homework 5-4 midseg... of triangles question 13, 5.4.15 x is the midpoint of $\overline{uv}$, y is the midpoint of $\overline{uw}$, and t is the midpoint of $\overline{vw}$. if $m\angle w = 63^\circ$ and $m\angle u = 40^\circ$, find $m\angle tyw$. $m\angle tyw = \square^\circ$ (simplify your answer.)

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# Explanation:
## Step1: Recall midsegment theorem
The midsegment of a triangle is parallel to the third side and half its length. So, $ TY \parallel UV $ because $ T $ and $ Y $ are midpoints of $ VW $ and $ UW $ respectively.
## Step2: Use corresponding angles
Since $ TY \parallel UV $, $ \angle TYW $ and $ \angle U $ are corresponding angles. Wait, no, wait. Wait, let's check the triangle angles. First, find $ \angle V $ in $ \triangle UVW $. The sum of angles in a triangle is $ 180^\circ $. So $ m\angle V = 180^\circ - m\angle U - m\angle W = 180 - 40 - 63 = 77^\circ $? Wait, no, maybe I made a mistake. Wait, $ TY $ is a midsegment, so $ TY \parallel UV $, so $ \angle TYW $ should be equal to $ \angle U $? Wait, no, let's look at the sides. $ Y $ is midpoint of $ UW $, $ T $ is midpoint of $ VW $, so $ TY \parallel UV $ (midsegment theorem: the segment connecting midpoints of two sides is parallel to the third side). So $ \angle TYW $ and $ \angle U $ are corresponding angles? Wait, no, $ UV $ and $ TY $ are parallel, and transversal is $ UW $. So $ \angle U $ and $ \angle TYW $ are alternate interior angles? Wait, $ U $ is at vertex $ U $, $ Y $ is on $ UW $, $ T $ is on $ VW $. So line $ TY $ is parallel to $ UV $, so $ \angle U $ (at $ U $, between $ UV $ and $ UW $) and $ \angle TYW $ (at $ Y $, between $ TY $ and $ YW $) are alternate interior angles, so they should be equal? Wait, no, maybe I mixed up. Wait, let's recast. The midsegment $ TY $ is parallel to $ UV $, so the corresponding angle to $ \angle U $ would be $ \angle TYW $? Wait, no, let's calculate $ \angle V $ first. In $ \triangle UVW $, $ m\angle U = 40^\circ $, $ m\angle W = 63^\circ $, so $ m\angle V = 180 - 40 - 63 = 77^\circ $. Wait, but $ TY \parallel UV $, so $ \angle TYW $ should be equal to $ \angle U $? Wait, no, maybe I got the transversal wrong. Wait, $ UW $ is the transversal cutting $ UV $ and $ TY $. So $ \angle U $ (at $ U $, between $ UV $ and $ UW $) and $ \angle TYW $ (at $ Y $, between $ TY $ and $ YW $): since $ UV \parallel TY $, alternate interior angles are equal. So $ \angle U = \angle TYW $? Wait, but $ \angle U $ is $ 40^\circ $, so $ m\angle TYW = 40^\circ $? Wait, that seems off. Wait, no, maybe $ TY \parallel UV $, so $ \angle TYW $ is equal to $ \angle U $ because they are corresponding angles. Wait, let's draw the triangle: $ U $ at top, $ V $ and $ W $ at bottom. $ X $ midpoint of $ UV $, $ Y $ midpoint of $ UW $, $ T $ midpoint of $ VW $. So $ TY $ connects midpoints of $ UW $ and $ VW $, so it's parallel to $ UV $. So $ UV \parallel TY $, so the angle at $ Y $ ( $ \angle TYW $) and angle at $ U $ ( $ \angle U $) are equal because they are corresponding angles (since $ UW $ is the transversal). So yes, $ m\angle TYW = m\angle U = 40^\circ $? Wait, but let's check the sum of angles. Wait, maybe I made a mistake. Wait, $ \angle W $ is $ 63^\circ $, $ \angle U $ is $ 40^\circ $, so $ \angle V $ is $ 77^\circ $. But $ TY \parallel UV $, so $ \angle YTW = \angle V = 77^\circ $ (corresponding angles). But we need $ \angle TYW $. In $ \triangle TYW $, angles sum to $ 180^\circ $. $ \angle W = 63^\circ $, $ \angle YTW = 77^\circ $, so $ \angle TYW = 180 - 63 - 77 = 40^\circ $. Ah, there we go. So using midsegment theorem, $ TY \parallel UV $, so $ \angle YTW = \angle V $ (corresponding angles). Then in $ \triangle TYW $, $ m\angle TYW = 180 - m\angle W - m\angle YTW $. We know $ m\angle W = 63^\circ $, $ m\angle YTW = m\angle V = 180 - 40 - 63 = 77^\circ $. So $ m\angle TYW = 180 - 63 - 77 = 40^\circ $. So that's correct.
# Answer:
$ 40 $

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QUESTION IMAGE

Question

Explanation:

Step1: Recall midsegment theorem

Step2: Use corresponding angles

Answer: