Vertical Angles meet at a vertex but are on opposite sides of the intersecting lines (definition).

Theorem – If two lines intersect then vertical angles are congruent

In the above diagram, vertical angles are

Angle 1 and angle 3

Angle 2 and angle 4

So,

m∠1 ≅ m∠3 —-> m∠1 = m∠3

m∠2 ≅ m∠4 —-> m∠2 = m∠4

Example 1 :

Look at the picture shown below and answer the following questions.

(i) Are ∠1 and ∠3 vertical angles ?

(ii) Are ∠2 and ∠4 vertical angles ?

Solution :

Solution (i) :

No. The angles are adjacent but their non-common sides are not opposite rays.

Solution (ii) :

Yes. The angles are adjacent and their non-common sides are opposite rays.

Solution (iii) :

No. The sides of the angles do not form two pairs of opposite rays.

Solution (iv) :

No. The sides of the angles do not form two pairs of opposite rays.

Example 2 :

In the diagram shown below, using Linear Pair Postulate, solve for x and y. Then, find the angle measures and analyze your results with Vertical Angles Theorem.

Solution :

Use the fact that the sum of the measures of angles that form a linear pair is 180°.

Solving for x :

∠AED and ∠DEB form a linear pair.

m∠AED + m∠DEB = 180°

Substitute m∠AED = (3x + 5)° and m∠DEB = (x + 15)°.

(3x + 5)° + (x + 15)° = 180°

Simplify.

4x + 20 = 180

Subtract 20 from each side.

4x = 160

Divide each side by 4.

x = 40

Solving for y :

∠AEC and ∠CEB form a linear pair.

m∠AEC + m∠CEB = 180°

Substitute m∠AEC = (y + 20)° and m∠CEB = (4y – 15)°.

(y + 20)° + (4y – 15)° = 180°

Simplify.

5y + 5 = 180

Subtract 5 from each side.

5y = 175

Divide each side by 5.

y = 35

Use substitution to find the angle measures :

m∠AED = (3x + 5)° = (3 • 40 + 5)° = 125°

m∠DEB = (x + 15)° = (40 + 15)° = 55°

m∠AEC = ( y + 20)° = (35 + 20)° = 55°

m∠CEB = (4y – 15)° = (4 • 35 – 15)° = 125°

So, the angle measures are 125°, 55°, 55°, and 125°. Because the vertical angles are congruent, the result is reasonable.

Example 3 :

In the stair railing shown at the right,if∠6 has a measure of 130°, find the measures of the other three angles.

Solution :

∠6 and ∠8 are vertical angles, they are equal.

m∠8 = m∠6

Substitute m∠6 = 130°.

m∠8 = 130°

∠5 and ∠6 form a linear pair, they are supplementary.

m∠5 + m∠6 = 180°

Substitute m∠6 = 130°.

m∠5 + 130° = 180°

Subtract 130° from both sides.

m∠5 = 50°

∠5 and ∠7 are vertical angles, they are equal.

m∠7 = m∠5

Substitute m∠5 = 50°.

m∠7 = 50°

Therefore,

m∠5 = 50°

m∠7 = 50°

m∠8 = 130°

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