As shown in the figure, in triangle \(ABC\),
\(\angle ABC = 45°\),
\(\angle ACB = 61°\),
extend \(BC\) to \(E\) such that \(CE = AC\),
extend \(CB\) to \(D\) such that \(DB = AB\),
find the measure of \(\angle DAE\).
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In this problem, we have triangle ABC with angle ABC equal to 45 degrees and angle ACB equal to 61 degrees. We extend BC to point E such that CE equals AC, and we extend CB to point D such that DB equals AB. Our task is to find the measure of angle DAE.
Let's start by finding angle BAC in the original triangle. Using the fact that angles in a triangle sum to 180 degrees, we can calculate angle BAC as 180 degrees minus angle ABC minus angle ACB. That's 180 minus 45 minus 61, which equals 74 degrees.
Now we'll analyze the two isosceles triangles. In triangle ABD, since DB equals AB, it's isosceles. The angle ABD is the exterior angle to ABC, so it equals 180 minus 45, which is 135 degrees. In an isosceles triangle, the base angles are equal, so angle DAB equals 22.5 degrees. Similarly, in triangle ACE, since CE equals AC, it's also isosceles. The angle ACE equals 180 minus 61, which is 119 degrees. Therefore, angle CAE equals 30.5 degrees.
Now we can find angle DAE by adding the three component angles. Since the points D, B, C, and E are collinear, angle DAE consists of angle DAB, plus angle BAC, plus angle CAE. That's 22.5 degrees plus 74 degrees plus 30.5 degrees, which equals 127 degrees. Therefore, the measure of angle DAE is 127 degrees.
To summarize what we've learned: First, we found angle BAC equals 74 degrees by using the fact that angles in a triangle sum to 180 degrees. Second, we recognized that triangle ABD is isosceles because AB equals DB, which gave us angle DAB equals 22.5 degrees. Similarly, triangle ACE is isosceles because AC equals CE, giving us angle CAE equals 30.5 degrees. Finally, we calculated angle DAE by adding the three component angles: 22.5 plus 74 plus 30.5, which equals 127 degrees. This problem demonstrates how to use triangle properties and angle relationships to find unknown angles in geometric figures.