Question

6 a) in the diagram below, x = b) in the diagram below, m∠adc = °

Explanation:

Part (a)

Step1: Identify Arc and Angle Relationship

In a circle, the measure of a central angle is equal to the measure of its intercepted arc. Here, arc \( AC \) is \( 75^\circ \), and \( \angle ABC \) is a central angle intercepting arc \( AC \). Also, \( \angle ABD = 40^\circ \) and \( \angle DBC = x^\circ \), so \( \angle ABC=\angle ABD + \angle DBC \), which means \( 40^\circ+x^\circ = 75^\circ \).

Step2: Solve for \( x \)

Subtract \( 40^\circ \) from both sides: \( x = 75 - 40 \).
\( x = 35 \)

Step1: Identify Angle Addition

To find \( m\angle ADC \), we add the two given angles at \( D \), which are \( 21^\circ \) and \( 65^\circ \). So \( m\angle ADC=21^\circ + 65^\circ \).

Step2: Calculate the Sum

\( 21 + 65 = 86 \), so \( m\angle ADC = 86^\circ \).

Answer:

\( 35 \)

Part (b)