QS bisects <PQR. If m<PQS = 3x and m<RQS =2x + 6, then m<PQR =?
the two halves are equal, so
3x = 2x+6
x=6
PQR is their sum: 18+18 = 36
Since QS bisects ∠PQR, we can use the angle bisector theorem to find the measure of angle ∠PQR.
According to the angle bisector theorem, the ratio of the two segments formed by an angle bisector is equal to the ratio of the two angles opposite those segments.
Let's assume the measure of angle ∠PQS is "3x".
Therefore, the measure of angle ∠RQS is "2x + 6".
Since QS bisects ∠PQR, we have the following ratio:
∠PQS / ∠RQS = PQ / QR
By substituting the given angles, we get:
(3x) / (2x + 6) = PQ / QR
To solve for x, we can cross-multiply:
(3x) * QR = (2x + 6) * PQ
3x * QR = 2x * PQ + 6 * PQ
3x * QR - 2x * PQ = 6 * PQ
Now, we can simplify and solve for x by matching the coefficients of QR and PQ:
QR / PQ = 2x / 3x
QR / PQ = 2 / 3
Since QS bisects ∠PQR, the angles opposite the segments PQ and QR are equal.
Therefore, the ratio QR / PQ is also equal to 1.
So, we have:
1 = 2 / 3
Now, we can solve for x:
2 / 3 = 1
Multiply both sides by 3:
2 = 3
Since the equation 2 = 3 is not true, there is no solution for x.
Therefore, the information given in the question is contradictory or not possible to solve.
As a result, we cannot determine the measure of angle ∠PQR with the information provided.
To find the measure of angle PQR, we can use the angle bisector theorem, which states that an angle bisector divides the opposite side of the triangle into segments that are proportional to the other two sides of the triangle.
Let's assume that the measure of angle PQS is 3x and the measure of angle RQS is 2x + 6. Since QS bisects angle PQR, we can set up a proportion:
PQ / QR = PS / SR
Since we are given that QS bisects angle PQR, we can say that PS = SR, which means the proportion can be simplified to:
PQ / QR = PS / PS
Simplifying further, we get:
PQ / QR = 1
This means that PQ is equal to QR. In other words, triangle PQR is an isosceles triangle.
In an isosceles triangle, the base angles (angles opposite the congruent sides) are equal. Therefore, angle PQS and angle RQS are congruent. So we can set up an equation:
3x = 2x + 6
Solving this equation, we find:
x = 6
Now that we know x, we can substitute it back into one of the given angle measures to find the measure of angle PQR:
m<PQR = 2x + 6
= 2(6) + 6
= 12 + 6
= 18
Therefore, the measure of angle PQR is 18 degrees.