Ray QR bisects PQM, where m PQR =3x-7 and m PQM = 4x+23. Find m RQM
Since QR bisects the angle, it is 1/2 of PQM. That is,
2(3x-7) = 4x+23
x = 18.5
So now you can calculate RQM, which is the same as PQR
Well, isn't Ray QR one lucky line! Not only does it bisect PQR, but it also magically splits PQM into two equal angles. Talk about a line with a sense of justice!
Now, let's get into some math fun. We have:
mPQR = 3x - 7
mPQM = 4x + 23
Since Ray QR bisects angle PQR, we know that mRQM is equal to mRQP. And since Ray QR magically splits angle PQM into two equal angles, mRQP is half of mPQM. So, let's solve for mRQP:
mRQP = mPQM / 2
mRQP = (4x + 23) / 2
mRQP = 2x + 11.5
And since mRQM = mRQP, we have:
mRQM = mRQP
mRQM = 2x + 11.5
So, there you have it! The measure of angle RQM is 2x + 11.5. Keep up the inquisitive spirit, my friend, and continue to discover the joys of geometry!
To find angle RQM, we can start by using the fact that Ray QR bisects angle PQM. This means that angle RQM is equal to angle PQM.
We are given that m PQR = 3x - 7 and m PQM = 4x + 23. Since Ray QR bisects angle PQM, we can set these two expressions equal to each other:
3x - 7 = 4x + 23
Next, let's solve for x:
3x - 4x = 23 + 7
-x = 30
x = -30
Now that we have found the value of x, we can substitute it back into one of the angle expressions to find the measure of angle RQM:
m RQM = m PQM = 4x + 23
m RQM = 4(-30) + 23
m RQM = -120 + 23
m RQM = -97
Therefore, the measure of angle RQM is -97 (degrees).
To find the measure of angle RQM, we need to understand that angle PQR is bisected by ray QR, which means that angle RQM and angle PQM are adjacent angles and their sum is equal to angle PQR.
Given that mPQR = 3x - 7 and mPQM = 4x + 23, we can set up an equation to find the value of x:
mPQR = mPQM + mRQM
Substituting the given values:
3x - 7 = 4x + 23 + mRQM
Now, we can isolate mRQM by subtracting 4x and 23 from both sides of the equation:
3x - 4x - 7 - 23 = mRQM
Simplifying:
-x - 30 = mRQM
Therefore, mRQM = -x - 30.
Since we are asked to find the measure of angle RQM, we need to express the answer solely in terms of x.