Question

proving and applying the asa and aas congruence criteria

1. in the figure shown, which theorem can be used to show that $\triangle abc \cong \triangle def$?

\\(\boldsymbol{\text{a}}\\) the triangles are not congruent.
\\(\boldsymbol{\text{b}}\\) sas triangle congruence theorem
\\(\boldsymbol{\text{c}}\\) aas triangle congruence theorem
\\(\boldsymbol{\text{d}}\\) asa triangle congruence theorem
triangles abc and def are shown with markings

1. which theorem is not a valid theorem to show that two triangles are congruent?

\\(\boldsymbol{\text{a}}\\) sas triangle congruence theorem
\\(\boldsymbol{\text{b}}\\) ssa triangle congruence theorem
\\(\boldsymbol{\text{c}}\\) asa triangle congruence theorem
\\(\boldsymbol{\text{d}}\\) aas triangle congruence theorem

1. in the figure shown, what additional information is needed to show that $\triangle abc \cong \triangle def$ by asa?

\\(\boldsymbol{\text{a}}\\) $m\angle c$
\\(\boldsymbol{\text{b}}\\) $\overline{ab} \cong \overline{de}$
\\(\boldsymbol{\text{c}}\\) $m\angle f$
\\(\boldsymbol{\text{d}}\\) $\overline{bc} \cong \overline{ef}$
triangles abc and def are shown with markings

1. select all the pairs of triangles that are congruent by asa.

\\(\boldsymbol{\text{a}}\\) triangle pair a
\\(\boldsymbol{\text{b}}\\) triangle pair b
\\(\boldsymbol{\text{c}}\\) triangle pair c
\\(\boldsymbol{\text{d}}\\) triangle pair d
\\(\boldsymbol{\text{e}}\\) triangle pair e
\\(\boldsymbol{\text{f}}\\) triangle pair f

1. in the figure shown, what is the value of $x$?

triangles abc and def are shown with markings; $bc = 13x - 12$

Explanation:

Step1: Analyze Question 1

In $\triangle ABC$ and $\triangle DEF$: $\angle B \cong \angle E$, $\angle A \cong \angle D$, $\overline{BC} \cong \overline{EF}$. This matches AAS (two angles and a non-included side congruent).

Step2: Analyze Question 2

SSA does not guarantee triangle congruence; SAS, ASA, AAS are valid congruence theorems.

Step3: Analyze Question 3

ASA requires two angles and the included side congruent. For $\triangle ABC$ and $\triangle DEF$, we have $\angle A \cong \angle E$, $\angle B \cong \angle D$. The included side for $\angle A$ and $\angle B$ is $\overline{AB}$, and for $\angle E$ and $\angle D$ is $\overline{DE}$. So we need $\overline{AB} \cong \overline{DE}$.

Step4: Analyze Question 4

ASA needs two angles and the included side congruent.

• Option A: Matches ASA (two angles, included side congruent).
• Option D: Matches ASA (two angles, included side congruent).
• Option F: Matches ASA (two angles, included side congruent).

Step5: Analyze Question 5

$\triangle ABC \cong \triangle FED$ by SAS. Corresponding sides $\overline{BC} \cong \overline{ED}$, so set up equation:
$13x - 12 = 7x - 6$
Solve for $x$:
$13x - 7x = -6 + 12$
$6x = 6$
$x = 1$

Answer:

1. C. AAS Triangle Congruence Theorem
2. B. SSA Triangle Congruence Theorem
3. B. $\overline{AB} \cong \overline{DE}$
4. A, D, F
5. $x=1$