Question

Math

Posted 4 months ago

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Below are $\triangle A B C$ and $\triangle D E F$. We assume that $A B=D E$, $m \angle A=m \angle D$, and $m \angle B=m \angle E$.
Here is a rough outline of a proof that $\triangle A B C \cong \triangle D E F$ :
1. We can map $\triangle A B C$ using a sequence of rigid transformations so that $A^{\prime}=D$ and $B^{\prime}=E$. Show drawing.
2. If $C^{\prime}$ and $F$ are on the same side of $\overleftrightarrow{D E}$, then $C^{\prime}=F$. Show drawing.
3. If $C^{\prime}$ and $F$ are on opposite sides of $\overleftrightarrow{D E}$, then we reflect $\triangle A^{\prime} B^{\prime} C^{\prime}$ across $\stackrel{\longleftrightarrow}{D E}$ and then $C^{\prime \prime}=F, A^{\prime \prime}=D$ and $B^{\prime \prime}=E$. Show drawing.
Answer two questions about this proof.
1) How did we show that the triangles were congruent?
Choose 1 answer:
(A) We mapped one figure onto the other using rigid transformations.
(B) We mapped one figure onto the other using any kind of transformations.
(C) We showed that all corresponding sides had equal lengths and all corresponding angles had equal measures.
2) What triangles did we show are congruent?
Choose 1 answer:
(A) Triangles where 2 pairs of corresponding angles have the same measure, and the included corresponding sides have the same length
(B) Triangles where 2 pairs of corresponding angles have the same measure
(C) All triangles
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Answer from Sia

Posted 4 months ago

Solution by Steps

step 2

Rigid transformations include translations, rotations, and reflections, which preserve the size and shape of geometric figures

step 3

By mapping $A$ to $D$ and $B$ to $E$, and ensuring that $C'$ and $F$ are either coincident or can be made coincident through reflection, we establish congruence of the triangles

step 4

The congruence of $\triangle ABC$ and $\triangle DEF$ is shown by the fact that they can be superimposed through these transformations

1 Answer

A

Key Concept

Rigid transformations to prove congruence

Explanation

Rigid transformations preserve the size and shape of figures, so mapping one triangle onto another using these transformations proves congruence.

step 1

The triangles we showed to be congruent have two pairs of corresponding angles with the same measure and the included corresponding sides of the same length

step 2

This is based on the given information that $AB=DE$, $m\angle A=m\angle D$, and $m\angle B=m\angle E$

step 3

These conditions satisfy the Angle-Side-Angle (ASA) congruence criterion for triangles

2 Answer

A

Key Concept

ASA Congruence Criterion

Explanation

Two triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side of the other triangle.

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