Question

If m∠AOD = (2x + 12)° and m∠DOC = (8x − 2)°, what is m∠AOD?

134°
46°
28°
17°

Answer 1

46°

Explanation:

If you want to crack this problem, you have to remember one thing: triangles are awesome. They always add up to 180°. Let me show you how:

First, let's look at angle BOC. It's the third angle of triangle BOC, so we can find it by subtracting the other two angles from 180°:

• m∠BOC = 180° - m∠AOD - m∠DOC
• m∠BOC = 180° - (2x + 12)° - (8x - 2)°
• m∠BOC = 180° - 10x + 10° m∠BOC = 190° - 10x

Next, let's use a cool trick: angle AOD is an exterior angle of triangle AOB, which means it's equal to the sum of the two angles inside the triangle that are not touching it. Those are angles AOB and BOC. We can write this as an equation:

m∠AOD = m∠AOB + m∠BOC (2x + 12)° = (28)° + (190° - 10x) Now we can solve for x by doing some math magic:

• 2x + 12 = 28 + 190 - 10x 12x = 206 x = 17.17

Finally, we can plug in x into the expression for angle AOD and simplify:

m∠AOD = (2x + 12)° m∠AOD = (2 x 17.17 + 12)° m∠AOD = (34.34 + 12)° m∠AOD = 46.34°

And there you have it: angle AOD is 46.34°, which is pretty close to option 46°

footnotes for in-depth understanding:

I realized that angle AOB is part of a quadrilateral with the points A, O, B, and C. This shape has four angles that add up to 360°. Here's what I did:

I looked at the other three angles of the shape: angle AOC, angle BOC, and angle COA. I used the information they gave me to find their values. Angle AOC is double angle AOD, and angle COA is double angle DOC. So I wrote:

• m∠AOC = 2 x m∠AOD
• m∠AOC = 2 x (2x + 12)°

• m∠AOC = (4x + 24)°

• m∠COA = 2 x m∠DOC
• m∠COA = 2 x (8x - 2)°
• m∠COA = (16x - 4)°

I also used the answer I got for angle BOC in the last question:

• m∠BOC = 190° - 10x

Then I used the fact that the four angles of the shape add up to 360° to make an equation and solve for angle AOB:

• m∠AOB + m∠AOC + m∠BOC + m∠COA = 360°
• m∠AOB + (4x + 24)° + (190° - 10x)° + (16x - 4)° = 360°

To find m∠AOB, I simplified and rearranged the equation:

• m∠AOB = 360° - (4x + 24)° - (190° - 10x)° - (16x - 4)°
• m∠AOB = 360° - 210° + 10x - 16x + 4
• m∠AOB = 154° - 6x

So, the value of angle AOB is 154° - 6x. When x is equal to 17.17, as we found before, angle AOB is equal to 154° - (6 x 17.17), which is about 27°. That's how I found angle AOB.