https://file.aiquickdraw.com/mathbot/file/1726666986795iwyapdpr.jpg

In the given image, we need to find the missing angle measures \( \angle BCA \) and \( \angle ABC \) for the triangle \( \triangle ABC \).

### Step 1: Analyze the given angles

- The angle at \( A \), \( \angle DAB \), is a right angle and equals \( 90^\circ \).
- \( \angle BAC \) is given as \( 97^\circ \).
- \( \angle ACB \) is given as \( (2x + 2)^\circ \).
- \( \angle ABC \) is given as \( (14x - 1)^\circ \).

### Step 2: Use the triangle sum property
The sum of the interior angles of any triangle is always \( 180^\circ \). Thus, for \( \triangle ABC \):

\[
\angle BAC + \angle ABC + \angle BCA = 180^\circ
\]

Substitute the values and expressions for the angles:

\[
97^\circ + (14x - 1)^\circ + (2x + 2)^\circ = 180^\circ
\]

### Step 3: Simplify the equation
Combine like terms:

\[
97 + 14x - 1 + 2x + 2 = 180
\]

This simplifies to:

\[
14x + 2x + 98 = 180
\]

\[
16x + 98 = 180
\]

### Step 4: Solve for \( x \)

Subtract 98 from both sides:

\[
16x = 180 - 98
\]

\[
16x = 82
\]

Now, divide by 16:

\[
x = \frac{82}{16} = 5.125
\]

### Step 5: Find the angle measures

1. **Find \( \angle BCA \)**:
   - Using \( x = 5.125 \), substitute into \( (2x + 2) \):

\[
\angle BCA = 2(5.125) + 2 = 10.25 + 2 = 12.25^\circ
\]

2. **Find \( \angle ABC \)**:
   - Using \( x = 5.125 \), substitute into \( (14x - 1) \):

\[
\angle ABC = 14(5.125) - 1 = 71.75 - 1 = 70.75^\circ
\]

### Final answers:
- \( \angle BCA = 12.25^\circ \)
- \( \angle ABC = 70.75^\circ \)

Would you like further clarification on any of these steps or concepts?

### 5 Related Questions:
1. How do you find missing angles in a right triangle?
2. What is the sum of interior angles in any polygon?
3. How would the answer change if one of the angles were obtuse?
4. Can a triangle have more than one right angle?
5. What happens to the angle measures when a triangle is dilated?

**Tip:** Always remember that the sum of the angles in any triangle must equal \( 180^\circ \). This is a key property in solving for unknown angles.

Find the missing angle measures for the triangle ABC using the given expressions for the angles.

Learn how to solve for missing angles in a triangle using algebraic expressions and the Triangle Angle Sum Theorem. This example problem includes the use of linear equations and substitution to find angle measures. Suitable for students in Grades 8-10.

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