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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