Concepts I touch on that you should be familiar with:

• All the angles in a triangle add to 180º
• Adding a negative number is subtraction 

The distributive property applies when you multiply a number by a sum or difference of two numbers. For example: 5(1+2). You “distribute” or multiply the 5 to each of the numbers being summed, the 1 and the 2, creating a new expression: 5+10. You’ll notice 5+10=15. Given the original expression 5(1+2), you may approach it as an order of operations challenge. You start with the grouping symbols: the parentheses. Inside the parentheses you have 1+2 which is 3. Then, you multiply that sum by the 5 outside of the parenthesis and get 5*3 which is 15. Notice that you get 15 either way. 

Multiplication and addition (or adding by a negative which is better known as subtraction) are linked in our algebra. 3*2 is just 3+3. 3*3 is 3+3+3. If you have 5 of the (1+2)s, then you have (1+2)+(1+2)+(1+2)+(1+2)+(1+2). Since we are only adding there is no need for parentheses, and you’ll find all the 1s add up to 5 and all the 2s add up to 10. That’s how we get the 5+10 in our distribution process. 

What if I give you the expression -(2+4)? How can you have a negative amount of (2+4)s? Don’t discount negative numbers. A negative times a positive is a negative. 2+4=6 is positive, and it’s multiplied by a negative. It’s easy to think of the negative sign as simply modifying whatever is after it, as if the negative is an accessory. -(2+4) is just -(8) so -8, of course. But we want to use the distributive property to solve this expression. Imagine the negative at the front as a -1. We distribute the leading number in -1(2+4) to get (-2)+(-4), simply by multiplying both numbers in the parentheses by -1. If you add a negative you are subtracting, so -2-4 is our expression which equals -6. 

Why is this important? Let’s say you’re given two angles in a triangle, 60º and 30º, and you know those two angles and the third remaining angle must add to 180º. To find the unknown angle (part), you want to subtract the sum of 60º (part) and 30º (part) from 180º (total). So you write 180-60+30. 180-60 is 120 and 120+30 is 150. But 150+30+60 is nowhere near 180! The problem is the way we structured the expression in the first place. You are trying to find the remaining angle by subtracting both other angles together from 180. So really you need the sum of the angles first. Simply add parentheses to indicate the order to solve in 180-(60+30). Oh look at that! It looks like our -(2+4) problem from earlier. This can be written as 180+(-(60+30)) because subtracting is just adding a negative. Let’s look at -(60+30). We can distribute the negative and get (-60)+(-30). -60-30 is -90. Now we put that into the full equation and get 180-90 which equals 90! And we can check our work by seeing if 90+30+60=180, which it does. 

Always remember to distribute the negative. It will save you massive headaches later in your math journey!