# From the diagram, prove that the two equations m1 = 3x + 4 and m3 = x + 10 imply together that x = 3.

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### 2 Answers

The angles m1 and m3 are *opposite *angles.

This is *given* the fact that they are opposite each other on the diagram of the two intersecting lines. That they are opposite in this way implies that they are equal in size. (They are like a mirror image of each other, if you imagine a mirror placed between them). Since they are equal we can say that they are *congruent *as congruent is another way of saying *equal *or *identical.*

Since m1 and m3 are equal, we can equate the two *expressions* in x defining each one, giving an equation in the variable x:

3x + 4 = x + 10

Using the *subtraction property of equality, *which is that equality is maintained when the same amount is subtracted from both sides of the equation, this implies that

1) subtracting x from both sides gives us

3x + 4 - x = x + 10 - x, leading to

2x + 4 = 10

2) Similarly, subtracting 4 from both sides gives us

2x +4 - 4 = 10 - 4, leading to

2x = 6

Then, using the *division property of equality, *which is that equality is maintained when both sides of the equation are divided by the same amount, this implies that

3) dividing both sides by 2 gives us

2x `-:` 2 = 6 `-: ` 2, leading to

x = 3

as required.

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**Sources:**

Angle 1 and 3 are vertical/opposite angles meaning they are equal to each other. This means you can set the two equations given equal to each other and solve for x.

3x+4 = x+10

2x+4 = 10

2x = 6

*x = 3*

When you set the equations equal and solve, you get that x does, in fact, equal 3.