LINEAR PAIRS AND VERTICAL ANGLES

Linear Pair Theorem :

Linear pair of angles are formed when two lines intersect each other at a single point. The angles are said to be linear if they are adjacent to each other after the intersection of the two lines.

∠LAS and ∠PAS are adjacent angles.

The sum of angles of a linear pair is always equal to 180°.

Vertical Angle Theorem :

The vertical angle theorem states that the vertical angles that are formed when two lines intersect are congruent.

Problem 1 :

From the picture at the right, name the 4 sets of linear pair angles?

Solution: 

Linear pair angles :

∠1 and ∠3

∠1 and ∠2

∠4 and ∠3

∠4 and ∠2

Problem 2 :

From the picture at the right, name the 2 sets of vertical angles?

Solution:

Vertical angles :

∠1 = ∠4

∠2 = ∠3

Problem 3 :

Vertical angles are always ________.

Solution:

Vertical angles are always equal.

Problem 4 :

Linear pairs are always _____, which means they add up to _____.

Solution:

Linear pairs are always supplementary, which means they add up to 180º.

Problem 5 :

x = _____

Solution:

Here x and 63° are adjacent angles, so they add upto 180°.

x° + 63° = 180°

x = 180° - 63°

x = 117°

Problem 6 :

y = _____

Solution:

Vertical angles are always equal.

So, y = 150°

Problem 7 :

k = _____

Solution:

Here k and 103° are adjacent angles, they are supplementary.

k + 103° = 180°

k = 180° - 103°

k = 77°

Problem 8 :

a = _____

Solution:

4a + 120° = 180°

4a = 180° - 120°

4a = 60°

a = 15°

Problem 9 :

w = _____

Solution:

Vertical angles are always equal.

105° = w - 10°

w = 115°

Problem 10 :

d = _____

Solution:

(d + 13) and 95° are adjacent angle. So, they add upto 180°

95° + (d + 13)° = 180°

108° + d = 180°

d = 180° - 108°

d = 72°

Tell whether the statement is always, sometimes, or never true. Explain your reasoning.

Problem 11 :

Complementary angles are adjacent.

Solution :

Yes always, because the sum of the complementary angles will be 90 degree and it must be adjacent angles.

Problem 12 :

Angles in a linear pair are supplements of each other.

Solution:

Meaning of linear pair is two angles lie on the same line. Meaning of supplementary angles is the sum of the angles is 180 degree. 

So, the given statement is always true.

Problem 13 :

Vertical angles are adjacent.

Solution:

Vertical angles will face each other opposite direction. So, they won't be next to each other. So, they are not adjacent.

The given statement is never true.

Problem 14 :

Vertical angles are supplements of each other.

Solution:

Supplementary angles are the angles whose sum is 180 degree. Vertical angles will be positioned opposite. It is true always.

Problem 15 :

If an angle is acute, then its complement is greater than its supplement.

Solution:

Let x be the acute angle. Then, its 

• Complement will be 90 - x.
• Supplement will be 180 - x

Its complement will not be greater than its supplement. Then the given statement will become never true.

Problem 16 :

If two complementary angles are congruent, then the measure of each angle is 45°. 

Solution :

Let x be the angle, then its complement will be 90 - x

Since the angle and its complement both are congruent, we write 

x = 90 - x

x + x = 90

2x = 90

x = 90/2

x = 45

So, the given statement is always true.

Problem 17 :

Explain why the supplement of an acute angle must be obtuse. 

Solution:

Let x be the acute angle. Then, its supplement will be 180 - x. 

180 - x > x for any value of x. Then the given statement is always true.

Problem 18 :

Explain why an obtuse angle does not have a complement.

Solution:

The given statement will never become true, because choosing and obtuse angle x which is more than 90 degree. The sum of angle and its complement will become 90 degree always. 

Problem 19 :

The iron cross is a skiing trick in which the tips of the skis are crossed while the skier is airborne. Find the value of x in the iron cross shown.

Solution :

Since the shown angles are linear pair, the sum of the angle measures will become 180 degree.

127 + 2x + 41 = 180

2x + 168 = 180

2x = 180 - 168

2x = 12

x = 12/2

x = 6

So, the value of x is 6.

Write and solve an algebraic equation to find the measure of each angle based on the given description.

Problem 20 :

The measure of an angle is 9 more than twice its complement. 

Solution :

Let x be the angle meaure. Its complement will be 90 - x

x = 2(90 - x) + 9

x = 180 - 2x + 9

x + 2x = 189

3x = 189

x = 189/3

x = 63

So, the required angles are 63 and 27.

Problem 21 :

Two angles form a linear pair. The measure of one angle is four times the measure of the other angle.

Solution :

Let x be the other angle.

One angle = 4x

x + 4x = 180

5x = 180

x = 180/5

x = 36

4x = 4(36) ==> 144

So, the required angles are 36 and 144.