2-5 Reasoning in Algebra and Geometry

Quick Review

You use deductive reasoning and properties to solve equations and justify your reasoning.

A proof is a convincing argument that uses deductive reasoning. A two-column proof lists each statement on the left and the justification for each statement on the right.

Example

What is the name of the property that justifies going from the first line to the second line?

∠ A ≅ ∠ B  and  ∠ B ≅ ∠ C ∠ A ≅ ∠ C table with 2 rows and 1 column , row1 column 1 , angle eh approximately equal to angle b , and , angle b approximately equal to angle c , row2 column 1 , angle eh approximately equal to angle c , end table

Transitive Property of Congruence

Exercises

31. Algebra Fill in the reason that justifies each step.

    Given: Q S = 42 q s equals 42

    Prove: x = 13 x equals , 13

    

    Statements	Reasons
    1) Q S = 42 q s equals 42	1) a. ? ¯ modified question mark with under bar below
    2) Q R + R S = Q S q r plus r s equals q s	2) b. ? ¯ modified question mark with under bar below
    3) ( x + 3 ) + 2 x = 42 open x plus , 3 close plus , 2 x equals , 42	3) c. ? ¯ modified question mark with under bar below
    4) 3 x + 3 = 42 3 x plus . 3 equals , 42	4) d. ? ¯ modified question mark with under bar below
    5) 3 x = 39 3 x equals , 39	5) e. ? ¯ modified question mark with under bar below
    6) x = 13 x equals , 13	6) f. ? ¯ modified question mark with under bar below

Use the given property to complete the statement.

32. Division Property of Equality If 2 ( A X ) = 2 ( B Y ) , 2 open eh x close equals 2 open b y close comma  then A X = ? ¯ . eh x equals , modified question mark with under bar below , .
33. Distributive Property: 3 p − 6 q = 3 ( ? ¯ ) 3 p minus , 6 q equals , 3 open , modified question mark with under bar below , close

2-6 Proving Angles Congruent

Quick Review

A statement that you prove true is a theorem. A proof written as a paragraph is a paragraph proof. In geometry, each statement in a proof is justified by given information, a property, postulate, definition, or theorem.

Example

Write a paragraph proof.

Given: ∠ 1 ≅ ∠ 4 angle 1 approximately equal to , angle 4

Prove: ∠ 2 ≅ ∠ 3 angle 2 approximately equal to , angle 3

Image Long Description

∠ 1 ≅ ∠ 4 angle 1 approximately equal to angle 4  because it is given. ∠ 1 ≅ ∠ 2 angle 1 approximately equal to angle 2  because vertical angles are congruent. ∠ 4 ≅ ∠ 2 angle 4 approximately equal to angle 2  by the Transitive Property of Congruence. ∠ 4 ≅ ∠ 3 angle 4 approximately equal to angle 3  because vertical angles are congruent. ∠ 2 ≅ ∠ 3 angle 2 approximately equal to angle 3  by the Transitive Property of Congruence.

Exercises

Use the diagram for Exercises 34–37.

34. Find the value of y.
35. Find m ∠ A E C . m angle eh e c .
36. Find m ∠ B E D . m angle b e d .

37. Find m ∠ A E B . m angle eh e b .

38. Given: ∠ 1 angle 1  and ∠ 2 angle 2  are complementary. ∠ 3 angle 3  and ∠ 4 angle 4  are complementary. ∠ 2 ≅ ∠ 4 angle 2 approximately equal to , angle 4

    Prove: ∠ 1 ≅ ∠ 3 angle 1 approximately equal to , angle 3

    

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