**Q: Geometry and Angles **

**Ray ****RY bisects angle ****NRT
Measure of angle **

**NRY = 2x² + 6x**

Measure of angle

Measure of angle

**YRT = 10x + 6**

Find x and the measure of angle

Find x and the measure of angle

**YRT**

A: Ahhh, fun to answer without a picture. But, we can do it. We have an angle that is *bisecting*. Bisected means cut in half. Therefore, **angle NRY = angle YRT**:

Mathematically: 2x² + 6x = 10x + 6

Now, time to solve! Since I see an “x²”, I think to myself that we must factor this. First step to factoring? Set the equation equal to zero:

2x² + 6x = 10x + 6

2x² + 6x – 10x = 6

2x² + 6x – 10x – 6 = 0

2x² – 4x – 6 = 0

Now, this is a quadratic that needs solving. We can either solve using the **quadratic equation** or by factoring. Let’s factor (View Factoring 101 for the basics of factoring):

2x² – 4x – 6 = 0

(2x )(x ) = 0

(2x + )(x – ) = 0 [I know there is a – and a + because the two numbers must multiple to give -6… And, a – times a + equals a -, right? However, I do not know that the + goes with the 2x and the – goes with the x… This is just my best guess to start. Factoring can be treated as a guessing game – if you use guess and check, you get better at it with time.]

(2x + 2)(x – 3) = 0

So,

(2x + 2) = 0 **or **(x – 3) = 0

x = -1 **or **3

We have two potential answers. Take note, if x=-1, then we have:

**Angle NRY = 2x² + 6x** = **2(-1)² + 6(-1) = 2 – 6 = -4**

An angle cannot be negative, therefore x cannot be -1.

Therefore, x must equal 3:

**Angle YRT = 10x + 6**

So, if x = 3:

**Angle YRT **= 10(3) + 6 = 30 + 6 = 36.