The problem involves two angles that are complementary, meaning their sum is $90^\circ$. The angles are given as expressions involving $x$:

1. First angle: $(3x - 20)^\circ$
2. Second angle: $(7x + 10)^\circ$

Step 1: Set up the equation

Since the angles are complementary, their sum equals $90^\circ$. So, we write the equation:

$(3x - 20) + (7x + 10) = 90$

Step 2: Simplify the equation

Combine like terms:

$3x + 7x - 20 + 10 = 90$

$10x - 10 = 90$

Step 3: Solve for $x$

Add 10 to both sides:

$10x = 100$

Now, divide by 10:

$x = 10$

Step 4: Find the angles

Substitute $x = 10$ into each expression:

• First angle: $3x - 20 = 3(10) - 20 = 30 - 20 = 10^\circ$
• Second angle: $7x + 10 = 7(10) + 10 = 70 + 10 = 80^\circ$

Final Answer:

• Smaller angle: $10^\circ$
• Larger angle: $80^\circ$

Let me know if you'd like more details or have any questions! Here are 5 related questions:

1. What happens if the angles are supplementary instead of complementary?
2. How would you solve this problem if the angle expressions were in terms of different variables?
3. What other geometric shapes involve complementary angles?
4. How does the method change if we deal with three angles instead of two?
5. Can angles be complementary if one of them is obtuse?

Tip: When solving algebraic expressions for angles, always check whether they add up correctly based on the given condition (complementary, supplementary, etc.).