The problem shows two similar triangles, and you're asked to find the missing length $x$ and the missing angles $A$, $B$, and $C$ in the right triangle. Let's break this down step by step:

Step 1: Similarity of triangles

Since the triangles are similar, the corresponding angles are equal, and the sides are proportional. This gives us the following relationships:

1. The angle measures in both triangles should correspond to each other.
2. The side lengths between the triangles should be proportional.

Step 2: Use the given angles

• The large triangle has angles: $10^\circ$, $26^\circ$, and $144^\circ$.
• In the smaller triangle, we already know one angle: $\angle B = 26^\circ$.

Using triangle similarity, the corresponding angles for the smaller triangle are:

• $\angle B$ corresponds to $26^\circ$.
• The $10^\circ$ angle in the larger triangle should correspond to $\angle A$ in the smaller triangle, so $\angle A = 10^\circ$.

Now, the third angle in a triangle can be found using the sum of interior angles: $\angle C = 180^\circ - 26^\circ - 10^\circ = 144^\circ$

Thus, $\angle C = 144^\circ$, which matches the large triangle as well.

Step 3: Proportions to find the missing side $x$

The sides in similar triangles are proportional. Let's use the ratio of the corresponding sides: $\frac{42}{9} = \frac{x}{27}$

Solve for $x$: $x = \frac{27 \times 42}{9} = \frac{1134}{9} = 126$

Summary of the solution:

• $x = 126$
• $\angle A = 10^\circ$
• $\angle B = 26^\circ$
• $\angle C = 144^\circ$

Would you like further details or clarification on any step?

5 Follow-up Questions:

1. How can you check if two triangles are similar using only side lengths?
2. How does the Angle-Angle (AA) similarity criterion work in triangles?
3. Can two triangles be similar if their angles don't match exactly but their sides are proportional?
4. How do we know which angles in two triangles are corresponding?
5. What is the relationship between side ratios in similar triangles?

Tip:

Always remember that in similar triangles, the corresponding angles are equal, and the ratios of the lengths of corresponding sides are constant.