**Proof and perpendicular lines worksheet :**

Worksheet given in this section is much useful to the students who would like to practice problems on different types of proofs in geometry and results about perpendicular lines.

**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)° aaaaaa 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.

**Problem 1 :**

In the diagram given below,

∠5 and ∠6 are a linear pair

∠6 and ∠7 are a linear pair

Prove ∠5 ≅ ∠7 using two-column proof, paragraph proof and flow proof.

**Solution :**

**Two-column Proof :**

∠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 |
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 :**

**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.

**Solution :**

**Problem 3 :**

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

**Solution : **

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

**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.

**Solution :**

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°

After having gone through the stuff given above, we hope that the students would have understood "Proof and perpendicular lines worksheet".

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