Suppose is congruent to . List all the other correct ways of expressing this congruence.

Geometric Twins | FIO

Question 6

Suppose $\Delta \text{HEN}$ is congruent to $\Delta \text{BIG}$ . List all the other correct ways of expressing this congruence.

Solution

Understand the Question

When two triangles are congruent, the order of letters in their names indicates which vertices, sides, and angles correspond to each other.
Given $\Delta \text{HEN} \cong \Delta \text{BIG}$ $Δ HEN ≅ Δ BIG$ , the vertices correspond as:
- $\text{H} \leftrightarrow \text{B}$
- $\text{E} \leftrightarrow \text{I}$
- $\text{N} \leftrightarrow \text{G}$
Since a triangle has $3$ vertices, there are $3! = 6$ total possible orderings. Given $1$ notation, there are $5$ other valid ways to express this congruence by maintaining the same vertex correspondence.

Step 1 · Identify Corresponding Vertices

From the given congruence $\Delta \text{HEN} \cong \Delta \text{BIG}$ , the corresponding pairs of vertices are:

$\text{H} \leftrightarrow \text{B}$
$\text{E} \leftrightarrow \text{I}$
$\text{N} \leftrightarrow \text{G}

Step 2 · List All Other Congruence Statements

Rearranging the vertices of the first triangle and matching the corresponding vertices of the second triangle yields the other $5$ ways:

Starting with $\text{H}$ : $\Delta \text{HNE} \cong \Delta \text{BGI}$
Starting with $\text{E}$ : $\Delta \text{EHN} \cong \Delta \text{IBG}$ $\Delta \text{ENH} \cong \Delta \text{IGB}$
Starting with $\text{N}$ : $\Delta \text{NHE} \cong \Delta \text{GBI}$ $\Delta \text{NEH} \cong \Delta \text{GIB}$

Answer

The other $5$ correct ways of expressing the congruence are:

$\Delta \text{HNE} \cong \Delta \text{BGI}$
$\Delta \text{EHN} \cong \Delta \text{IBG}$
$\Delta \text{ENH} \cong \Delta \text{IGB}$
$\Delta \text{NHE} \cong \Delta \text{GBI}$
$\Delta \text{NEH} \cong \Delta \text{GIB}$

Common Mistakes

Ignoring Vertex Correspondence: Writing statements like $\Delta \text{HNE} \cong \Delta \text{BIG}$ where the vertices do not align correctly (e.g., matching $\text{N}$ with $\text{I}$ instead of $\text{G}$ ).
Missing Combinations: Forgetting that $3$ vertices produce $3! = 6$ total permutations, leading to omitting some of the $5$ alternative representations.

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Q14

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Q15

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Q16

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Q17

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Q19

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