Math Problem Statement

In ΔTUV, m, angle, T, equals, left parenthesis, 5, x, minus, 3, right parenthesis, degreesm∠T=(5x−3) ∘ , m, angle, U, equals, left parenthesis, 4, x, plus, 4, right parenthesis, degreesm∠U=(4x+4) ∘ , and m, angle, V, equals, left parenthesis, x, minus, 11, right parenthesis, degreesm∠V=(x−11) ∘ . Find m, angle, V, .m∠V.

Solution

To find the measure of angle VV (denoted as mVm\angle V) in triangle ΔTUV\Delta TUV, we can use the fact that the sum of the angles in any triangle is always 180180^\circ. So, we have the equation:

mT+mU+mV=180m\angle T + m\angle U + m\angle V = 180^\circ

The given values are: mT=(5x3),mU=(4x+4),mV=(x11)m\angle T = (5x - 3)^\circ, \quad m\angle U = (4x + 4)^\circ, \quad m\angle V = (x - 11)^\circ

Substitute these into the equation:

(5x3)+(4x+4)+(x11)=180(5x - 3) + (4x + 4) + (x - 11) = 180

Now, let's solve for xx:

Step 1: Combine like terms

5x+4x+x3+411=1805x + 4x + x - 3 + 4 - 11 = 180 10x10=18010x - 10 = 180

Step 2: Solve for xx

10x=19010x = 190 x=19x = 19

Step 3: Find mVm\angle V

Now that we know x=19x = 19, substitute it back into the expression for mV=(x11)m\angle V = (x - 11):

mV=1911=8m\angle V = 19 - 11 = 8^\circ

Thus, mV=8m\angle V = 8^\circ.

Let me know if you need further details or have any questions. Here are some related questions for further practice:

  1. What is the measure of mTm\angle T when x=19x = 19?
  2. What is the measure of mUm\angle U when x=19x = 19?
  3. Can the angles in a triangle ever sum to more than 180180^\circ?
  4. If ΔABC\Delta ABC has angles represented similarly, how would you solve for unknown angles?
  5. What is the significance of a triangle's angle sum theorem in geometry?

Tip: Always remember to check if the sum of your angles equals 180180^\circ when solving for unknowns in triangles.

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Math Problem Analysis

Mathematical Concepts

Algebra
Geometry
Triangle Angle Sum

Formulas

Sum of angles in a triangle: m∠T + m∠U + m∠V = 180°

Theorems

Triangle Angle Sum Theorem

Suitable Grade Level

Grades 8-10