Question

1. find ( mangle abc ).

diagram: points ( d ), ( b ), ( c ) on a straight line (( dc )), with point ( a ) above ( b ). angle between ( ba ) and ( bd ) is ( (2x)^circ ), angle between ( ba ) and ( bc ) is ( (5x + 5)^circ ). multiple - choice options: ( 50^circ ), ( 130^circ ), ( 60^circ ), ( 25^circ )

Explanation:

Step1: Identify supplementary angles

Angles \( \angle ABD = (2x)^\circ \) and \( \angle ABC = (5x + 5)^\circ \) are supplementary (they form a linear pair), so their sum is \( 180^\circ \).
\[ 2x + (5x + 5) = 180 \]

Step2: Solve for \( x \)

Combine like terms:
\[ 7x + 5 = 180 \]
Subtract 5 from both sides:
\[ 7x = 175 \]
Divide by 7:
\[ x = 25 \]

Step3: Find \( m\angle ABC \)

Substitute \( x = 25 \) into \( (5x + 5)^\circ \):
\[ 5(25) + 5 = 125 + 5 = 130 \]

Answer:

\( 130^\circ \) (corresponding to the option "130°")