Question

Given:$\overline{\mathrm{LF}}\cong \overline{\mathrm{KF}}\mathbf{;}\mathbf{}\overline{\mathrm{LA}}\cong \overline{\mathrm{KA}}$

Prove:$\overline{\mathrm{LJ}}\cong \overline{\mathrm{KJ}}$

a. List the key steps of a proof.

b. Write a proof in two-column form.

Short answer

a.

$\mathrm{\Delta }\mathbf{LAF}\cong \mathrm{\Delta }\mathbf{KAF}$	SSS property
$\angle \mathbf{LAF}\cong \angle \mathbf{KAF}$	Corr. Parts of$\cong \mathrm{\Delta }$ are $\cong$.
$\mathrm{\Delta }\mathbf{LAJ}\cong \mathrm{\Delta }\mathbf{KAJ}$	SAS postulate
$\overline{\mathrm{LJ}}\cong \overline{\mathrm{KJ}}$	Corr. Parts of$\cong \mathrm{\Delta }$ are $\cong$.

b.

Statements	Reason
$\overline{\mathrm{LF}}\cong \overline{\mathrm{KF}}\mathbf{;}\mathbf{}\overline{\mathrm{LA}}\cong \overline{\mathrm{KA}}$	Given
$\overline{\mathrm{AF}}\cong \overline{\mathrm{AF}}$	Reflexive property
$\mathrm{\Delta }\mathbf{LAF}\cong \mathrm{\Delta }\mathbf{KAF}$	SSS property
$\angle \mathbf{LAF}\cong \angle \mathbf{KAF}$	Corr. Parts of$\cong \mathrm{\Delta }$ are $\cong$.
$\mathrm{\Delta }\mathbf{LAJ}\cong \mathrm{\Delta }\mathbf{KAJ}$	SAS postulate
$\overline{\mathrm{LJ}}\cong \overline{\mathrm{KJ}}$	Corr. Parts of$\cong \mathrm{\Delta }$ are $\cong$.

Step-by-step solution

Part a. Step 1. Description of step.

Consider $\mathrm{\Delta }LAF$and $\mathrm{\Delta }KAF$, $\overline{AF}\cong \overline{AF}$by reflexive property. Also, by given information, $\overline{LF}\cong \overline{KF}$ and $\overline{LA}\cong \overline{KA}$. Therefore, by SSS postulate, $\mathrm{\Delta }LAF\cong \mathrm{\Delta }KAF$.

Part a. Step 2. Description of step.

As $\mathrm{\Delta }LAF\cong \mathrm{\Delta }KAF$, by corresponding parts of congruent triangles, $\angle LAF\cong \angle KAF$.

Part a. Step 3. Description of step.

Consider$\mathrm{\Delta }LAJ$ and $\mathrm{\Delta }KAJ$, $\overline{LA}\cong \overline{KA}$,$\overline{AF}\cong \overline{AF}$ and $\angle LAF\cong \angle KAF$, then by SAS postulate, $\mathrm{\Delta }LAJ\cong \mathrm{\Delta }KAJ$.

Part a. Step 4. Description of step.

As$\mathrm{\Delta }LAJ\cong \mathrm{\Delta }KAJ$, by corresponding parts of congruent triangles, $\overline{LJ}\cong \overline{KJ}$.

Part a. Step 5. Key steps of a proof.

Make a list of key steps of a proof.

$\mathrm{\Delta }LAF\cong \mathrm{\Delta }KAF$	SSS property
$\angle LAF\cong \angle KAF$	Corr. Parts of$\cong \mathrm{\Delta }$are $\cong$.
$\mathrm{\Delta }LAJ\cong \mathrm{\Delta }KAJ$	SAS postulate
$\overline{LJ}\cong \overline{KJ}$	Corr. Parts of$\cong \mathrm{\Delta }$ are $\cong$.

Part b. Step 1. Description of step.

Consider $\mathrm{\Delta }LAF$ andwidth="115" height="23" role="math">ΔKAF,AF¯≅AF¯ by reflexive property. Also, by given information,$\overline{LF}\cong \overline{KF}$ and $\overline{LA}\cong \overline{KA}$ . Therefore, by SSS postulate, $\mathrm{\Delta }LAF\cong \mathrm{\Delta }KAF$.

Part b. Step 2. Description of step.

As $\mathrm{\Delta }LAF\cong \mathrm{\Delta }KAF$, by corresponding parts of congruent triangles, $\angle LAF\cong \angle KAF$.

Part b. Step 3. Description of step.

Consider $\mathrm{\Delta }LAJ$ and$\mathrm{\Delta }KAJ,\overline{LA}\cong \overline{KA},\overline{AF}\cong \overline{AF}$ and $\angle LAF\cong \angle KAF$, then by SAS postulate, $\mathrm{\Delta }LAJ\cong \mathrm{\Delta }KAJ$.

Part b. Step 4. Description of step.

As$\mathrm{\Delta }LAJ\cong \mathrm{\Delta }KAJ$, by corresponding parts of congruent triangles, $\overline{LJ}\cong \overline{KJ}$.

Part b. Step 5. Two-column proof.

Write a two-column proof of the above explanation.

Statements	Reason
$\overline{LF}\cong \overline{KF};\overline{LA}\cong \overline{KA}$	Given
$\overline{AF}\cong \overline{AF}$	Reflexive property
$\mathrm{\Delta }LAF\cong \mathrm{\Delta }KAF$	SSS property
$\angle LAF\cong \angle KAF$	Corr. Parts of$\cong \mathrm{\Delta }$are $\cong$.
$\mathrm{\Delta }LAJ\cong \mathrm{\Delta }KAJ$	SAS postulate
$\overline{LJ}\cong \overline{KJ}$	Corr. Parts of$\cong \mathrm{\Delta }$are $\cong$.