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The problem involves two congruent isosceles triangles forming a parallelogram, where the largest angle of the parallelogram is given as \(116^\circ\). The task is to determine the angles labeled \(a\) and \(b\).

### Step-by-step Solution:

1. **Angles in a Parallelogram:**
   - In a parallelogram, opposite angles are equal, and adjacent angles are supplementary (i.e., they add up to \(180^\circ\)).
   - Given that the largest angle is \(116^\circ\), the adjacent angle must be \(180^\circ - 116^\circ = 64^\circ\).

2. **Relationship of Angles \(a\) and \(b\):**
   - Each isosceles triangle has two equal angles, labeled \(a\) in the diagram.
   - Since the sum of angles in a triangle is \(180^\circ\), the equation for one of the isosceles triangles would be:
     \[
     2a + b = 180^\circ
     \]
   - In the parallelogram, the angle formed by the adjacent side is \(b\), and from the geometry of the parallelogram, we know that:
     \[
     2b = 64^\circ
     \]
     (since the two \(b\) angles add up to \(64^\circ\)).

3. **Solve the system of equations:**
   - First, solve for \(b\) from the second equation:
     \[
     2b = 64^\circ \implies b = \frac{64^\circ}{2} = 32^\circ
     \]
   - Now substitute \(b = 32^\circ\) into the first equation:
     \[
     2a + 32^\circ = 180^\circ \implies 2a = 180^\circ - 32^\circ = 148^\circ
     \]
     \[
     a = \frac{148^\circ}{2} = 74^\circ
     \]

### Final Answer:
- \(a = 74^\circ\)
- \(b = 32^\circ\)

Let me know if you need further details or explanations!

Here are 5 related questions to expand on this topic:
1. What are the properties of a parallelogram?
2. How can you verify if a triangle is isosceles given only angle information?
3. What are the properties of congruent triangles?
4. How do you solve systems of linear equations in geometry problems?
5. Why do the angles of any triangle always add up to \(180^\circ\)?

**Tip:** For angle problems involving shapes like parallelograms, remember that adjacent angles are always supplementary, which helps break down complex problems!

Two congruent, isosceles triangles are joined, as shown, to form a parallelogram. The largest angle of the parallelogram is 116°. Write two equations. Solve them to find the value of a and the value of b.

This problem involves solving for angles in a parallelogram formed by two congruent isosceles triangles. Learn how to find the values of 'a' and 'b' using properties of triangles and parallelograms. Suitable for students in Grades 9-11.

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