$\delta yeg$ and $\delta pom$ are acute triangles. $\angle e \cong \angle o$. which statement will prove $\delta yeg$ is always congruent to $\delta pom$?

select one:
a. $\angle y \cong \angle p$ and $\angle g \cong \angle m$
b. there is a sequence of rigid motions that maps $\angle g$ onto $\angle m$ and $\overline{ye}$ onto $\overline{po}$.
c. there is a sequence of rigid motions that maps point $y$ onto point $p$ and $\overline{yg}$ onto $\overline{pm}$.
d. $\overline{yg} \cong \overline{pm}$ and $\overline{ye} \cong \overline{po}$

Question

$\delta yeg$ and $\delta pom$ are acute triangles. $\angle e \cong \angle o$. which statement will prove $\delta yeg$ is always congruent to $\delta pom$?

select one:
a. $\angle y \cong \angle p$ and $\angle g \cong \angle m$
b. there is a sequence of rigid motions that maps $\angle g$ onto $\angle m$ and $\overline{ye}$ onto $\overline{po}$.
c. there is a sequence of rigid motions that maps point $y$ onto point $p$ and $\overline{yg}$ onto $\overline{pm}$.
d. $\overline{yg} \cong \overline{pm}$ and $\overline{ye} \cong \overline{po}$

Accepted Answer

# Explanation:

⚡ Using what you learned: congruence criteria for triangles (sss, sas, asa, aas, hl)

## Step 1: Analyze the given information

We are given two acute triangles, $\Delta YEG$ and $\Delta POM$, with one pair of congruent corresponding angles:
$$ \angle E \cong \angle O $$

We need to find which statement guarantees that $\Delta YEG \cong \Delta POM$.

## Step 2: Evaluate the options

* **Option a**: $\angle Y \cong \angle P$ and $\angle G \cong \angle M$
  This gives us three pairs of congruent angles (AAA). AAA shows similarity, not necessarily congruence.

* **Option b**: There is a sequence of rigid motions that maps $\angle G$ onto $\angle M$ and $\overline{YE}$ onto $\overline{PO}$.
  Rigid motions preserve angle measures and segment lengths. This implies:
  $$ \angle G \cong \angle M $$
  $$ \overline{YE} \cong \overline{PO} $$
  Combined with the given $\angle E \cong \angle O$, we have:
  * Angle: $\angle G \cong \angle M$
  * Angle: $\angle E \cong \angle O$
  * Non-included Side: $\overline{YE} \cong \overline{PO}$

This satisfies the Angle-Angle-Side (AAS) congruence criterion, which proves $\Delta YEG \cong \Delta POM$.

* **Option c**: There is a sequence of rigid motions that maps Point $Y$ onto Point $P$ and $\overline{YG}$ onto $\overline{PM}$.
  This only establishes $\overline{YG} \cong \overline{PM}$. Combined with $\angle E \cong \angle O$, we have Side-Angle (SSA), which is not a valid congruence criterion.

* **Option d**: $\overline{YG} \cong \overline{PM}$ and $\overline{YE} \cong \overline{PO}$
  Combined with $\angle E \cong \angle O$, this gives Side-Side-Angle (SSA) using the non-included angle $\angle E$ and $\angle O$. SSA does not guarantee congruence.

# Answer:

b. There is a sequence of rigid motions that maps $\angle G$ onto $\angle M$ and $\overline{YE}$ onto $\overline{PO}$.

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Question

Explanation:

Step 1: Analyze the given information

Step 2: Evaluate the options

Answer: