The missing measures of angles ∠PRQ, ∠RPQ, and ∠PQR are:
a) ∠PRQ = 78 degrees
b) ∠RPQ = 30 degrees
c) ∠PQR = 72 degrees.
To find the missing measures of angles ∠PRQ, ∠RPQ, and ∠PQR, we can use the fact that the sum of the angles in a triangle is 180 degrees. Let's set up equations using the given information and solve for x.
We know that:
m∠PRQ = 8x - 10
m∠RPQ = 4x - 14
m∠PQR = 7x - 5
To find x, we can add the three angles and set the sum equal to 180:
(8x - 10) + (4x - 14) + (7x - 5) = 180
Simplifying the equation:
8x + 4x + 7x - 10 - 14 - 5 = 180
19x - 29 = 180
Now, let's solve for x:
19x = 180 + 29
19x = 209
x = 209 / 19
x ≈ 11
Now that we have found x, we can substitute it back into the expressions for each angle to find their measures:
a) ∠PRQ = 8x - 10
∠PRQ = 8(11) - 10
∠PRQ = 88 - 10
∠PRQ = 78 degrees
b) ∠RPQ = 4x - 14
∠RPQ = 4(11) - 14
∠RPQ = 44 - 14
∠RPQ = 30 degrees
c) ∠PQR = 7x - 5
∠PQR = 7(11) - 5
∠PQR = 77 - 5
∠PQR = 72 degrees
Therefore, the missing measures of angles ∠PRQ, ∠RPQ, and ∠PQR are:
a) ∠PRQ = 78 degrees
b) ∠RPQ = 30 degrees
c) ∠PQR = 72 degrees.