The diagram shows a triangle with two given angles: $\angle ABC = 46^\circ$ and $\angle ACB = 52^\circ$. We are asked to find the measure of $\angle CAB$ and $\angle CAE$.

Step 1: Find $\angle CAB$

In any triangle, the sum of the interior angles is $180^\circ$. Therefore, to find $\angle CAB$, we use the following equation:

$\angle CAB + \angle ABC + \angle ACB = 180^\circ$

Substituting the known values:

$\angle CAB + 46^\circ + 52^\circ = 180^\circ$

$\angle CAB = 180^\circ - 46^\circ - 52^\circ = 82^\circ$

Thus, $\angle CAB = 82^\circ$.

Step 2: Find $\angle CAE$

Since $AE$ is shown as a straight line, $\angle CAE$ is a supplementary angle to $\angle CAB$, meaning:

$\angle CAE + \angle CAB = 180^\circ$

Substituting the value of $\angle CAB = 82^\circ$:

$\angle CAE = 180^\circ - 82^\circ = 98^\circ$

Thus, $\angle CAE = 98^\circ$.

Final Answers:

• $\angle CAB = 82^\circ$
• $\angle CAE = 98^\circ$

Would you like further details on any of the steps or any other questions? Here are five related questions that could expand your understanding:

1. How do we prove that the sum of interior angles in a triangle is always $180^\circ$?
2. What are supplementary angles and how are they used in geometry problems?
3. Can you explain how exterior angles of a triangle relate to the interior angles?
4. What is the difference between complementary and supplementary angles?
5. How do straight angles help in finding unknown angles in complex figures?

Tip: Always double-check your calculations, especially when working with angle sums and supplementary relationships, as small errors can lead to incorrect conclusions.