Geometry

Question Analysis

The main focus is using the equal - length markings (suggesting  is an angle - bisector) and the given angle expressions along with the Angle Bisector Equidistant Theorem to find the value of  and then determine the measure of ∠DCB.

 

Key Concept Explanation

The Angle Bisector Equidistant Theorem states that if a ray bisects an angle, then any point on that ray is equidistant from the two sides of the angle. The equal - length markings from  to  and  to  imply that  bisects ∠DCB. So, ∠DCA = ∠BCA.

 

Step - by - Step Solution

1. Since  bisects ∠DCB, by the Angle Bisector Equidistant Theorem, ∠DCA = ∠BCA.

We know that ∠DCA=(4x)° and ∠BCA=(x + 27)°. So, we set up the equation

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