Question

given: ∠dba is inscribed in circle c, with diameter bx. prove: m∠dba = 1/2 mda⌢. proof: let m∠abx = x° and m∠dbx = y°. drag the choices into the boxes to form a correct proof. show hints draw ac and dc. △abc and △dbc are triangles because radii of a circle are congruent, so ac = bc = dc. then m∠bac = x° and m∠bdc = by the isosceles triangle theorem. m∠acx = 2x° and m∠dcx = by the exterior angle theorem.

Explanation:

Step1: Recall circle - related properties

Since \(BX\) is a diameter of circle \(C\), and \(\triangle ABC\) and \(\triangle DBC\) have \(AC = BC=DC\) (radii of the circle). In \(\triangle ABC\), if \(AC = BC\), by the Isosceles - Triangle Theorem, if \(\angle ABX=x^{\circ}\), then \(\angle BAC = x^{\circ}\). Similarly, in \(\triangle DBC\), if \(BC = DC\), and \(\angle DBX = y^{\circ}\), then \(\angle BDC=y^{\circ}\).

Step2: Use the Exterior - Angle Theorem

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles. \(\angle ACX\) is an exterior angle of \(\triangle ABC\). So, \(m\angle ACX=m\angle BAC + m\angle ABC\). Given \(m\angle ACX = 2x^{\circ}\), and \(m\angle BAC=x^{\circ}\), which is consistent with the Isosceles - Triangle and Exterior - Angle Theorems.

Step3: Prove the inscribed - angle relationship

The measure of an inscribed angle \(\angle DBA\) in a circle is half of the measure of the intercepted arc \(\overset{\frown}{DA}\). Let's consider the central angle and inscribed angle relationship. The central angle corresponding to arc \(\overset{\frown}{DA}\) is \(2\angle DBA\). We know that the sum of angles in the relevant triangles formed by the radii and chords in the circle and using angle - chasing based on isosceles triangles (due to equal radii) and exterior - angle properties, we can prove that \(m\angle DBA=\frac{1}{2}m\overset{\frown}{DA}\).

Answer:

The proof is completed by using the properties of isosceles triangles (because radii of a circle are equal) and the exterior - angle theorem of triangles, along with the inscribed - angle theorem in a circle.