Question

geometry - hellis
topic 3: readiness assessment
lines ( ell ) and ( m ) are intersected by transversal ( t ) and ( ell parallel m ).
select all the angles that are supplementary to ( angle 7 ).
a. ( angle 5 )
b. ( angle 2 )
c. ( angle 6 )
d. ( angle 8 )
e. ( angle 1 )
f. ( angle 3 )

Explanation:

Step1: Recall the definition of supplementary angles

Supplementary angles are two angles whose sum is \(180^{\circ}\).

Step2: Analyze angle - angle relationships

• For \(\angle5\): \(\angle5+\angle7 = 180^{\circ}\) (linear - pair of angles, since they are adjacent and form a straight line).
• For \(\angle6\): Since \(\ell\parallel m\) and \(t\) is a transversal, \(\angle7\) and \(\angle6\) are consecutive interior angles. By the consecutive - interior - angles theorem, \(\angle7+\angle6=180^{\circ}\) (because \(\ell\parallel m\)).
• For \(\angle2\): \(\angle2\cong\angle6\) (alternate - interior angles as \(\ell\parallel m\) and \(t\) is a transversal). Since \(\angle7+\angle6 = 180^{\circ}\), then \(\angle7+\angle2=180^{\circ}\) (substitution property).
• For \(\angle8\): \(\angle7\) and \(\angle8\) are vertical angles. \(\angle7=\angle8\) (vertical - angles theorem), so \(\angle7+\angle8

eq180^{\circ}\) (they are equal, not supplementary unless they are right angles, which is not indicated).

• For \(\angle1\): \(\angle1\cong\angle5\) (corresponding angles as \(\ell\parallel m\) and \(t\) is a transversal). Since \(\angle7+\angle5 = 180^{\circ}\), \(\angle7+\angle1 = 180^{\circ}\) (substitution property).
• For \(\angle3\): \(\angle3\cong\angle7\) (corresponding angles as \(\ell\parallel m\) and \(t\) is a transversal). So \(\angle7+\angle3

eq180^{\circ}\) (they are equal, not supplementary unless they are right angles, which is not indicated).

Answer:

A. \(\angle5\), B. \(\angle2\), C. \(\angle6\), E. \(\angle1\)