In the figure below, ABC DEF, A = 45^, C = 85^. Find the measures of D and E.

a) If two triangles are congruent, then they have exactly same shape and size (True or False).

• True. Congruent triangles are defined as having the same shape and size.

b) Any two equilateral triangles are congruent (True or False).

• False. While all equilateral triangles have the same shape (all angles are $60^\circ$), their sizes can differ. For example, an equilateral triangle with side length 2 cm is not congruent to an equilateral triangle with side length 5 cm.

c) In the figure below, $\triangle ABC \equiv \triangle DEF$, $\angle A = 45^\circ$, $\angle C = 85^\circ$. Find the measures of $\angle D$ and $\angle E$.

Step 1: Use the property of congruent triangles to find $\angle D$. Since $\triangle ABC \equiv \triangle DEF$, their corresponding angles are equal. Therefore, $\angle A = \angle D$. Given $\angle A = 45^\circ$. $\angle D = 45^\circ$

Step 2: Find $\angle B$ in $\triangle ABC$. The sum of angles in a triangle is $180^\circ$. $\angle A + \angle B + \angle C = 180^\circ$ $45^\circ + \angle B + 85^\circ = 180^\circ$ $130^\circ + \angle B = 180^\circ$ $\angle B = 180^\circ - 130^\circ$ $\angle B = 50^\circ$

Step 3: Use the property of congruent triangles to find $\angle E$. Since $\triangle ABC \equiv \triangle DEF$, their corresponding angles are equal. Therefore, $\angle B = \angle E$. From Step 2, $\angle B = 50^\circ$. $\angle E = 50^\circ$

The measures are: $\boxed{\angle D = 45^\circ, \angle E = 50^\circ}$