Geometry - How to find if of acute / obtuse isosceles triangles are congruent

Thingy

Refer to the above diagram. $\overline{AD} \cong \overline{BC}$ and $\overline{AC} \cong \overline{BD}$ . By what statement does it follow that $\bigtriangleup ADC \cong \bigtriangleup BCD$ ?

The Side-Side-Side Postulate

The Hinge Theorem

The Converse of the Isosceles Triangle Theorem

The Isosceles Triangle Theorem

The Side-Angle-Side Postulate

Explanation

In addition to the fact that $\overline{AD} \cong \overline{BC}$ and $\overline{AC} \cong \overline{BD}$ , we also have that $\overline{DC} \cong \overline{DC}$ , since, by the Reflexive Property of Congruence, any segment is congruent to itself. We can restate this in a more usable form as $\overline{DC} \cong \overline{CD}$ ; since we have three side congruences between triangles, it follows from the Side-Side-Side Postulate that $\bigtriangleup ADC \cong \bigtriangleup BCD$ .

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ such that

$\overline{AB} \cong \overline{AC}$

$\overline{DE} \cong \overline{DF}$

$\overline{BC} \cong \overline{EF}$

Which statement(s) must be true?

(a) $\bigtriangleup ABC \sim \bigtriangleup DEF$

(b) $\bigtriangleup ABC \cong \bigtriangleup DEF$

Neither (a) nor (b)

(a) but not (b)

(b) but not (a)

(a) and (b)

Explanation

Neither similarity nor congruence of the two triangles follows from the statements given, as can be seen from the figure below:

Isosceles 1

$\overline{AB} \cong \overline{AC}$ , $\overline{DE} \cong \overline{DF}$ , and $\overline{BC} \cong \overline{EF}$ . However, the triangles are not similar, as