Question

b. complete the proof demonstrating that the 2 triangles you created are congruent.
given: $overline{ab} \parallel \overline{dc}$ and $overline{bc} \parallel \overline{ad}$
prove: $\triangle abc \cong \triangle cda$

statement	reason
$\angle bac \cong \angle dca$
$overline{bc} \parallel \overline{ad}$	given
$\angle acb \cong \angle cad$
	reflexive property of congruence
$\triangle abc \cong \triangle cda$

1. triangle hmc and kwr are shown. determine whether the given information is sufficient to prove that the triangles are congruent. if so, state the appropriate congruence theorem or postulate to support that.

given information	congruent?	theorem or postulate used
$\overline{hc} \cong \overline{kr}$, $\overline{hm} \cong \overline{kw}$, $\angle mhc \cong \angle wkr$
$\overline{hc} \cong \overline{kr}$, $\overline{hm} \cong \overline{kw}$, $\angle hcm \cong \angle kwr$

Explanation:

First Problem (Proof Completion)

Step1: Identify alternate interior angles

$\overline{AB} \parallel \overline{DC}$ creates alternate interior angles $\angle BAC \cong \angle DCA$, so the reason is Alternate Interior Angles Theorem

Step2: Identify alternate interior angles

$\overline{BC} \parallel \overline{AD}$ creates alternate interior angles $\angle ACB \cong \angle CAD$, so the reason is Alternate Interior Angles Theorem

Step3: State reflexive congruent side

The shared side $\overline{AC} \cong \overline{CA}$ matches the reflexive property, so the statement is $\overline{AC} \cong \overline{CA}$

Step4: Apply ASA congruence

Two pairs of congruent angles and the included congruent side prove $\triangle ABC \cong \triangle CDA$, so the reason is ASA Congruence Postulate

1. For $\angle CMH \cong \angle RWK$, $\angle MHC \cong \angle WKR$, $\overline{HM} \cong \overline{KW}$: Two pairs of congruent angles and a non-included congruent side match the AAS theorem.
2. For $\overline{HC} \cong \overline{KR}$, $\overline{HM} \cong \overline{KW}$, $\angle MHC \cong \angle WKR$: Two pairs of congruent sides and the included congruent angle match the SAS postulate.
3. For $\overline{HC} \cong \overline{KR}$, $\overline{HM} \cong \overline{KW}$, $\angle HCM \cong \angle KWR$: Two congruent sides and a non-included congruent angle (SSA) do not prove triangle congruence.

Answer:

Statement	Reason
$\angle BAC \cong \angle DCA$	Alternate Interior Angles Theorem
$\overline{BC} \parallel \overline{AD}$	Given
$\angle ACB \cong \angle CAD$	Alternate Interior Angles Theorem
$\overline{AC} \cong \overline{CA}$	Reflexive property of congruence
$\triangle ABC \cong \triangle CDA$	ASA Congruence Postulate

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Second Problem (Triangle Congruence Check)