in trapezoid mode, diagonal $ \overline { md } $ is perpendicular to $ \overline { om } $, and $ om = me = ed = 5 $ in. find $ m \angle odm, m \angle mod $, and $ m \angle med $. answer: $ m \angle odm = $ $ m \angle mod = $ $ m \angle med = $

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# Explanation:
## Step1: Prove $ \triangle OMD\cong\triangle EDM $
Since $ MODE $ is a trapezoid, $ OM\parallel ED $. Given $ OM = ME=ED = 5$ in.
By $ SSS $ ( $ OM = ED$, $ MD=DM$, $ ME = OM$ ), $ \triangle OMD\cong\triangle EDM $. Then $ \angle ODM=\angle DME $.
Because $ OM\parallel ED $, $ \angle OMD+\angle DME = 90^{\circ}$ ( $ \angle OMD + \angle ODM=90^{\circ}$ as $ \angle OMD = 90^{\circ}$ )
Also, $ ME = ED$, so $ \triangle MED $ is isosceles. Let $ \angle ODM = x$, then $ \angle DME=x$, $ \angle MED = 180 - 2x$.
Since $ OM\parallel ED $, $ \angle MOD+\angle ODE = 180^{\circ}$. And $ \triangle OMD$ is right - angled ($ \angle OMD = 90^{\circ}$), $ OM = MD$ ( $ OM = ME = ED$, from $ \triangle OMD\cong\triangle EDM$, $ MD = OM$ )
## Step2: Find $ \angle ODM$
In right - angled $ \triangle OMD$, $ OM = MD$, so $ \angle ODM = 45^{\circ}$ (using the property of a right - angled isosceles triangle: if $ a = b$ in $ \triangle ABC$ with $ \angle C = 90^{\circ}$, then $ \angle A=\angle B = 45^{\circ}$, here $ a = OM$, $ b = MD$, $ \angle OMD = 90^{\circ}$)
## Step3: Find $ \angle MOD$
In right - angled $ \triangle OMD$, $ \angle MOD=45^{\circ}$ (sum of angles in a triangle: $ \angle MOD+\angle ODM+\angle OMD=180^{\circ}$, $ \angle OMD = 90^{\circ}$, $ \angle ODM = 45^{\circ}$)
## Step4: Find $ \angle MED$
Since $ OM\parallel ED $, $ \angle MOD+\angle ODE = 180^{\circ}$. $ \angle ODE=\angle ODM+\angle MDE$. From $ \triangle OMD\cong\triangle EDM$, $ \angle MDE=\angle MOD = 45^{\circ}$. So $ \angle ODE = 90^{\circ}$.
Since $ ME = ED$, $ \angle MED=90^{\circ}$ (using the property of parallel lines and congruent triangles. Another way: extend $ OM$ and draw a line parallel to $ MD$ through $ E$. Or, since $ \triangle OMD\cong\triangle EDM$, $ \angle DEM=\angle MOD + \angle ODM=90^{\circ}$)

# Answer:
$ m\angle ODM = 45^{\circ}$
$ m\angle MOD=45^{\circ}$
$ m\angle MED = 90^{\circ}$

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QUESTION IMAGE

Question

Explanation:

Step1: Prove \( \triangle OMD\cong\triangle EDM \)

Step2: Find \( \angle ODM\)

Step3: Find \( \angle MOD\)

Step4: Find \( \angle MED\)

Answer: