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

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Question

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ .

$\angle A \cong \angle D$

$\angle B \cong \angle E$

$\angle C \cong \angle F$

True or false: It follows from the information given that $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Answer

The congruence of corresponding angles of two triangles does not alone prove that the triangles are congruent. For example, see the figures below:

The three angle congruence statements are true, but the sides are not congruent, so the triangles are not congruent. The statement is false.

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Question

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ .

$\overline{AB} \cong \overline{DE}$

$\overline{AC} \cong \overline{DF}$

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

True or false: It follows from the given information that $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Answer

As we are establishing whether or not $\bigtriangleup ABC \sim \bigtriangleup DEF$ , then , , and correspond respectively to , , and .

By the Side-Side-Side Congruence Postulate (SSS), if all three pairs of corresponding sides of two triangles are congruent, then the triangles themselves are congruent. Between $\bigtriangleup ABC$ and $\bigtriangleup DEF$ , $\overline{AB}$ and $\overline{DE}$ are corresponding sides, their congruence is given. The other two congruences between corresponding sides are given, so the conditions of SSS are satisfied. $\bigtriangleup ABC \cong \bigtriangleup DEF$ is indeed true.

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Question

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ .

$\overline{AB} \cong \overline{DE}$

$\overline{AC} \cong \overline{EF}$

$\overline{BC}\cong \overline{DF}$

True or false: It follows from the given information that $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Answer

As we are establishing whether or not $\bigtriangleup ABC \sim \bigtriangleup DEF$ , then , , and correspond respectively to , , and .

By the Side-Side-Side Congruence Postulate (SSS), if all three pairs of corresponding sides of two triangles are congruent, then the triangles themselves are congruent. However, if we restate the first side congruence as

$\overline{AB} \cong \overline{ED}$

and examine it with the other two:

$\overline{AC} \cong \overline{EF}$

$\overline{BC}\cong \overline{DF}$

We see that while we can invoke SSS, the points correspond to , respectively. The triangle congruence that follows is therefore $\bigtriangleup ABC \cong \bigtriangleup EDF$ .

The answer is therefore false.

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Question

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ .

$\overline{AB} \cong \overline{DE}$

$\overline{AC} \cong \overline{DF}$

$\angle A \cong \angle D$

True or false: It follows from the given information that $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Answer

As we are establishing whether or not $\bigtriangleup ABC \cong \bigtriangleup DEF$ , then , , and correspond respectively to , , and .

By the Side-Angle-Side Congruence Postulate (SAS), if two pairs of corresponding sides and the included angle of one triangle are congruent to the corresponding parts of a second, the triangles are congruent. $\overline{AB} \cong \overline{DE}$ and $\overline{AC} \cong \overline{DF}$ , indicating congruence between corresponding sides, and $\angle A \cong \angle D$ , indicating congruence between corresponding included angles. This satisfies the conditions of SAS, so $\bigtriangleup ABC \cong \bigtriangleup DEF$ is true.

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Question

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ .

$\overline{AB} \cong \overline{DE}$

$\overline{AC} \cong \overline{DF}$

$\angle C \cong \angle F$

True or false: It follows from the given information that $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Answer

Examine the diagram below.

$\overline{AB} \cong \overline{DE}$ , $\overline{AC} \cong \overline{DF}$ , and $\angle C \cong \angle F$ , but $\overline{BC} cong \overline{EF}$ . As a result, it is not true that $\bigtriangleup ABC \cong \bigtriangleup DEF$ . Therefore, the statement is false.

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Question

Refer to the above two triangles. By what statement does it follow that $\bigtriangleup ABC \cong \bigtriangleup DEF$ ?

Answer

We are given that two angles of $\bigtriangleup ABC$ - $\angle B$ and $\angle C$ - and a nonincluded side $\overline{AC}$ are congruent to their corresponding parts, $\angle E$ , $\angle F$ , and $\overline{DF}$ of $\bigtriangleup DEF$ . It follows from the Angle-Angle-Side Theorem that $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

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Question

Refer to the above two triangles. By what statement does it follow that $\bigtriangleup ABC \cong \bigtriangleup DEF$ ?

Answer

We are given that two sides of $\bigtriangleup ABC$ - sides $\overline{AB}$ and $\overline{BC}$ - and their included angle $\angle B$ are congruent to their corresponding parts, sides $\overline{DE}$ and $\overline{EF}$ and $\angle E$ of $\bigtriangleup DEF$ . It follows from the Side-Angle-Side Postulate that $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

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