triangle jkl is formed by connecting the midpoints of the side of triangle ghi. the measures of the interior angles of triangle ghi are shown. find the measure of ∠glj. figures not necessarily drawn to scale.

Question

Accepted Answer

# Explanation:
## Step1: Recall the Midline Theorem (Triangle Midsegment Theorem)
The midline theorem states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half as long. So, if $ L $ and $ J $ are midpoints, $ LJ \parallel HI $.

## Step2: Use Corresponding Angles (Alternate Interior Angles)
Since $ LJ \parallel HI $ and $ GI $ is a transversal, $ \angle GLJ $ and $ \angle I $ are corresponding angles (alternate interior angles) and thus equal. Wait, no, wait. Wait, let's check the angles of triangle $ GHI $. The angles of $ \triangle GHI $ are $ \angle I = 37^\circ $, $ \angle H = 97^\circ $, $ \angle G = 46^\circ $. Wait, but $ L $ is the midpoint of $ GI $? Wait, no, $ JKL $ is formed by midpoints. So $ L $ is midpoint of $ GI $, $ J $ is midpoint of $ GH $, $ K $ is midpoint of $ HI $? Wait, maybe $ LJ $ is parallel to $ HI $. So $ \angle GLJ $ and $ \angle I $? Wait, no, maybe $ \angle GLJ $ is equal to $ \angle H $? Wait, no, let's re-examine. Wait, the sum of angles in a triangle is $ 180^\circ $. Wait, $ \angle I = 37^\circ $, $ \angle G = 46^\circ $, $ \angle H = 97^\circ $ (since $ 37 + 46 + 97 = 180 $, $ 37+46=83 $, $ 180 - 83 = 97 $, correct). Now, since $ L $ and $ J $ are midpoints, $ LJ \parallel HI $ (by midline theorem). So the line $ LJ $ is parallel to $ HI $, and $ GI $ is a transversal. Therefore, $ \angle GLJ $ and $ \angle I $ are alternate interior angles? Wait, no, $ \angle I $ is at vertex $ I $, between $ GI $ and $ HI $. $ \angle GLJ $ is at vertex $ L $, between $ GI $ and $ LJ $. Since $ LJ \parallel HI $, alternate interior angles would be $ \angle GLJ $ and $ \angle I $? Wait, no, maybe $ \angle GLJ $ is equal to $ \angle H $? Wait, no, let's think again. Wait, $ J $ is the midpoint of $ GH $, $ L $ is the midpoint of $ GI $, so $ LJ $ is midline, so $ LJ \parallel HI $ and $ LJ = \frac{1}{2}HI $. Therefore, the corresponding angles: $ \angle GLJ $ and $ \angle I $? Wait, no, $ \angle I $ is $ 37^\circ $, but wait, maybe $ \angle GLJ $ is equal to $ \angle H $? Wait, no, let's check the angles. Wait, $ \angle H $ is $ 97^\circ $, but that seems too big. Wait, no, maybe I made a mistake. Wait, the midline theorem: the segment connecting midpoints of two sides is parallel to the third side. So if $ L $ is midpoint of $ GI $ and $ J $ is midpoint of $ GH $, then $ LJ \parallel HI $. So $ LJ \parallel HI $, so $ \angle GLJ $ and $ \angle I $ are alternate interior angles, so they should be equal. Wait, $ \angle I = 37^\circ $, but wait, no, $ \angle G $ is $ 46^\circ $, $ \angle I $ is $ 37^\circ $, $ \angle H $ is $ 97^\circ $. Wait, maybe $ LJ $ is parallel to $ HI $, so $ \angle GLJ = \angle I = 37^\circ $? No, that doesn't seem right. Wait, maybe $ LJ $ is parallel to $ HI $, so $ \angle GLJ $ is equal to $ \angle H $? Wait, no, $ \angle H $ is at $ H $, between $ GH $ and $ HI $. $ \angle GLJ $ is at $ L $, between $ GI $ and $ LJ $. Wait, maybe I mixed up the sides. Wait, let's label the triangle: $ G $ at the bottom, $ I $ at the top left, $ H $ at the top right. So $ GI $ is the left side, $ GH $ is the right side, $ HI $ is the top side. Midpoints: $ L $ on $ GI $, $ J $ on $ GH $, $ K $ on $ HI $. Then $ LJ $ connects midpoints of $ GI $ and $ GH $, so $ LJ \parallel HI $ (midline theorem). Therefore, $ LJ \parallel HI $, so the transversal $ GI $ creates alternate interior angles $ \angle GLJ $ and $ \angle I $. Wait, $ \angle I $ is $ 37^\circ $, so $ \angle GLJ = 37^\circ $? But wait, let's check the sum. Wait, no, maybe $ \angle GLJ $ is equal to $ \angle H $? Wait, $ \angle H $ is $ 97^\circ $, but that would be if $ LJ $ is parallel to $ GI $, but no. Wait, maybe I made a mistake. Wait, the midline theorem: midsegment is parallel to the third side. So if $ L $ is midpoint of $ GI $ and $ J $ is midpoint of $ GH $, then midsegment $ LJ $ is parallel to $ HI $. Therefore, $ LJ \parallel HI $, so $ \angle GLJ $ and $ \angle I $ are alternate interior angles, so $ \angle GLJ = \angle I = 37^\circ $? But wait, let's check the angles again. Wait, $ \angle G = 46^\circ $, $ \angle I = 37^\circ $, $ \angle H = 97^\circ $. Wait, maybe the correct angle is $ \angle H = 97^\circ $? Wait, no, maybe $ LJ $ is parallel to $ HI $, so $ \angle GLJ $ is equal to $ \angle H $? Wait, no, $ \angle H $ is at $ H $, between $ GH $ and $ HI $. $ \angle GLJ $ is at $ L $, between $ GI $ and $ LJ $. Wait, maybe the transversal is $ GH $? No, $ GI $ is the transversal. Wait, let's draw this mentally. $ GI $ is a side from $ G $ (bottom) to $ I $ (top left). $ GH $ is from $ G $ (bottom) to $ H $ (top right). $ HI $ is from $ H $ (top right) to $ I $ (top left). Midpoints: $ L $ on $ GI $, $ J $ on $ GH $, $ K $ on $ HI $. So $ LJ $ connects $ L $ (mid $ GI $) to $ J $ (mid $ GH $). So $ LJ $ is parallel to $ HI $ (midline theorem). So $ LJ \parallel HI $, so the angle between $ GI $ and $ LJ $ (which is $ \angle GLJ $) should be equal to the angle between $ GI $ and $ HI $ (which is $ \angle I = 37^\circ $)? Wait, no, $ \angle I $ is between $ GI $ and $ HI $, so if $ LJ \parallel HI $, then $ \angle GLJ $ (between $ GI $ and $ LJ $) is equal to $ \angle I $ (between $ GI $ and $ HI $) because they are alternate interior angles. So $ \angle GLJ = 37^\circ $? But wait, that seems small. Wait, maybe I got the midpoints wrong. Maybe $ L $ is midpoint of $ GI $, $ J $ is midpoint of $ HI $? No, the problem says $ JKL $ is formed by connecting midpoints of $ GHI $. So $ J $ is midpoint of $ GH $, $ K $ midpoint of $ HI $, $ L $ midpoint of $ GI $. So $ LJ $ is midline of $ \triangle GHI $, parallel to $ HI $. Therefore, $ \angle GLJ = \angle I = 37^\circ $? Wait, but let's check the sum of angles in $ \triangle GLJ $. Wait, $ \angle G = 46^\circ $, if $ \angle GLJ = 37^\circ $, then $ \angle GJL = 180 - 46 - 37 = 97^\circ $, which is equal to $ \angle H $, which makes sense because $ LJ \parallel HI $, so $ \angle GJL = \angle H $ (corresponding angles). Ah, there we go. So $ \angle GLJ $ is equal to $ \angle I = 37^\circ $? Wait, no, $ \angle GJL $ is equal to $ \angle H = 97^\circ $, and $ \angle GLJ $ is equal to $ \angle I = 37^\circ $, and $ \angle G = 46^\circ $, so $ 37 + 46 + 97 = 180 $, correct. So yes, $ \angle GLJ = 37^\circ $? Wait, no, wait, $ \angle I $ is $ 37^\circ $, so $ \angle GLJ = 37^\circ $. Wait, but let's confirm the midline theorem. The midline is parallel to the third side, so $ LJ \parallel HI $, so alternate interior angles: $ \angle GLJ $ and $ \angle I $ are alternate interior angles, so they are equal. Therefore, $ \angle GLJ = 37^\circ $. Wait, but I think I made a mistake earlier. Wait, $ \angle I $ is $ 37^\circ $, so $ \angle GLJ = 37^\circ $.

Wait, no, wait, maybe $ \angle GLJ $ is equal to $ \angle H $. Wait, no, let's re-express. Let's denote:

- $ L $ is midpoint of $ GI $
- $ J $ is midpoint of $ GH $

Therefore, by midline theorem, $ LJ \parallel HI $ and $ LJ = \frac{1}{2}HI $.

Therefore, the line $ LJ $ is parallel to $ HI $, so the transversal $ GI $ creates alternate interior angles $ \angle GLJ $ and $ \angle I $. Therefore, $ \angle GLJ = \angle I = 37^\circ $.

Yes, that makes sense. So the measure of $ \angle GLJ $ is $ 37^\circ $.

# Answer:
$ \boxed{37^\circ} $

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Recall the Midline Theorem (Triangle Midsegment Theorem)

Step2: Use Corresponding Angles (Alternate Interior Angles)

Answer: