Joseph A. Montagna
Complementary angles
are two angles whose measure equal 90 degrees.
Supplementary angles
have a sum of their measures equal to 180 degrees.
Vertical angles
are angles that are opposite one another when two lines intersect.
The final lessons of this unit concern geometry proofs. We will look at three theorems of geometry that relate to the three types of angles above. A
theorem
is a statement that has been proved by a logical reasoning process. It is hoped that students will understand how a proof proceeds through its steps, and that they may be able to write a proof of their own. Students should know that it is helpful to remember these and other theorems.
Given Information:
(figure available in print form)
Conclusion: 3= 4
To reach the above conclusion we take the following steps:
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STATEMENTS
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REASONS
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1. ABC = 90 degrees
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by definition of lines
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2. PQR = 90 degrees
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by definition of lines
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3. 1 + 3 = 2 + 4
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each sum = 90 degrees
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4. 1 = 2
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given
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5. 3 = 4
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Subtraction Postulate
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The conclusion is reached in step 5.
A theorem that results from this proof is:
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Theorem 1: If two angles are complements of equal angles, then the two angles are equal.
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Given information:
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1 and 2 are supplements
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3 and 4 are supplements
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1 equals 3
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Conclusion:
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2 = 4
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STATEMENTS
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REASONS
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1. 1 + 2=180 degrees
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by definition of supplements
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2. 3 + 4=180 degrees
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by definition of supplements
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3. 1 + 2 = 3 + 4
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the sums of each = 180 degrees
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4. 1 = 3
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given
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5. 2 = 4
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Subtraction Postulate
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Theorem 2: If two angles are supplements of equal angles, then the two angles are equal.
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Theorem 3: Vertical angles are equal.
Have the class try to prove theorem 3. This is an easy proof, comprised of three steps.
Geometry books are loaded with exercises on writing proofs. Give students an opportunity to work on these kinds of exercises. Below is a list of reasons that are used in proofs.
REASONS USED IN PROOFS
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* The information is given
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* Definitions
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* Postulates
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* Theorems
The limitation of space makes it necessary to cut these lessons at this point. If one were to continue beyond this point the major topics to be covered should be: triangles and other polygons, areas of polygons, proportions, circles, solid geometry, and the areas and volumes of solids. Some of these topics are covered by objectives in math for 7th and 8th graders, while others are appropriate for high school level.
