the measure of the second angle of a triangle is 2 degrees less than twice the measure of the first angle. the measure of the third angle is 25 degrees more than half the measure of the second angle. find the measure of each angle.

Let's denote the measures of the three angles of the triangle as follows:

- \( A_1 \) = measure of the first angle
- \( A_2 \) = measure of the second angle
- \( A_3 \) = measure of the third angle

### Given Relationships:
1. The second angle is 2 degrees less than twice the first angle: 
   \[
   A_2 = 2A_1 - 2
   \]
2. The third angle is 25 degrees more than half the second angle: 
   \[
   A_3 = \frac{1}{2}A_2 + 25
   \]
3. The sum of the angles in a triangle is always 180 degrees:
   \[
   A_1 + A_2 + A_3 = 180
   \]

### Step 1: Express \( A_3 \) in terms of \( A_1 \)
Using the equation for \( A_2 \):
\[
A_3 = \frac{1}{2}(2A_1 - 2) + 25
\]
Simplify \( A_3 \):
\[
A_3 = A_1 - 1 + 25 = A_1 + 24
\]

### Step 2: Substitute \( A_2 \) and \( A_3 \) in the sum equation
Substitute the expressions for \( A_2 \) and \( A_3 \) into the equation \( A_1 + A_2 + A_3 = 180 \):
\[
A_1 + (2A_1 - 2) + (A_1 + 24) = 180
\]

### Step 3: Solve for \( A_1 \)
Simplify the equation:
\[
A_1 + 2A_1 - 2 + A_1 + 24 = 180
\]
\[
4A_1 + 22 = 180
\]
Subtract 22 from both sides:
\[
4A_1 = 158
\]
Divide by 4:
\[
A_1 = 39.5
\]

### Step 4: Find \( A_2 \) and \( A_3 \)
- \( A_2 = 2A_1 - 2 = 2(39.5) - 2 = 79 - 2 = 77 \)
- \( A_3 = A_1 + 24 = 39.5 + 24 = 63.5 \)

### Conclusion
The measures of the angles are:
- \( A_1 = 39.5^\circ \)
- \( A_2 = 77^\circ \)
- \( A_3 = 63.5^\circ \)

Would you like more details on this solution or have any questions?

Here are five related questions:
1. How do you verify the sum of angles in a triangle?
2. What if the second angle was 3 degrees less than twice the first angle?
3. How does changing the relationship between the angles affect the solution?
4. How can you determine if a triangle is acute, obtuse, or right-angled based on its angles?
5. Can this method be used to solve for angles in non-Euclidean geometries?

**Tip:** Always check the sum of your angles to ensure they add up to 180 degrees when solving triangle problems.

The measure of the second angle of a triangle is 2 degrees less than twice the measure of the first angle. The measure of the third angle is 25 degrees more than half the measure of the second angle. Find the measure of each angle.

Learn how to solve for the angles of a triangle where the second angle is 2 degrees less than twice the first, and the third is 25 degrees more than half the second. This solution includes step-by-step algebraic expressions and uses the sum of angles in a triangle theorem. Ideal for students in Grades 8-10.

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