Let's break this problem down step by step.

We are given:

• The measure of the supplement of an angle exceeds twice the measure of the complement of the angle by 40.
• We need to find half of the measure of the complement.

Definitions:

• Supplement of an angle $x$ is $180^\circ - x$.
• Complement of an angle $x$ is $90^\circ - x$.

Step 1: Set up the equation.

Let the angle be $x$.

• The supplement is $180^\circ - x$.
• The complement is $90^\circ - x$.

According to the problem, the supplement exceeds twice the complement by 40: $180^\circ - x = 2(90^\circ - x) + 40$

Step 2: Solve the equation.

Expand the right-hand side: $180^\circ - x = 180^\circ - 2x + 40$

Simplify: $180^\circ - x = 220^\circ - 2x$

Now, solve for $x$: $-x + 2x = 220^\circ - 180^\circ$ $x = 40^\circ$

Step 3: Find the complement.

The complement of the angle is: $90^\circ - 40^\circ = 50^\circ$

Step 4: Find half the complement.

Half of the complement is: $\frac{50^\circ}{2} = 25^\circ$

Final Answer: Half the measure of the complement is 25 degrees.

Let me know if you need more details or have any questions!

Relative questions:

1. What is the relationship between complementary and supplementary angles?
2. How do you set up an equation based on word problems involving angles?
3. How does knowing the complement and supplement help in solving unknown angles?
4. What are some typical examples where these angle concepts are used in geometry?
5. How would the solution change if the supplement was less than twice the complement by a certain number?

Tip:

Always remember to check the logic of word problems by carefully analyzing the conditions and relationships given between quantities.