Question

angle - side - angle
quadrilateral abcd is a parallelogram. by definition, that means that ab is parallel to dc, and bc is parallel to ad.
an auxiliary line, ac was created to show how abcd can be decomposed into 2 triangles.

1. if we have two parallel lines cut by the transversal (ac), what angles are congruent?
2. is there a side that is shared with the two triangles? what is the name of the side?
3. do we have enough information to prove the triangles are congruent? using what theorem?

prove △abc≅△adc
statements reasons
angle - side - angle triangle congruence theorem: in two triangles, if two pairs of corresponding angles, and the pair of corresponding sides between the angles, are congruent, then the triangles must be congruent. (theorem)
∠a≅∠c, ae≅ec, ∠dea≅∠bec, so △dea≅△bec

Explanation:

Step1: Identify congruent angles

Since \(AB\parallel DC\) and \(AC\) is a transversal, \(\angle BAC\cong\angle DCA\) (alternate - interior angles) and since \(BC\parallel AD\) and \(AC\) is a transversal, \(\angle ACB\cong\angle CAD\) (alternate - interior angles).

Step2: Identify shared side

The side \(AC\) is shared by \(\triangle ABC\) and \(\triangle ADC\).

Step3: Determine congruence

We have two pairs of congruent angles (\(\angle BAC\cong\angle DCA\), \(\angle ACB\cong\angle CAD\)) and the included side \(AC\) is congruent to itself. By the Angle - Side - Angle (ASA) congruence theorem, \(\triangle ABC\cong\triangle ADC\).

For the proof table:

Statements	Reasons
\(\angle BAC\cong\angle DCA\)	Alternate - interior angles of parallel lines (\(AB\parallel DC\) and \(AC\) is transversal)
\(\angle ACB\cong\angle CAD\)	Alternate - interior angles of parallel lines (\(BC\parallel AD\) and \(AC\) is transversal)
\(AC = AC\)	Reflexive property of equality
\(\triangle ABC\cong\triangle ADC\)	Angle - Side - Angle (ASA) congruence theorem

Answer:

1. \(\angle BAC\) and \(\angle DCA\), \(\angle ACB\) and \(\angle CAD\) are congruent as they are alternate - interior angles formed by parallel lines and a transversal.
2. The shared side is \(AC\).
3. Yes, we have enough information to prove the triangles are congruent. We use the Angle - Side - Angle (ASA) congruence theorem.