in triangle PQR,line segment PQ is = and congruent to line segment QR. if m<p = 4x + 50, m<Q =2x + 60 and m<R = 14x + 30, find the measure of each angle.

You have conflicting information.

First of all, the 3 angles must add up to 180°, so
4x+50 + 2x+60 + 14x + 30 = 180
20x = 40
x = 2
and the 3 angles would be 58°, 64° , and 42°.

BUT, in the first sentence you said that PQ = QR, making it isosceles.

Contradiction!
Two of the angles would have to be equal.

To find the measure of each angle in triangle PQR, we can use the fact that the sum of all angles in a triangle is 180 degrees.

Since line segment PQ is congruent to line segment QR, we can conclude that angle P and angle R are congruent.

Let's solve for x first using the given angle measures:

m<P = 4x + 50
m<Q = 2x + 60
m<R = 14x + 30

Since angle P and angle R are congruent, we can set their measures equal to each other:

4x + 50 = 14x + 30

Solving this equation for x:

4x - 14x = 30 - 50
-10x = -20
x = -20 / -10
x = 2

Now, let's substitute the value of x back into the angle measures to find the measures of angles P, Q, and R:

m<P = 4x + 50
m<P = 4(2) + 50
m<P = 8 + 50
m<P = 58 degrees

m<Q = 2x + 60
m<Q = 2(2) + 60
m<Q = 4 + 60
m<Q = 64 degrees

m<R = 14x + 30
m<R = 14(2) + 30
m<R = 28 + 30
m<R = 58 degrees

Therefore, the measure of angle P is 58 degrees, the measure of angle Q is 64 degrees, and the measure of angle R is 58 degrees.

To find the measures of the angles in triangle PQR, we can use the fact that the sum of the measures of the angles in any triangle is always 180 degrees.

Let's start by assigning variables to the given angles:
m<P = 4x + 50
m<Q = 2x + 60
m<R = 14x + 30

Now, we can write an equation by summing up the measures of all the angles:
(m<P) + (m<Q) + (m<R) = 180

Substituting the given values:
(4x + 50) + (2x + 60) + (14x + 30) = 180

Now, let's solve for x:

4x + 50 + 2x + 60 + 14x + 30 = 180
20x + 140 = 180
20x = 180 - 140
20x = 40
x = 40 / 20
x = 2

Now that we have found the value of x, we can substitute it back into the expressions for m<P, m<Q, and m<R to find their respective measures:

m<P = 4x + 50 = 4(2) + 50 = 8 + 50 = 58 degrees

m<Q = 2x + 60 = 2(2) + 60 = 4 + 60 = 64 degrees

m<R = 14x + 30 = 14(2) + 30 = 28 + 30 = 58 degrees

Therefore, the measures of the angles in triangle PQR are:
Angle P: 58 degrees
Angle Q: 64 degrees
Angle R: 58 degrees