Question

prove (in writing) the side - angle - side triangle congruence theorem.
prove (in writing) the isosceles triangle theorem.
learning targets:
i can explain why the side - angle - side triangle congruence theorem works
i can use the side - angle - side triangle congruence theorem in a proof.
supplies: straitedge tool, pencil, tracing paper
warm up: information overload?
highlight each piece of given information that is used in the proof, and each line in the proof where that piece of information is used.
given:

• $overline{ab}congoverline{de}$ - $angle acongangle d$
• $overline{ac}congoverline{df}$ - $angle bcongangle e$
• $overline{bc}congoverline{ef}$ - $angle ccongangle f$

Explanation:

Step1: Recall SAS congruence theorem

The SAS (Side - Angle - Side) congruence theorem states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Step2: Identify relevant given information

We are given $\overline{AB}\cong\overline{DE}$, $\angle A\cong\angle D$, and $\overline{AC}\cong\overline{DF}$. Here, $\angle A$ is the included angle between $\overline{AB}$ and $\overline{AC}$ in $\triangle ABC$, and $\angle D$ is the included angle between $\overline{DE}$ and $\overline{DF}$ in $\triangle DEF$.

Step3: Apply the SAS theorem

By the Side - Angle - Side (SAS) congruence theorem, since two sides ($\overline{AB}\cong\overline{DE}$ and $\overline{AC}\cong\overline{DF}$) and the included angle ($\angle A\cong\angle D$) of $\triangle ABC$ are congruent to two sides and the included angle of $\triangle DEF$, we can conclude that $\triangle ABC\cong\triangle DEF$.

Answer:

$\triangle ABC\cong\triangle DEF$ by the Side - Angle - Side (SAS) congruence theorem.