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.