Solve this question for me---Question Stem:
20. In the diagram below, sides AC and AB of triangle ABC are congruent. If the measure of angle DCA is 115 degrees, what is the measure in degrees of angle A?
Options:
None
Chart/Diagram Description:
Type: Geometric figure (triangle and a line).
Main Elements:
Points: Labeled points A, B, C, D.
Lines: Straight line segment DCB (D, C, B are collinear in that order). Segments AC and AB form a triangle ABC with vertex A.
Shape: A triangle ABC is shown, with point A at the top and base BC. Point D is on the line containing BC, to the left of point C.
Labels and Annotations: Points A, B, C, D are labeled. The diagram illustrates triangle ABC and the line segment DCB. The question states that sides AC and AB of triangle ABC are congruent (AC = AB), and the measure of angle DCA is 115 degrees. The question asks for the measure of angle A (angle BAC).
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We have triangle ABC where sides AC and AB are congruent. Point D lies on the extension of line BC. We're given that angle DCA measures 115 degrees, and we need to find the measure of angle A.
First, we identify that angles DCA and ACB form a linear pair since they are adjacent angles on a straight line. Linear pair angles always sum to 180 degrees. Since angle DCA is 115 degrees, we can calculate angle ACB as 180 minus 115, which equals 65 degrees.
Since triangle ABC is isosceles with AC equal to AB, we know that the base angles are equal. In an isosceles triangle, the angles opposite the equal sides are congruent. Therefore, angle ABC equals angle ACB, which we found to be 65 degrees.
Now we use the triangle angle sum theorem. The sum of all angles in any triangle equals 180 degrees. We have angle A plus angle ABC plus angle ACB equals 180 degrees. Substituting our known values: angle A plus 65 degrees plus 65 degrees equals 180 degrees. This gives us angle A plus 130 degrees equals 180 degrees. Therefore, angle A equals 50 degrees.
Let's summarize our solution. First, we used the linear pair property to find that angle ACB equals 65 degrees. Then, we applied the isosceles triangle theorem to determine that angle ABC also equals 65 degrees. Finally, using the triangle angle sum theorem, we calculated that angle A equals 50 degrees. Therefore, the answer is 50 degrees.