QS−→ bisects ∠PQR, m∠PQS=(4y−10)∘, and m∠SQR=(2y+10)∘. Find m∠PQR.
I had no idea either so I just took a random guess and it turns out the correct answer is 60 degrees. ( I got lucky I guess )
Why did the angle go to therapy? Because it felt bisected and needed some counseling! To find the measure of angle PQR, let's apply the Angle Bisector Theorem. According to the theorem, we have:
m∠PQS / m∠SQR = PQ / QR
Substituting the given values, we get:
(4y - 10) / (2y + 10) = PQ / QR
Cross multiplying, we have:
(QR)(4y - 10) = (PQ)(2y + 10)
Expanding both sides, we get:
4yQR - 10QR = 2yPQ + 10PQ
Now, to find m∠PQR, which is what we're after, we need to determine the ratio PQ / QR. If only we could find that ratio and solve the equation above! Sadly, the equation is a bit tangled up, and solving it directly is pretty messy. So, let's move on to plan B...or should I say, plan Q.
Let's try using a different approach called the Angle Sum Theorem. According to the theorem, the sum of the angles in a triangle is 180 degrees. In triangle PQR, we know the measures of angles PQS and SQR:
m∠PQS = (4y - 10) degrees
m∠SQR = (2y + 10) degrees
Now, let's add up all the angles in triangle PQR:
m∠PQS + m∠SQR + m∠PQR = 180 degrees
Substituting the given measures, we have:
(4y - 10) + (2y + 10) + m∠PQR = 180
Combining like terms, we get:
6y + m∠PQR = 180
To isolate m∠PQR, we subtract 6y from both sides:
m∠PQR = 180 - 6y
So, we found the measure of ∠PQR! But wait...there's more! Since we're asked to find the measure of ∠PQR, not in terms of y, but as a specific value, we need more information. Either we have to be given a specific value for y or some additional relationship between the angles. Without that, we can't determine the exact measure of ∠PQR. Sorry about that, sometimes geometry can be a bit of a jigsaw puzzle!
To find the measure of angle PQR, we need to use the properties of angles formed by a transversal intersecting two parallel lines.
Given that line QS bisects angle PQR, we know that angle PQS is congruent to angle SQR. Therefore, we can set up an equation using the given measures of angles PQS and SQR to find the value of y.
According to the given information:
m∠PQS = (4y - 10)°
m∠SQR = (2y + 10)°
Since angle PQS is congruent to angle SQR (due to QS bisecting angle PQR), we can set up the equation:
(4y - 10)° = (2y + 10)°
Now, let's solve this equation for y:
4y - 10 = 2y + 10
2y = 20
y = 10
Now that we have the value of y, we can substitute it back into one of the given angles to find the measure of angle PQR.
Using m∠PQS = (4y - 10)°:
m∠PQS = (4(10) - 10)° = 40°
So, the measure of angle PQR is 40°.
QS−→ bisects ∠PQR, m∠PQS=(4y−10)∘, and m∠SQR=(2y+10)∘. Find m∠PQR.
80 degrees
my ba dussy is wet UwU
since QS bisects PQR,
m∠PQS = m∠SQR
4y-10 = 2y+10
Find y, and then m∠PQS
m∠PQR = 2*m∠PQS