if m<ABC=7x-4 and m<CBX=7-x, find the value of x.

Question

Please explain the steps

Answer 1

Hi Alia,

I believe there must be a picture of angle(s) that accompanies the question. From your question, there can be several possibility on where point X can be. Based on the location of point X and the relationship between (< for angle) <ABC and <CBX, we can write an equation to solve for X.

Case 1: Point X is somewhere on ray BA, then m<ABC = m<CBX. By substitution, 7x-4 = 7-x. Add x to both sides, we get 8x-4 = 7. Add 4 to both sides, we get 8x = 11. Divide both sides by 8, we get x= 11/8 or 1 3/8.

Case 2: Point X is on another ray with starting point at point B, forming an angle <CBX sharing one side, BC, with <ABC. If <ABX is an right angle (90 degrees), then m<ABC + m<CBX = 90. By substitution, (7x-4) + (7-x) = 90. Combining like terms, we get 6x+3 = 90. Subtract 3 from both sides, we get 6x = 87. Divide both sides by 6, we get x=87/6 or 14 1/2 or 14.5 degrees. If <ABX is a straight angle (180 degrees), then m<ABC + m<CBX = 180. By substitution, (7x-4) + (7-x) = 180. Combining like terms, 6x + 3 = 180. Subtract 3 from both sides, 6x = 177. Divide both sides by 6, we get x = 177/6 or 29 3/6 or 29 1/2 or 29.5 degrees.

Because I don't see the picture, these are the only cases I can easily imagine. If they don't match the picture you have, please let me know. With a better description of the drawn picture/figure you have, we can then go from there. Let me know also if one of these cases matches your picture, then let me know if the solution I provided is of help to you.

Faye Huey-Ming H.

Answer 2

I am going to presume that ABX is a straight line. If not, please describe the figure.

In other words, the figure should look something like this:

C

A______/_______X

B

If line segment ABX is not explicitly stated as being a straight line in the problem, you cannot presume it.

By the angle addition postulate,

m∠ABC + m∠CBX = m∠ABX = 180º

since ∠ABX is a straight line.

Given:

m∠ABC = (7x-4)º

m∠CBX = (7-x)º

Using the substitution property of equality,

(7x-4)º + (7-x)º = 180º

For simplicity, let's drop the degree marks while solving this equation.

7x-4 + 7-x = 180

We can use the commutative property of addition (a+b= b+a) while we are combining like terms:

7x + 3 -x = 180

Subtract 3 from both sides:

6x = 177

Now divide both sides of the equation by the coefficient of x:

6x/6 = 177/ 6 = 29.5

x = 29.5

Since x is inside the parentheses above and the degree mark is outside the parenentheses, x is not given a degree mark. You already have it (outside the parentheses).

The next step in such problems is usually to ask for the measures of the angles. You weren't asked for it here. But you should check your answer, and to do that, you can also find the measure of the angles.

m∠ABC = 7x - 4 = 7×(29.5) - 4 = 206.5 - 4 = 202.5º

Note that I put the degree mark back on. Angles are typically measured in degrees, and you should always include the units. (Angles can also be measured in radians, turns, and gradians. Don't worry about these alternative units right now.)

m∠BCX = 7-x = 7-29.5 = -22.5º

Negative angles go clockwise, and positive angles go counter-clockwise. HOWEVER, I think your teacher just made up this problem quickly, without thinking it through. OR, MY DIAGRAM DOES NOT CORRESPOND TO THE ACTUAL PROBLEM.

I am going to finish checking in the hopes that my initial assumption was correct.

202.5º + (-22.5º) = 180º

and that does check out.

If my diagram was not correct, or if you have any questions, please let me know.