in length?
A) ASA B) SSS C) AAS D) SAS
A) is not correct. The side that is congruent is not included between the congruent angles.
B) is not correct. Only one side is marked congruent.
C) is correct. The base angles are congruent, the right angles are congruent, and the middle piece of
wood is congruent to itself. The triangles are congruent by the AAS postulate.
D) is not correct. Only one side is marked congruent.
The following is a fun activity to help reinforce the concept of triangle congruency. Take a walk with
the student and observe where triangles occur in the neighborhood. Look at houses, buildings, parks,
and landscapes and pick out triangles that look congruent. If possible, take a ruler or tape measure and a
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protractor along to a public place or a place where it will not matter if measurements are taken. Take
measurements of the sides and angles of triangular objects in pairs. Verify that they are congruent by
comparing their measures. After the walk, have the student build a roof rafter out of toothpicks. Use a
ruler and a protractor to measure the sides and angles. Verify that all the triangles are the same using the
congruency postulates. Ask the student questions like, "What would happen if the triangles were not
congruent? What would happen if engineers built bridges with triangles that were not congruent?"
Have the student research on the Internet or at the library why congruent triangles are important in real
life situations.
Symmetry - C
Symmetry occurs all around us, in nature, at home, and at school. It occurs in crystals, plants, animals, and
even in some letters of the alphabet. This study guide will help students determine whether a figure is
symmetric about a point or a line.
For this study guide, students need to be familiar with the coordinate plane.
Determining Line Symmetry
Line Symmetry: A figure is symmetric about a line when one half of the object is the mirror image of
the other.
In order to determine if an object is symmetric about a line, try to find the line of symmetry, the line that
divides the figure into two mirror images. Line symmetry is sometimes referred to as reflection
symmetry.
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Once symmetry has been determined, it is necessary to name the line of symmetry. If the line is vertical,
it will have the form x = a constant (a number). If it is horizontal, it will have the form y = a constant.
Recognizing Horizontal and Vertical Lines
•If the line crosses the x-axis, the line will be vertical. The intersection point names the line. For
example, if the line crosses the x-axis at 7, then the line is named x = 7.
•If the line crosses the y-axis, the line will be horizontal. The intersection point names the line. If the
line crosses the y-axis at 5, then the line is named y = 5.
Example 1: About which line is the figure symmetric?
Step 1:
Step 2:
Step 3:
Step 4:
Draw a line where the figure is symmetrical and imagine folding the paper along the line.
The opposing sides are the same, so they are symmetrical.
Name the line. Since this is a vertical line it has the form x = a constant.
The line crosses the x-axis at (2, 0) so the line is x = 2.
Answer: x = 2.
Example 2: About which line are the triangles symmetric?
A. x = - 4
B. y = - 4
C. x = - 2
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D. y = - 2
Step 1: In order for the triangles to be symmetric, the line of symmetry must pass through the midpoint
of the segment between the two triangles. Determine the endpoints of the segment between the
triangles: (- 6, 0) and (- 6, - 4).
Step 2: Use the midpoint formula (below) to
determine the coordinates of the midpoint.
Substitute the correct values into the formula and
simplify to determine the coordinates of the midpoint: (- 6, - 2).
Step 3: In this case, there are two lines that would pass through this midpoint and separate the figure into
two equal parts. Those lines are: x = - 6 and y = - 2.
Step 4: Since the question provides four answer
choices, x = - 6 can be eliminated because it is NOT one of the four choices. The line y = - 2 is an
answer choice, so must be the line of symmetry. Test the answer by folding the graph along the line y =
- 2. The triangles should match up with each other. NOTE: If the problem did not give answer choices,
there would be two lines of symmetry.
Answer: D. y = - 2
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Determining Point Symmetry:
Point Symmetry: A figure is symmetric about a point if it is built around a single point, called the
center. That is, for each point in the figure, there is another one directly opposite on the other side of the
center.
A simple test to determine point symmetry is to turn the figure upside down. If it does not change, point
symmetry is achieved.
The following figure is symmetric about a center point.
Example 3: Which of the following figures is not symmetric about a point?
a) is not correct. Each way the figure is turned, there is a point exactly opposite any given point. It is
symmetric about a point.
b) is correct. Turned upside down, the flower does not look the same. It is not symmetric about a point.
c) is not correct. Each way the figure is turned, there is a point exactly opposite any given point. It is
symmetric about a point.
d) is not correct. Each way the figure is turned, there is a point exactly opposite any given point. It is
symmetric about a point.
There are many activities that a student could do to reinforce symmetry. Such as...
Have the student stand in front of a full length mirror and study the symmetry in his or her face and
body. The image on the mirror is an exact reflection so there is a line of symmetry between the student
and the mirror.
Take a piece of dot or graph paper and draw a line splitting it in half (horizontally, vertically, or
diagonally). Create a symmetrical figure by drawing geometrical shapes on one side and reflecting them
onto the other side. Color may be added to enhance the lines of symmetry.
Cut out pictures in a magazine that look symmetrical. Fold them horizontally, vertically, or diagonally
to determine if they are symmetrical.
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Midpoint
This study guide will focus on using a formula to determine the midpoint of a line segment.
Midpoint: The point halfway between the end points of a line segment. A midpoint divides a line
segment into two equal parts.
A formula is commonly used in order to find the midpoint of a line segment.
Coordinate point: A point in the coordinate plane denoted by (x, y) where x moves in the horizontal
direction and y moves in the vertical direction.
The student will be given two points and/or a graph from which he or she will have to determine the
points needed for the midpoint formula. The student will have to substitute the two points into the
formula and evaluate using the order of operations.
Example 1: Calculate the midpoint of line segment AB if A = (-1, -11) and B = (10, -3).
Step 1: Draw a diagram if one is not given. Use the diagram in the end to verify the answer.
Step 4: The
answer should appear in the following form:
(new x coordinate, new y coordinate)
Answer: (4.5, -7)
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The following is a fun and active way to reinforce this concept. Take two pieces of rope, string, or yarn
at least 20 feet in length. Construct a coordinate plane on the ground (the driveway is a great place to do
this) by placing these lengths perpendicular to each other (the ropes will cross at a 90º angle). Mark off
two feet intervals with tape or permanent marker. If chalk is available, write the scale on the ground
where the x-axis ranges from -5 to 5, and the y axis has the same range. See below:
Stand on one coordinate point and have a second person stand on another. Have the student try to
determine the midpoint of the imaginary line segment by estimation. Have the student use the formula
to verify the midpoint.
Triangles - B
A triangle is a three-sided figure that has three angles. The sum of the three angle measures of a triangle is 180º
. When we are given the other two angle measures, we can find the unknown angle measure by adding the two
known angle measures and subtracting that number from 180º .
Example 1: What is the value of y?
Add the two known angles: 60º + 70º = 130º
Since the sum of the angle measures must equal 180º , subtract: 180º - 130º = 50º .
The unknown angle measure is 50º .
Example 2: What is the measure of
C?
(1) 110º + m A = 180º ; 180º - 110º = 70º ; m A = 70º
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(2) 180º - (70º + 80º ) =
(3) 180º - 150º = 30º
Step 1: We must find the measure of
180º . The measure of A = 70º .
C
A. We know that the sum of the angles on a straight line equal
Step 2: Subtract the sum of the angles (70º + 80º = 150º ) from 180º .
Step 3: The measure of A is 30º
The Pythagorean Theorem allows us to determine the lengths of sides of a right triangle. (A right
triangle is a triangle with a 90º angle). The Pythagorean Theorem states that the square of the
hypotenuse (the longest side, and side opposite the 90º angle) of a right triangle is equal to the sum of
the squares of the legs (the two shorter sides) of that triangle. The formula in written form is:
where a and b are the short sides and c is the long side (the hypotenuse).
Example 3: The lengths of two legs of a right triangle are 4 and 3. What is the length of the hypotenuse
c?
Step 1: a = 4 and b = 3. Substitute these values into the Pythagorean theorem.
Step 2: Evaluate the values of the square terms. 4 x 4 = 16 and 3 x 3 = 9.
Step 3: Add 16 and 9 to get 25.
Step 4: Take the square root of both sides of the equation to isolate the variable c.
Step 5: We only need the positive value of the square root because we are talking about a distance.
The hypotenuse of the triangle is 5.
Angles - B
An angle is created by two rays with the same endpoint. That endpoint is called the vertex.
