https://file.aiquickdraw.com/mathbot/file/1730999333926vn3sqqom.jpg

Let's go through each question one by one.

### Problem 10
Given:
- \(\triangle ABC\) is equilateral, meaning all angles are \(60^\circ\).
- \(\angle C = 8x - 44\).

Since \(\angle C = 60^\circ\), we set up the equation:
\[
8x - 44 = 60
\]
Solving for \(x\):
1. Add \(44\) to both sides:
   \[
   8x = 104
   \]
2. Divide by \(8\):
   \[
   x = 13
   \]

### Problem 11
Given:
- \(\triangle RST\) is isosceles, meaning two angles are equal.
- \(\angle R = 15x - 31\) and \(\angle T = 9x + 11\).

In an isosceles triangle, if two angles are equal, then the third angle can be calculated using the triangle angle sum property. Assume that \(\angle R = \angle T\).

Setting up the equation:
\[
15x - 31 = 9x + 11
\]
Solving for \(x\):
1. Subtract \(9x\) from both sides:
   \[
   6x - 31 = 11
   \]
2. Add \(31\) to both sides:
   \[
   6x = 42
   \]
3. Divide by \(6\):
   \[
   x = 7
   \]

Now, substitute \(x = 7\) to find \(m\angle S\):
\[
m\angle R = 15(7) - 31 = 105 - 31 = 74^\circ
\]
\[
m\angle T = 9(7) + 11 = 63 + 11 = 74^\circ
\]

Since \(\angle R = \angle T\), we calculate \(\angle S\) as:
\[
m\angle S = 180^\circ - 74^\circ - 74^\circ = 32^\circ
\]

### Problem 12
Given:
- \(m\angle P = 14\) less than five times \(x\): \(m\angle P = 5x - 14\).
- \(m\angle Q = 5\) less than \(x\): \(m\angle Q = x - 5\).
- \(m\angle R = 9\) less than twice \(x\): \(m\angle R = 2x - 9\).

Using the triangle angle sum property:
\[
m\angle P + m\angle Q + m\angle R = 180
\]
Substitute the expressions:
\[
(5x - 14) + (x - 5) + (2x - 9) = 180
\]
Combine terms:
\[
8x - 28 = 180
\]
Add \(28\) to both sides:
\[
8x = 208
\]
Divide by \(8\):
\[
x = 26
\]

Now, substitute \(x = 26\) to find each angle:
\[
m\angle P = 5(26) - 14 = 130 - 14 = 116^\circ
\]
\[
m\angle Q = 26 - 5 = 21^\circ
\]
\[
m\angle R = 2(26) - 9 = 52 - 9 = 43^\circ
\]

---

### Summary of Solutions
1. **Problem 10**: \(x = 13\)
2. **Problem 11**: \(m\angle S = 32^\circ\)
3. **Problem 12**: \(x = 26\), \(m\angle P = 116^\circ\), \(m\angle Q = 21^\circ\), \(m\angle R = 43^\circ\)

Would you like further details on any of these problems, or do you have any questions?

---

### Related Questions
1. How would the solution change if \(\triangle ABC\) in Problem 10 was not equilateral?
2. Can you solve for \(x\) in an isosceles triangle where one angle expression is given?
3. How do you verify that two angles in a triangle are equal if it’s not specified?
4. What other properties of triangles are useful for solving angle-related problems?
5. How would you solve if \(\angle R\) and \(\angle S\) were equal instead in Problem 11?

### Tip
Always double-check the properties of the triangle (isosceles, equilateral, etc.) as they provide key insights for setting up equations.

Solve the following problems:

10. If \(\triangle ABC\) is equilateral, solve for \(x\).
11. If \(\triangle RST\) is isosceles, find \(m\angle S\).
12. In \(\triangle PQR\), if \(m\angle P\) is 14 less than five times \(x\), \(m\angle Q\) is five less than \(x\), and \(m\angle R\) is nine less than twice \(x\), find \(x\) and the measure of each angle.

Learn how to solve for unknown variables and angles in equilateral and isosceles triangles. This problem covers the triangle angle sum property, algebraic manipulation, and understanding of equilateral and isosceles triangle characteristics. Suitable for Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation