Question

question 22
1 pts
in the figure, ( bd parallel eg ) and the measure of ( angle cfe = 65^circ ). what is the measure of ( angle acd )?
figure with lines and points: ( a )---( c )---( f )---( h ) (horizontal line), ( b )---( c )---( d ) (line), ( e )---( f )---( g ) (line), ( bd parallel eg )
options:
( 35^circ )
( 65^circ )
( 115^circ )
( 135^circ )
question 23
1 pts

Explanation:

Step1: Identify Corresponding Angles

Since \( BD \parallel EG \) and \( AH \) is a transversal, \( \angle BCF \) and \( \angle CFE \) are corresponding angles? Wait, no, actually \( \angle ACD \) and \( \angle CFE \) – wait, let's correct. \( BD \parallel EG \), and \( AH \) is a transversal. The angle \( \angle CFE = 65^\circ \), and \( \angle ACD \) and \( \angle CFE \) – wait, no, \( \angle ACD \) and \( \angle CFE \): actually, \( \angle ACD \) and \( \angle CFE \) are same - side? No, wait, \( \angle ACD \) and \( \angle CFE \): let's see, \( BD \parallel EG \), so \( \angle BCF \) (which is adjacent to \( \angle ACD \)) and \( \angle CFE \) are corresponding angles? Wait, no, \( \angle ACD \) and \( \angle CFE \): actually, \( \angle ACD \) and \( \angle CFE \) are supplementary? Wait, no, let's think again. \( \angle ACD \) and \( \angle CFE \): since \( BD \parallel EG \), and \( AH \) is a transversal, \( \angle ACD \) and \( \angle CFE \) – wait, \( \angle ACD \) is an exterior angle? Wait, no, \( \angle ACD \) and \( \angle CFE \): let's look at the lines. \( BD \parallel EG \), so the consecutive interior angles? No, wait, \( \angle ACD \) and \( \angle CFE \): actually, \( \angle ACD \) and \( \angle CFE \) are same - side? No, wait, \( \angle ACD \) and \( \angle CFE \): let's calculate. The sum of \( \angle ACD \) and \( \angle CFE \) should be \( 180^\circ \)? Wait, no, \( \angle CFE = 65^\circ \), so \( \angle ACD = 180^\circ - 65^\circ=115^\circ \)? Wait, no, wait, maybe I made a mistake. Wait, \( BD \parallel EG \), and \( AH \) is a transversal. \( \angle CFE = 65^\circ \), and \( \angle ACD \) and \( \angle CFE \): actually, \( \angle ACD \) and \( \angle CFE \) are same - side interior angles? Wait, no, \( \angle ACD \) is adjacent to \( \angle BCA \), which is a straight line. Wait, let's start over.

\( BD \parallel EG \), \( AH \) is a transversal. So \( \angle BCF \) (which is equal to \( \angle CFE \) because they are corresponding angles) is \( 65^\circ \). Then \( \angle ACD \) and \( \angle BCF \) are supplementary because they form a linear pair (they are on a straight line \( AH \)). So \( \angle ACD + \angle BCF=180^\circ \). Since \( \angle BCF = \angle CFE = 65^\circ \) (corresponding angles), then \( \angle ACD = 180^\circ - 65^\circ = 115^\circ \).

Step2: Verify the Calculation

We know that when two parallel lines are cut by a transversal, corresponding angles are equal. So \( \angle BCF=\angle CFE = 65^\circ \) (corresponding angles). Then, since \( \angle ACD \) and \( \angle BCF \) are supplementary (they form a linear pair, so their sum is \( 180^\circ \)), we calculate \( \angle ACD=180^\circ - 65^\circ = 115^\circ \).

Answer:

\( 115^\circ \) (the option with \( 115^\circ \))