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.