Question

enter the given angle measure into the diagram. (given)

• eq bisects ∠lec.
• m∠lec = 160°.

enter each measure required to calculate m∠leg, starting with the given information.
the measure of ∠leg can be calculated.
m∠leg = °
the measure of ∠leg can not be calculated.
diagram notes:

Explanation:

Step1: Understand Angle Bisector

An angle bisector divides an angle into two equal parts. So, if \( EQ \) bisects \( \angle LEC \), then \( \angle LEQ = \angle QEC \), and each is half of \( \angle LEC \). But wait, first, let's check the straight line. Wait, \( GQ \) and \( EC \)? Wait, no, looking at the diagram, \( G \) and \( Q \) are on a straight line, \( L \) and \( C \) – wait, maybe \( \angle LEC = 160^\circ \), and \( EQ \) bisects it, but also, \( \angle LEG \) and \( \angle LEQ \) – wait, no, maybe \( GQ \) is a straight line (180°), and \( \angle LEC = 160^\circ \), so the adjacent angle? Wait, no, let's re-examine. Wait, the problem says \( EQ \) bisects \( \angle LEC \), so \( \angle LEQ = \angle QEC = \frac{160^\circ}{2} = 80^\circ \). Then, since \( G \) and \( Q \) are on a straight line (so \( \angle GEQ = 180^\circ \)? No, wait, \( G \), \( E \), \( Q \) are colinear? So \( \angle LEG + \angle LEQ = 180^\circ \)? Wait, no, if \( G \), \( E \), \( Q \) are on a straight line, then \( \angle GEQ = 180^\circ \). But \( \angle LEQ \) is 80°, so \( \angle LEG = 180^\circ - 80^\circ = 100^\circ \)? Wait, no, maybe I messed up. Wait, let's start over.

Wait, the given is \( m\angle LEC = 160^\circ \), and \( EQ \) bisects \( \angle LEC \). So first, find \( \angle LEQ \): since bisector, \( \angle LEQ = \frac{1}{2} \times 160^\circ = 80^\circ \). Then, \( G \) and \( Q \) are on a straight line, so \( \angle LEG + \angle LEQ = 180^\circ \) (because they are supplementary, forming a straight line). Therefore, \( \angle LEG = 180^\circ - 80^\circ = 100^\circ \)? Wait, no, that can't be. Wait, maybe \( G \), \( E \), \( C \)? No, the diagram shows \( G \) and \( Q \) on a horizontal line, \( L \) and \( C \) on a vertical? Wait, no, the diagram: points \( G \), \( E \), \( Q \) are colinear (horizontal line), and \( L \), \( E \), \( C \) – wait, \( \angle LEC = 160^\circ \), \( EQ \) bisects it, so \( \angle LEQ = 80^\circ \). Then, since \( G \), \( E \), \( Q \) are colinear (straight line, 180°), \( \angle LEG + \angle LEQ = 180^\circ \). Therefore, \( \angle LEG = 180^\circ - 80^\circ = 100^\circ \)? Wait, but that seems off. Wait, maybe I made a mistake. Wait, no, let's check again.

Wait, the problem says \( m\angle LEC = 160^\circ \), \( EQ \) bisects \( \angle LEC \), so \( \angle LEQ = \angle QEC = 80^\circ \). Now, looking at the diagram, \( G \), \( E \), \( Q \) are on a straight line (so \( \angle GEQ = 180^\circ \)). Then, \( \angle LEG \) and \( \angle LEQ \) are adjacent angles forming a linear pair, so their sum is 180°. Therefore, \( m\angle LEG = 180^\circ - m\angle LEQ = 180^\circ - 80^\circ = 100^\circ \)? Wait, but that would mean \( \angle LEG = 100^\circ \). Wait, but maybe the diagram is different. Wait, maybe \( \angle LEC = 160^\circ \), and \( G \), \( E \), \( C \) are on a straight line? No, the diagram shows \( G \) and \( Q \) on one line, \( L \) and \( C \) on another. Wait, perhaps the correct approach is:

1. Since \( EQ \) bisects \( \angle LEC \), \( \angle LEQ = \frac{1}{2} \times 160^\circ = 80^\circ \).
2. \( \angle LEG \) and \( \angle LEQ \) are supplementary (because \( G \), \( E \), \( Q \) are colinear, so they form a straight line, 180°).
3. Therefore, \( m\angle LEG = 180^\circ - 80^\circ = 100^\circ \). Wait, but that seems high. Wait, maybe I got the angle wrong. Wait, maybe \( \angle LEC = 160^\circ \), and \( \angle LEG \) is adjacent to \( \angle LEQ \) but on the other side. Wait, no, let's check the diagram again. The diagram has point \( E \), with lines \( EL \), \( EC \), \( EQ \), \( EG \). \( G \) and \( Q \) are on a horizontal line (so \( \angle GEQ = 180^\circ \)), \( L \) is above, \( C \) is below. \( \angle LEC = 160^\circ \), so the angle between \( EL \) and \( EC \) is 160°, \( EQ \) bisects that, so \( \angle LEQ = 80^\circ \), then \( \angle LEG \) is the angle between \( EL \) and \( EG \), and since \( EG \) and \( EQ \) are a straight line (180°), \( \angle LEG = 180° - 80° = 100° \). Yes, that makes sense.

Step2: Calculate \( m\angle LEG \)

Using the angle bisector theorem (dividing the angle into two equal parts) and linear pair (supplementary angles sum to 180°):

- \( m\angle LEQ = \frac{1}{2} \times m\angle LEC = \frac{1}{2} \times 160^\circ = 80^\circ \)
- \( m\angle LEG + m\angle LEQ = 180^\circ \) (linear pair)
- Therefore, \( m\angle LEG = 180^\circ - 80^\circ = 100^\circ \)

Answer:

\( 100 \)