PROOF AND PERPENDICULAR LINES WORKSHEET

Problem 1 :

In the diagram given below,

∠5 and ∠6 are a linear pair

∠6 and ∠7 are a linear pair

Problem 2 :

In the diagram given below, ∠1 and ∠2 are congruent and also a linear pair. Using flow proof, prove that the lines g and h are perpendicular.

Problem 3 :

If two sides of the adjacent acute angles (2x + 3)° and (4x - 6)° are perpendicular, find the value of 'x'.

Problem 4 :

In the diagram given below, the lines m and n are perpendicular. Find the measures of the angles ∠1, ∠2, ∠3 and ∠4.

1. Answer :

Two-column Proof :

Statements

∠5 and ∠6 are a linear pair

∠6 and ∠7 are a linear pair

∠5 and ∠6 are complementary

∠6 and ∠7 are complementary

∠5  ∠7

Reasons


Given


Linear pair postulate

Congruence Supplements theorem

Paragraph Proof :

Because ∠5 and ∠6 are a linear pair, the linear pair postulate says that ∠5 and ∠6 are supplementary. The same reasoning shows that ∠6 and ∠7 are supplementary. Because ∠5 and ∠7 are both supplementary to ∠6, the congruent supplements theorem says that ∠5  ∠7.

Flow Proof :

2. Answer :

3. Answer :

According to result 2, if two sides of two adjacent acute angles are perpendicular, then the angles are complementary.

So, we have 

(x+3)° + (2x-6)°  =  90°

x + 3 + 2x - 6  =  90 

Simplify. 

3x - 3  =  90

Add 3 to both sides. 

3x  =  93

Divide both sides by 3. 

x  =  31

4. Answer :

According to result 3, if two lines are perpendicular then they intersect to form four right angles. 

So, we have 

m∠1  =  90°

m∠2  =  90°

m∠3  =  90°

m∠4  =  90°

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