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Geometry
Lesson Activities
Proving the Laws of Sines and Cosines
The Lesson Activities will help you meet these educational goals:
 Content Knowledge—You will prove the Laws of Sines and Cosines and use them to
solve problems.
 Mathematical Practices—You will make sense of problems and solve them.
Directions
You will evaluate some of these activities yourself, and your teacher may evaluate others.
Please save this document before beginning the lesson and keep the document open for
reference during the lesson. Type your answers directly in this document for all activities.
_________________________________________________________________________
Self-Checked Activities
Read the instructions for the following activities and type in your responses. At the end of the
lesson, click the link to open the Student Answer Sheet. Use the answers or sample
responses to evaluate your own work.
1. Law of Sines
In this activity, you will explore how trigonometric ratios can be used to solve a non-right
triangle.
a. To solve a triangle in which one side length and two angle measures of a triangle are
known, you will use the GeoGebra geometry tool to find a relationship between angle
measures and opposite side lengths in a triangle. Go to triangle one side two angles,
and complete each step below. If you need help, follow these instructions for using
GeoGebra.
i.
Calculate mABC using the Triangle Sum Theorem.
Type your response here:
ii. Construct a line through B that is perpendicular to AC . Label the intersection of
this line and AC as point D. Based on their angle measures, how can you classify
ΔABD and ΔCBD?
Type your response here:
iii. Considering BD as a side of ΔABD, express its length in terms of variables
representing side lengths and angle measures in ΔABD. Show your work.
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Type your response here:
iv. Considering BD as a side of ΔCBD, express its length in terms of variables
representing side lengths and angle measures in ΔCBD. Show your work.
Type your response here:
v. Combine the two expressions found in parts iii and iv to relate a, c, A, and C.
Rewrite the expression as an equality between two ratios. Show your work.
Type your response here:
vi. Use the equation you found in part v to calculate a to two decimal places. Show
your work.
Type your response here:
b. Construct a line through A that is perpendicular to BC. Label the point of intersection
of this line and BC as point E.
i.
Based on their angle measures, how can you classify ΔABE and ΔACE?
Type your response here:
ii. Considering AE as a side of ΔAEC, express its length in terms of variables
representing side lengths and angle measures in ΔAEC. Show your work.
Type your response here:
iii. Considering AE as a side of ΔABE, express its length in terms of variables
representing side lengths and angle measures in ΔABE. Show your work.
Type your response here:
iv. Combine the two expressions found in parts ii and iii to relate b, c, B, and C.
Rewrite the expression as an equality between two ratios. Show your work.
Type your response here:
v. Use the equation you found in part iv to calculate b to two decimal places. Show
your work.
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Type your response here:
vi. Combine the equations you found in parts a v and b iv to write a relationship
between pairs of measurements (angles and opposite sides) in a triangle. Explain
how this relationship can be used to solve for all the measurements on a triangle
when one side and two angles are given. You’ll need to list more than one step.
Type your response here:
c. To solve a triangle in which two side lengths and one angle measure of a triangle are
known initially, you will use GeoGebra to find a relationship between angle measures
and opposite side lengths in a triangle. Go to triangle two sides one angle, and
complete each step below.
i.
Construct a line through B that is perpendicular to AC. Label the point of
intersection of this line and AC as D. Based on their angle measures, how can you
classify ΔABD and ΔBCD?
Type your response here:
ii. Considering BD as a side of ΔABD, express its length in terms of variables
representing side lengths and angle measures in ΔABD. Show your work.
Type your response here:
iii. Considering BD as a side of ΔBCD, express its length in terms of variables
representing side lengths and angle measures in ΔBCD. Show your work.
Type your response here:
iv. Combine the two expressions you found in parts ii and iii to relate a, c, A, and C.
Rewrite the expression as an equality between two ratios. Show your work.
Type your response here:
v. Use the equation you found in part iv to calculate mC to two decimal places.
Show your work.
Type your response here:
vi. Calculate mB to two decimal places using the Triangle Sum Theorem.
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Type your response here:
vii. Similarly, by constructing a line through point A that is perpendicular to and passes
b
c

through BC , you get
. Using this relationship, what is the length of
sin B sinC
side b to two decimal places?
Type your response here:
viii. What relationship can you derive from part iv and the relationship you used in part
vii? How can you use this relationship to solve for all the measurements on any
triangle when the length of two sides and the measure of an angle opposite one of
the two sides are given? You’ll need to list more than one step.
Type your response here:
How did you do? Check a box below.
Nailed It!—I included all of the same ideas as the model response on the Student Answer Sheet.
Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.
Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.
2. Law of Cosines
You will use the GeoGebra geometry tool to study the interior angles of triangles. Go to
triangle squares and defects, and complete each step below.
a. Verify that ΔABC is a right triangle by showing that the sum of the areas of the squares
formed with sides a and b is equal the area of the square formed with side c. (Hint: If
ΔABC is a right triangle, its sides must satisfy the Pythagorean Theorem.)
Type your response here:
b. Change ΔABC by moving point B to set mBCA to the values in the table. As you
move point B, you’ll notice that a defect, or a rectangular area, is formed either within
or next to square a2 and square b2. Calculate and record the areas listed in the table to
two decimal places for each measure of BCA.
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Type your response here:
Area of
Area of
2
Square b2
mBCA Square a
Area of
Defect1
Area of
Defect2
Area of
Square c2
60°
80°
100°
120°
c. Now examine the relationship that exists among the areas in the table above. Fill in
the first two columns of the table below by performing the operations shown in the
column headers. You can copy the areas in the last column from the table in part b.
Type your response here:
Area of Square a2 + Area
of Square b2 – Area of
Defect1 – Area of Defect2
mBCA
Area of Square a2 + Area
of Square b2 + Area of
Defect1 + Area of Defect2
Area of
Square c2
60°
80°
100°
120°
d. What relationships between the areas do you observe when mBCA < 90°, and when
mBCA > 90°?
Type your response here:
e. Now you’ll try to write formulas for the areas of defect1 and defect2. Start by moving B
to make C acute. Notice that ΔBCD is a right triangle. Use a trigonometric ratio to
relate b1 to a.
Type your response here:
f. Use the equation you wrote in part e to express the area of defect2 in terms of the
measures of ΔABC. The variable b1 should not appear in the final expression.
(Hint: Use the formula for the area of a rectangle, area = length × width.)
Type your response here:
g. If C is acute, ΔARC is also a right triangle. Use a trigonometric ratio to relate a1 to b.
Type your response here:
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h. Use the equation you wrote in part g to express the area of defect1 in terms of the
measures of ΔABC. (The variable a1 should not appear in the final expression.)
Type your response here:
i.
Using the relationship found in part d, when C is acute, area of square c2 = area of
square a2 + area of square b2 − area of defect1 − area of defect2. Now combine this
observation with the formulas for the areas of defect1 and defect2 that you wrote in
parts f and h. The result is an equation relating one side of ΔABC to the other two
sides and the angle those two sides include.
Type your response here:
j.
If you follow a similar approach by drawing a line through point C that is perpendicular
to AB and using the associated defects for A or B instead of C, what are the
resulting equations? (Do not go through the complete derivation again, just think about
the result you came up with in part i and what similar formulas for A or B would look
like.)
Type your response here:
k. In what kinds of problems can you apply the equations in parts i and j to solve for the
side lengths and angle measurements in triangles? Explain. (Consider the measures
of the triangle that are given at the onset of a problem.)
Type your response here:
How did you do? Check a box below.
Nailed It!—I included all of the same ideas as the model response on the Student Answer Sheet.
Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.
Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.
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