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An interesting method for improving the student's understanding of angles is to have him or her draw the
various types of angles. Then, develop a series of flash cards. On one side of the card, draw the figure.
On the other side of the card, write the name. The following are definitions to help get you started:
Obtuse Angle - an angle with a measure greater than 90 degrees and less than 180 degrees
Right Angle - an angle with a measure equal to 90 degrees
Acute Angle - an angle with a measure greater than 0 degrees and less than 90 degrees
Adjacent angles - two angles with a common vertex and a common side
Complementary angles - two angles whose measures have a sum of 90 degrees
Supplementary angles - two angles whose measures have a sum of 180 degrees
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Vertical angles - opposite angles formed by two intersecting lines; vertical angles are congruent
Congruency - C
Congruent figures have the same shape and size.
It may be beneficial to verify that the student understands the definition of similar and congruent figures.
To help understand congruence, draw two triangles exactly the same shape and size (use graph paper or
a copy machine to ensure the figures are the same). Draw an additional triangle of a different shape and
size. Cut out each figure. Arrange the figures on a table and ask the student to find the congruent
triangles. Remember, two figures are congruent if they are the same shape and size.
Symmetry:
A figure is said to have a line of symmetry when it can be folded in half along that line so that the two
halves are congruent. One artistic way to help the student understand symmetry is for you to draw half
of a simple shape (such as a square, circle, triangle, or heart) on a piece of paper and have the student
draw the other half. Fold the paper on the halfway line and compare to see if both sides match.
A vertical line of symmetry crosses through the figure vertically. A horizontal line of symmetry crosses
through the figure horizontally. A diagonal line of symmetry crosses through the figure at a diagonal.
The following three figures show the lines of symmetry (dotted lines):
Symmetry - D
The graph of an equation can be symmetric about a line or a point. The lines of symmetry that are important to
consider when graphing are the x- and y-axes. Often, the origin is the point of symmetry that is considered
when graphing a specific equation. Knowledge about the symmetric properties of a graph is helpful to speed up
the actual process of graphing an equation.
Symmetry with respect to the x- and y-axes:
The graph of an equation is symmetric with respect to the y-axis if both ordered pairs (x, y) and (-x, y)
are solutions of the equation. That is, if the point (2, 4) lies on the graph of an equation that is symmetric
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with respect to the y-axis, then (-2, 4) must also lie on the graph. Pictorially, a graph that is symmetric
with respect to the y-axis can be "folded" along the y-axis such that each point that lies to the right of the
y-axis will coincide with its corresponding point to the left of the y-axis. An example of a graph that is
symmetric with respect to the y-axis is shown below.
Note that for any point that is chosen that lies to the right of the y-axis, such as (2, 4), its corresponding
point, (-2, 4), also lies on the graph.
The graph of an equation is symmetric with respect to the x-axis if both ordered pairs (x, y) and (x, -y)
are solutions of the equation. That is, if the point (2, 4) lies on the graph of an equation that is symmetric
with respect to the x-axis, then (2, -4) must also lie on the graph. Pictorially, a graph that is symmetric
with respect to the x-axis can be "folded" along the x-axis such that each point that lies above the x-axis
will coincide with its corresponding point below the x-axis. An example of a graph that is symmetric
with respect to the x-axis is shown below.
Note that for any point that is chosen that lies above the x-axis, such as (4, 2), its corresponding point,
(4, -2), also lies on the graph.
A figure or equation can be symmetric with respect to both axes. The following figure is symmetric
about the y-axis because the graph to the right of the y-axis is the mirror image of the graph to the left of
the y-axis. The graph of the following figure is also symmetric about the x-axis since the graph above
the x-axis is the mirror image of the graph below the x-axis.
Example 1: Is the following figure symmetric with respect to the x-axis, y-axis, both axes, or neither of
the axes?
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Because each half of the graph is a mirror image with respect to the x-axis, the equation exhibits
symmetry with respect to the x-axis.
Example 2: Is the following figure symmetric with respect to the x-axis, y-axis, both axes, or neither of
the axes?
Because each half of the graph is a mirror image with respect to the y-axis, the equation exhibits
symmetry with respect to the y-axis.
Example 3: Is the following figure symmetric with respect to the x-axis, y-axis, both axes, or neither of
the axes?
Because each half of the graph is a mirror image with respect to both the x- and y-axes, the equation
exhibits symmetry with respect to both the x- and y-axes.
Example 4: Is the following figure symmetric with respect to the x-axis, y-axis, both axes, or neither of
the axes?
Because neither half of the graph is a mirror image about either of the axes, the figure does not exhibit
x-axis or y-axis symmetry.
Example 5: Point A and point B have symmetry with respect to the x-axis. What are the coordinates of
point B when point A is the point (3, -5)?
To find the coordinates of point B, simply find the opposite of the given y-coordinate of point A. The
opposite of -5 is 5. Do not change the x-coordinates because points that are symmetric with respect to
the x-axis have the same x-coordinate. Thus point B is (3, 5).
Example 6: Point A and point B have symmetry with respect to the y-axis. What are the coordinates of
point B when point A is the point (3, -5)?
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To find the coordinates of point B, simply find the opposite of the given x-coordinate of point A. The
opposite of 3 is -3. Do not change the y-coordinates because points that are symmetric with respect to
the y-axis have the same y-coordinate. Thus point B is (-3, -5).
Symmetry with respect to a point:
If two points, A and B, are symmetric to a given point, C, then the given point, C, is the midpoint of the
line segment that joins points A and B. The midpoint formula given below is used to determine the
coordinates of the midpoint of the line segment joining the two points
Example 7: The points A and B have symmetry with respect to a point C. If point A is (4, 6) and point
B is (-8, 10), what are the coordinates of point C?
Step 1: Since points A and B are symmetric with respect to C, C is the midpoint of segment AB. Write
down the midpoint formula.
Step 2: Substitute the given values:
Step 3: Simplify the fractions to get the coordinates of the ordered pair.
Therefore point C is (-2, 8).
Spatial Relationships - C
Spatial relationships include understanding geometric transformations as well as recognizing the projection of a
three-dimensional object into two dimensions.
Transformations:
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If a figure has changed its size, direction, or position, then the figure has been transformed. The
following examples will discuss four different transformations.
Rotation of a figure:
A rotation of a figure changes the direction that a figure is facing. The new figure is found by rotating
the original figure about a fixed point for a given number of degrees. The fixed point may be located on
or off the original figure.
Reflection of a figure:
If two figures are reflections, then they are mirror images of each other. A line of reflection can be
drawn between the figures such that if the paper upon which the figures had been drawn was folded on
this line, the two figures would coincide. That is, all of the points of one figure would lie upon all of the
points of the second figure. If any parts of the figures do not coincide, the figures are not reflections of
each other.
Translation of a figure:
A translation takes a figure and moves it in its entirety along a line from one position to another. Note
that after a figure is translated, the figure does not change the direction it is facing, its size, or its shape;
only its location has been changed. Watching a train engine move shows a translation in action. As the
engine passes you and moves down the track, it changes position, but the size and shape of the train
engine do not change. In the following example, the circle has been translated because its position has
changed.
Dilation of a figure:
A dilation of a figure can be thought of as the transformation that shrinks or stretches a figure. In the
following examples, the hexagon has shrunk in its size whereas the parabola has become "wider."
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How to determine the type of transformation:
To determine the type of transformation that has taken place, ask the following questions:
(1) Has the figure changed its size?
If yes, the transformation is probably a dilation.
(2) Has the figure changed its position by travelling along a linear (straight) path?
If yes, the transformation is probably a translation.
(3) Do the figures appear to be mirror images of each other?
If yes, the transformation is probably a reflection.
(4) Has the direction the figure was facing changed?
If yes, the transformation is probably a rotation.
It is possible to combine different types of transformations to make changes to the original figure. The
following examples will only involve one transformation.
Example 1: Which transformation was performed on the following figure?
Since the figure has changed its size by getting smaller, the transformation is a dilation.
Example 2: Which transformation was performed on the following figure?
Since the figure has not changed shape, but is facing a different direction, the transformation is a
rotation.
Example 3: Which transformation was performed on the following figure?
Since the triangle has shifted to the right along the x-axis, the transformation is a translation.
Example 4: Which transformation was performed on the following figure?
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Since the figure appears to have been "flipped" across the y-axis to get its mirror image, the
transformation is a reflection.
Spatial Relationships (Projection of a three-dimensional object into two dimensions:
The following problems will require the interpretation of two-dimensional pictures drawn of a threedimensional building. Two pictures of either the top, frontal, right, or left view will be presented and
then questions will be asked about what a viewer sees from one of the other sides.
A top view that looks like figure A below reflects a building that has one change in the height of the
building. Figure B shows a building that has two changes in the height of the building.
A frontal view such as the one in the diagram below better shows the differing heights of the building. In
this example, the building is shown to have three different heights from middle to low to high.
The following are the front and top views of a building to be used for Examples 5 and 6.
Example 5: What is the view from the right side of the building?
From the right side of the building, the viewer can only see the highest wall. The viewer has no idea if
there are any differing heights on the other side of the wall because the view is blocked by the high wall.
Thus, the view from the right side of the building is:
Example 6: What is the view from the left side of the building?
From the left side of the building, the viewer can see two different heights of the building, the middle
and highest heights. The "dip" in the building's height cannot be seen by the viewer. Thus the view from
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the left side of the building is:
Example 7: The following are the right and left views of a building. Which of the following could be
the frontal view?
The right view indicates to the viewer that there is at least one change of height with the known change
of height at a lower part of the building. The left view indicates to the viewer that there is at least one
change of height with the known change of height at a higher level of the building. The viewer cannot
tell whether there is more than one change of height because a top view is not presented to confirm the
exact number of height changes. Thus the only choice among the four options is Choice C. Note that
Choice A does show a height change on the left side, but shows no height change on the right side.
Choice B shows no height changes on either the right or left sides. Choice D shows no height change on
the left side.
Transformations
A transformation moves every point of a geometric figure to a new position in the coordinate plane.
It may be beneficial to develop a coordinate plane with graph paper. Cut out a figure, such as a
parallelogram, and plot it on the graph. Move (transform) the parallelogram to another area and plot
those points.
For example, a figure with coordinate points (1, 3), (5, 1), (4, 8), (2, 2) can be moved 5 places to the
right to have coordinate points of (6, 3), (10, 1), (9, 8), (7, 2). Help the student plot all of these points to
show him or her how the figure transforms.
A reflection is a transformation in which a figure is flipped over a line. Each point in the reflection is the
same distance from the line as the original point.
Example 1: The x-axis is the line of symmetry for figure ABCD. What is the reflection point of point
B?
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Answer: (-2, 2). Point B is represented by the coordinates (-2, -2). Point B is 2 units from the x-axis (the
line of symmetry). The figure will be reflected across the x-axis. The reflection point of point B will be
2 units above the x-axis at (-2, 2).
A rotation involves rotating a figure around a point called the point of rotation.
Example 2: What will the coordinates of point B be if the figure ABCD is rotated around point Y so that
point A is at (4, -1)?
Answer: (4, -3). Since figure ABCD is a square (each side is 2 units), when point A is rotated to the
position of point B, then point B will be rotated to the position of point C.
Congruence (AAS/ASA/SAS) - A
Congruent angles are two angles which have the same measure. Side-angle-side (SAS) congruence and angleside-angle (ASA) congruence are used to show that two triangles are congruent.
Each of the formulas are explained below.
The theorem for SAS:
If two sides and the included angle (the angle between the two sides) of one triangle are
congruent to the two sides and the included angle of a second triangle, then the two triangles are
congruent.
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The theorem for ASA:
If two angles and the included side (the side between the two angles) of one triangle are
congruent to the two angles and the included side of a second triangle, then the two triangles are
congruent.
The triangle sum theorem states that the sum of the three angles in a triangle is 180º .
Graphing Equations - A
Students must graph an equation (such as y = 3x - 5) on the coordinate plane.
Example 1: Graph: y = x + 3
Step 1: When given an equation, such as y = x + 3, the first step is to make a table of ordered pairs.
Select a list of values for x and then calculate the values of y.
Step 2: Graph the ordered pairs you calculated in Step 1. Remember, the x-value moves left (negative)
or right (positive) from zero and the y-value moves up (positive) or down (negative) from the x-value.
If you connect the points, this equation will create a straight line.
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In order to determine whether a set of ordered pairs is a solution to the graph of a line, you substitute
those values in for the variables.
Example 2: Which of the following points is a solution to the graph of the line, y = 2x + 4?
A. (2,7)
B. (1,6)
C. (1,3)
Substitute the x- and y-values from the ordered pairs into the equation y = 2x + 4.
(2, 7): 7 = 2(2) + 4
7 = 8
false
(1, 6): 6 = 2(1) + 4
6 = 6
true
(1, 3): 3 = 2(1) + 4
3 = 6
false
The answer is: (1, 6) is a solution to the graph because its values of x and y make the equation true.
An equation that represents a line has an infinite number of solutions because there are two variables in
the equation. If one of the variables is fixed, then there will only be one solution to the equation.
Example 3: If the equation y = 3x + 1creates a straight line, how many solutions to the equation are
there?
Solution: There are an infinite number of solutions because for every value of x, a different value of y
can be found.
Example 4: If 2y = x - 4 and x = 1, how many solutions to the equation are possible?
Solution: There is only one possible solution because if x = 1, 2y = 1 - 4, so y = -3/2. For each value of
x, there is only one value for y.
Graphing Equations - B
Students must graph an equation (such as y = 3x - 5) on the coordinate plane.
Example 1: Graph: y = x + 3
Step 1: When given an equation, such as y = x + 3, the first step is to make a table of ordered pairs.
Select a list of values for x and then calculate the values of y.
Step 2: Graph the ordered pairs you calculated in Step 1. Remember, the x-value moves left (negative)
or right (positive) from zero and the y-value moves up (positive) or down (negative) from the x-value.
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If you connect the points, this equation will create a straight line.
In order to solve a system of equations by graphing, create a table for each equation and graph both
equations on one coordinate plane. The following example illustrates how to do this.
Example 2: Solve the system of equations by graphing.
y = 6x - 4
y = 2x
Step 1: Make a table for each equation. Choose values for the variable x and solve the equation for the
value of y.
Step 2: Graph both equations on one coordinate plane.
The solution is (1, 2) because that coordinate point is where the graphs of the two lines intersect.
Example 4: The difference of six times a number (x) and four is another number (y). One number (y) is
twice the other number (x). What are the two numbers?
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(1) 6x - 4 = y
(2) y = 2x
Step 1: "The difference of six times a number (x) and four is another number (y)" can be written as an
equation by replacing word names with variables and numbers.
Step 2: "One number (y) is twice the other number (x)" can be written as an equation by replacing word
names with variables and numbers.
To finish solving this example, refer to Example 3 to see how to solve the equations by graphing.
Pythagorean Theorem
The Pythagorean theorem is used to find the lengths of the sides and hypotenuse of a right triangle.
The Pythagorean theorem states that the square of the hypotenuse (the longest side) of a right triangle (a
triangle with one 90 degree angle) is equal to the sum of the squares of the legs of that triangle (the two
shorter sides).
Looking at the diagram above, we can note that a and b always denote legs of a right triangle and c
always denotes the hypotenuse of a right triangle.
Example: The lengths of two legs of a right triangle are 3 and 4. What is the length of the hypotenuse?
Step 1: Determine the values of a, b, and c. In this case, a and b are known and c is the unknown length
of the hypotenuse.
Step 2: Substitute the values of a and b into the Pythagorean theorem.
Step 3: Square 3 (3 x 3 = 9) and square 4 (4 x 4 = 16).
Step 4: Add 9 and 16 to get 25.
Step 5: Take the square root of each side of the equation. The square root of 25 is 5 and the square root
of c-squared is c.
Answer: The length of the hypotenuse of this triangle is 5.
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Distance Formula
The distance formula can be used to find the distance between two points on a coordinate plane.
If you need to find the distance, d, between Point A and Point B on a coordinate plane, you can use the
distance formula:
Some distance problems can be solved by using the Pythagorean theorem. Pythagoras was a Greek
mathematician, philosopher, and theologian who lived around 580 to 500 B.C. He proved the universal
validity of a theorem later called the Pythagorean theorem.
The Distance Formula can be applied to find the distance between any two points.
Example: On a coordinate plane, point D is (3, 7), and point E is (6, 2). How far is point D from point
E?
Step 1: Substitute the values of the variables into the distance formula.
Step 2: Simplify (6 - 3) and (2 - 7).
Step 3: Square the two results of step 2.
Step 4: Simplify the radicand (the expression under the radical sign).
Answer:
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