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

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Example Question #41 : Acute / Obtuse Isosceles Triangles

Thingy

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

Possible Answers:

The Converse of the Isosceles Triangle Theorem

The Side-Angle-Side Postulate

The Isosceles Triangle Theorem

The Hinge Theorem

The Side-Side-Side Postulate

Correct answer:

The Side-Side-Side Postulate

Explanation:

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

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Example Question #1 : How To Find If Of Acute / Obtuse Isosceles Triangles Are Congruent

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

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

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

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

Which statement(s) must be true?

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

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

Possible Answers:

Neither (a) nor (b)

(a) and (b)

(a) but not (b)

(b) but not (a)

Correct answer:

Neither (a) nor (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

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

$\frac{EF}{BC} = \frac{10}{10}= 1$ $\displaystyle \frac{EF}{BC} = \frac{10}{10}= 1$

and

$\frac{DE}{AB} = \frac{15}{10} = \frac{3}{2}$ $\displaystyle \frac{DE}{AB} = \frac{15}{10} = \frac{3}{2}$

$\frac{DE}{AB} \ne \frac{EF}{BC}$ $\displaystyle \frac{DE}{AB} \ne \frac{EF}{BC}$, so at least one pair of corresponding sides is not in proportion. Therefore,

$\bigtriangleup ABC \nsim \bigtriangleup DEF$ $\displaystyle \bigtriangleup ABC \nsim \bigtriangleup DEF$

The triangles are not similar, and thus cannot be congruent either, so neither statement holds.

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

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #41 : Acute / Obtuse Isosceles Triangles

Example Question #1 : How To Find If Of Acute / Obtuse Isosceles Triangles Are Congruent

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