6. If $$ riangle C A T \cong riangle D O G $$, and $$ \angle T=90^{\circ} $$ and $$ \angle D=45^{\circ} $$, then $$ \angle O= $$

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6. If $$ 	riangle C A T \cong 	riangle D O G $$, and $$ \angle T=90^{\circ} $$ and $$ \angle D=45^{\circ} $$, then $$ \angle O= $$

UpStudy AI · Accepted Answer

Given that $$\triangle CAT \cong \triangle DOG$$, the corresponding angles and sides of the triangles are equal. Here's the step-by-step reasoning:

1. **Correspondence of Angles:**
   - $$ \angle C $$ corresponds to $$ \angle D $$.
   - $$ \angle A $$ corresponds to $$ \angle O $$.
   - $$ \angle T $$ corresponds to $$ \angle G $$.

2. **Given Angles:**
   - $$ \angle T = 90^\circ $$ in $$\triangle CAT$$, so $$ \angle G = 90^\circ $$ in $$\triangle DOG$$.
   - $$ \angle D = 45^\circ $$ in $$\triangle DOG$$, which means $$ \angle C = 45^\circ $$ in $$\triangle CAT$$.

3. **Finding $$ \angle O $$:**
   - In $$\triangle CAT$$, the sum of angles is $$ 180^\circ $$:
     $$
     \angle C + \angle A + \angle T = 180^\circ \
     45^\circ + \angle A + 90^\circ = 180^\circ \
     \angle A = 45^\circ
     $$
   - Since $$ \angle A $$ corresponds to $$ \angle O $$:
     $$
     \angle O = 45^\circ
     $$

**Answer:** $$ \boxed{45^{\circ}} $$
$$ \angle O = 45^\circ $$

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