Based on the description you provided, I understand that you have two right triangles touching at the tips. One triangle has a 90 degree angle on the bottom left and a 33 degree angle on the bottom right. The other triangle has a right angle on the top right, but the other two angles are not given.
To find the measure of angle A, you need to use the information given about the extended sides and the angles formed. Let's break it down step by step:
1. In the left triangle, you have a 90 degree angle on the bottom left and a 33 degree angle on the bottom right. Therefore, the remaining angle can be found by subtracting the sum of the other two angles from 180 degrees:
Angle A = 180 - 90 - 33 = 57 degrees
2. Now, focus on the right triangle where the right angle is on the top right. You mentioned that when you extend the right side of the right triangle, it forms an angle A. We can work with that.
3. You stated that the shared vertex between the two triangles forms an angle of 33 degrees. Since angles opposite to each other are equal (vertical angles theorem), angle A in the right triangle is also 33 degrees.
4. The last piece of information is that an angle of 110 degrees is formed from the extended sides of the hypotenuse of the left triangle and a leg of the right triangle.
Now, here's where you made a slight mistake:
5. The two triangles are not congruent just because their angles are equal. To prove congruence, we need to have equal angles and equal sides. Since the sides are not given in this case, we cannot assume that the triangles are congruent.
6. However, we know that the angle opposite to angle A in the right triangle is 33 degrees, which we found in step 3. Therefore, angle A is indeed 33 degrees.
To summarize, the measure of angle A in the right triangle is 33 degrees, not 123 degrees as you initially thought.