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2.4 Reasoning in Algebra Properties Utilize the following properties throughout your studies in this section. The last three properties may seem new, but are very useful and handy. Example #1 Solve for X by filling in the property that justifies each step. So, starting off with the first step. We can justify a. without looking at the numbers and variables given. You simply add up the angles to get 180 degrees, a straight line. The justification would be Angle Addition Postulate ….. Therefore, a. Angle Addition Postulate. Second Step (b): The justification has to be a property where something replaces another, otherwise known as ….. Subsitution property. Clearly in the diagram, angle CDE is also x and angle EDF is (3x+20). Third Step (c): Try to solve this step………………………… You may think that this is addition property, but you’re wrong. Because the variables are the same, the parentheses are unneeded so this is simply a case of Simplification. Fourth Step (d): In this step, you need to reduce 4x+20=180 to 4x=160. This means you need to subtract 20 from both sides. This calls for the Subtraction Property of Equality. Fifth Step (e): Last Step! You need to produce x= 40 from 4x=160. With basic algebra, you can realize this takes division. Hence, the Division Property of Equality. Now a bit harder and longer, Example #2 A bit more difficult, Example #3. a. Segment Addition Postulate* b. Substitution Property of Equality c. Distributive Property d. Simplify e. Subtraction Property of Equality f. Division Property of Equality *If three points, A, B, and C, are collinear, and B is between points A and C, then AB+BC=AC. Last Example! #3 Which property justifies this statement? In this situation, the BC is being added to the right side and subtracted from the left. This is known as Addition Property of congruence. Now, do it on your own with these 3 practice problems! 1) 2) 3) Answer Key 1) D 2) A. mEPF+mFPG=mEPG by Angle Addition Postulate B. mEPF=mFPG by Definition of Angle Bisector C. 4x+2=6x-10 by Substitution Property of Equality D. 4x+12=6x by Addition Property of Equality E. 12=2x by Subtraction Property of Equality F. 6=x by Division Property of Equality 3) In the fifth step, the Distributive Property should have simplified to ab-a2=b2-a2. 2.5 Proving Angles Congruent Definitions Theorem: A conjecture (a conclusion reached by using inductive reasoning) that is proven Paragraph proof: a proof written as sentences in a paragraph Theorems Theorem 2.1 Vertical Angles Theorem Vertical angles are congruent. Theorem 2.2 Congruent Supplements Theorem If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent. Theorem 2.3 Congruent Complements Theorem If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent. Theorem 2.4 All right angles are congruent. Theorem 2.5 If two angles are congruent and supplementary, then each is a right angle. Remember, in a proof, information in the “Given” supplies what you know. The “Prove” section supplies what you need to show, ultimately completing your proof. Example #1 using Vertical Angles Theorem Solve for x. Example #2 a. 90 b. 90 X+10=4x-35 Vertical Angles are congruent 10=3x-35 Subtraction Property of Equality 45=3x Addition Property of Equality 15=x Division Property of Equality c. Substitution Property of Equality d. m3 Example #3 using Theorem 2.5 a. V b. 180 (Definition of supplementary angles) c. Division (NOT MULTIPLICATION!) d. right Now time for practice! 1) Find the value of x. 2) Write a paragraph proof with the following information. 3)Find the value of each variable and the measure of each labeled angle. Answer Key 1) X=15 2) Because of Vertical Angles Theorem, angles 2 and 3 are congruent. Since the Given states that angle 1 is complementary to angle 2, you can say that angle 1 is complementary to angle 3 by Substitution Property. Also since angle 3 is complementary to angle 4, and angle 1 and 3 are complements, then bby Congruent Complements Theorem, you can state that angles 1 and 4 are congruent. Angle 5 is supplementary to angle 1 by Definition of Supplementary Angles and so are angles 4 and 6. Interchanging angles 1 and 4 by Substitution property, angles 5 and 6 are both supplementary to angle 4. Angles 5 and 6 are congruent by Congruent Supplements Theorem. 3)Y = 70, X=35, 2X= 70, X+Y+5=110