Question

4-6 congruence in overlapping triangles
use the diagram shown for items 1 and 2.

1. which is a needed step to prove that $\triangle abf \cong \triangle edg$?

\\(\boldsymbol{\text{a}}\\) $\angle bcd \cong \angle bcd$ \\(\boldsymbol{\text{c}}\\) $\angle bcg \cong \angle dcf$
\\(\boldsymbol{\text{b}}\\) $\overline{gf} \cong \overline{gf}$ \\(\boldsymbol{\text{d}}\\) $\triangle cfg$ is isosceles.

1. if a proof shows $\overline{af} \cong \overline{eg}$, is it possible to show that $\triangle abf \cong \triangle edg$? explain.

yes no are are not asa aas sas hl
$\square$; the triangles $\square$ congruent by $\square$.

1. how do you justify $\overline{gf} \cong \overline{gf}$ as a step in a proof? explain.

\\(\boldsymbol{\text{a}}\\) since $\overline{gf}$ is the same length in both triangles, the sas triangle congruence theorem can be used to justify $\overline{gf} \cong \overline{gf}$.
\\(\boldsymbol{\text{b}}\\) since $\overline{gf}$ is a segment shared by both triangles, the reflexive property of congruence can be used to justify $\overline{gf} \cong \overline{gf}$.
\\(\boldsymbol{\text{c}}\\) since $\overline{gf}$ is on a leg of both triangles, the hl triangle congruence theorem can be used to justify $\overline{gf} \cong \overline{gf}$.
\\(\boldsymbol{\text{d}}\\) since $\overline{gf}$ is in both triangles, $\triangle cfg$ is an isosceles triangle and the sss triangle congruence theorem can be used to justify $\overline{gf} \cong \overline{gf}$.
use the figure shown for items 4 and 5.

1. how can you justify that $\angle abe \cong \angle cde$? explain.

$\overline{ad}$ $\overline{bc}$ $\overline{cd}$ vertical angles
alternate interior angles alternate exterior angles
$\overline{ab}$ and $\square$ are parallel, so $\angle abe$ and $\angle cde$ form congruent $\square$.

1. select all the lines in the proof of $\triangle abc \cong \triangle cda$ that have the correct justification.

$\square$ a. $\overline{ab} \parallel \overline{cd}$ and $\overline{bc} \parallel \overline{ad}$, asa triangle congruence theorem
$\square$ b. $\overline{ac} \cong \overline{ca}$, figures are congruent to themselves
$\square$ c. $\angle acb \cong \angle cad$, alternate interior angles of parallel lines
$\square$ d. $\angle bac \cong \angle dca$, alternate interior angles of parallel lines
$\square$ e. $\triangle abc \cong \triangle cda$, asa triangle congruence theorem

Explanation:

Step1: Analyze △ABF ≅ △EDG requirements

From the diagram: $\angle A = \angle E = 90^\circ$, $\overline{AB} \cong \overline{ED}$. To prove congruence, we need either another pair of sides or angles. Option C: $\angle BCG \cong \angle DCF$ implies $\angle ABF \cong \angle EDG$ (vertical angles' supplements), which gives AAS. Other options are irrelevant: A is reflexive but not useful, B is reflexive for a non-included segment, D does not help with the target triangles.

Step2: Solve Item 2 with $\overline{AF} \cong \overline{EG}$

We know $\angle A = \angle E = 90^\circ$, $\overline{AB} \cong \overline{ED}$, $\overline{AF} \cong \overline{EG}$. This fits HL (right triangles with hypotenuse and leg congruent) or SAS.

Step3: Justify $\overline{GF} \cong \overline{GF}$

A segment congruent to itself uses the Reflexive Property of Congruence, which matches option B.

Step4: Solve Item 4 for $\angle ABE \cong \angle CDE$

In the parallelogram, $\overline{AB} \parallel \overline{CD}$. $\angle ABE$ and $\angle CDE$ are alternate interior angles formed by transversal $\overline{BD}$, so they are congruent.

Step5: Evaluate Item 5 proof lines

• A: Incorrect, ASA is not the justification for parallel sides.
• B: Correct, $\overline{AC} \cong \overline{CA}$ is reflexive.
• C: Incorrect, $\angle ACB$ and $\angle CAD$ are not alternate interior angles for the given parallel sides.
• D: Correct, $\angle BAC$ and $\angle DCA$ are alternate interior angles from $\overline{AB} \parallel \overline{CD}$.
• E: Correct, after establishing two angles and included side, ASA proves congruence.

Answer:

1. C. $\angle BCG \cong \angle DCF$
2. Yes; the triangles are congruent by HL (or SAS).
3. B. Since $\overline{GF}$ is a segment shared by both triangles, the Reflexive Property of Congruence can be used to justify $\overline{GF} \cong \overline{GF}$.
4. $\overline{AB}$ and $\overline{CD}$ are parallel, so $\angle ABE$ and $\angle CDE$ form congruent alternate interior angles.
5. B. $\overline{AC} \cong \overline{CA}$, figures are congruent to themselves

D. $\angle BAC \cong \angle DCA$, alternate interior angles of parallel lines
E. $\triangle ABC \cong \triangle CDA$, ASA Triangle Congruence Theorem