Find the value of x. The diagram is not to scale.
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
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.
To find the value of x, we need to use the given information:
From the given information, we know that segment RS is congruent to segment ST. This means that the lengths of these two segments are equal.
Now, let's look at the given angle measures:
Angle RST = 7x - 54
Angle STU = 8x
Triangle RST is a triangle, and the sum of the angles in a triangle is always 180 degrees. So we can set up an equation:
Angle RST + Angle STU + Angle RTS = 180
Substitute the given angle measures into the equation:
(7x - 54) + 8x + Angle RTS = 180
Simplify the equation:
15x - 54 + Angle RTS = 180
Angle RTS is the missing angle measure, and in a triangle, the sum of the angles is 180 degrees. So we can replace Angle RTS with its measure:
15x - 54 + 180 = 180
Now, let's solve for x:
15x - 54 + 180 = 180
15x + 126 = 180
15x = 180 - 126
15x = 54
x = 54/15
x = 3.6
Therefore, the value of x is 3.6.
To find the value of x, we can use the fact that the sum of the angles in a triangle is 180 degrees.
The given information tells us that angle RST is equal to 7x - 54, and angle STU is equal to 8x.
Since segment RS is congruent to segment ST, we can conclude that angle RST and angle RTS are congruent. Therefore, angle RTS is also equal to 7x - 54.
And since angle RST and angle RTU are corresponding angles, and segment RT is extended to point U, we can conclude that angle RTU is also equal to 7x - 54.
Now, let's use the fact that the sum of the angles in a triangle is 180 degrees.
angle RST + angle RTS + angle RTU = 180
(7x - 54) + (7x - 54) + 8x = 180
Combine like terms:
14x - 108 + 8x = 180
Combine like terms again:
22x - 108 = 180
Now, let's isolate x. Add 108 to both sides of the equation:
22x = 288
Finally, divide both sides of the equation by 22 to solve for x:
x = 288 / 22
Simplifying the right side:
x ≈ 13.09
Therefore, the value of x is approximately 13.09.