Find the value of x. The diagram is not to scale.

Question

Given: Segment R S is congruent to Segment S T, mangleRST = 7x - 54, mangleSTU = 8x. Triangle R S T is shown. Side R T is extended as a ray past T through point U on the exterior of the triangle.
(1 point)
Responses

14
14

152
152

16
16

19

Answer 1

To solve for x, we can set up an equation using the given angle measurements in the triangle:

m∠RST + m∠STU + m∠UTR = 180 (Triangle Angle Sum Theorem)

Substituting the given angle measurements:

7x - 54 + 8x + m∠UTR = 180

Combining like terms:

15x - 54 + m∠UTR = 180

Simplifying:

15x + m∠UTR = 234 (equation 1)

Since segment RS is congruent to ST, we also have:

m∠TRS = m∠TSR (Alternate Interior Angles Theorem)

Substituting in the angle measurements:

m∠UTR = m∠TRS = 8x

Substituting this into equation 1:

15x + 8x = 234

Combining like terms:

23x = 234

Solving for x:

x = 234 / 23

x ≈ 10.17

So the value of x is approximately 10.17.