Mathematics
Grade9
Easy
Question
Use the Vertical Angles Congruence Theorem to find the measure of each angle in the diagram at the right.
Hint:
We are given a figure of two intersecting lines. As the lines are intersecting, four angles are formed. We are asked to find all the angles. We have to use the vertical angles congruence theorem to solve the question. The angles which are opposite to each other are called as adjacent angles. The angles opposite to each other are vertically opposite angles.
The correct answer is:
The intersecting lines are PQ and LM. Intersection point is O.
The given angles and the values are as follows:
∠POM = 16y°
∠MOQ = 6(x + 2)°
∠LOM = (18y – 8)°
∠POL = 10(x – 4)°
Vertical angles congruence theorem states that, the angles which are vertically opposite to each other are congruent.
For the given figure, ∠POM and ∠LOM are congruent.
Similarly, ∠MOQ and ∠POL are congruent.
We will equate the values of the given angles.
∠POM = ∠LOM
16y = 18y – 8
Adding 8 to both the sides and rearranging the equation we get,
18y – 8 + 8 = 16y + 8
18y = 16y + 8
Subtracting 16y from both the sides.
18y – 16y = 16y – 16y + 8
2y = 8
y = 4
Similarly,
∠MOQ = ∠POL
6(x + 2) = 10(x – 4)
6x + 12 = 10x – 40.
10x – 40 = 6x + 12
We will take the variables to one side and constants to the other sides.
10x – 6x = 12 + 40
4x = 52
x = 13
Now, we will substitute the values of x and y in the equations of angles.
∠POM = 16y°
= 16(4)
= 64°
∠MOQ = 6(x + 2)°
= 6(13 + 2)
= 6(15)
= 90°
The values of the angles are as follows:
∠POM = ∠LOM = 64°
∠MOQ = ∠POL = 90°
For such questions, we should know properties of vertical angles. We should know the rules of algebra to solve the equations. The alternate way to solve the question will be using supplementary angles. When two angles intersect, the sum of adjacent angles is 180°.
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