Question

what else must you know to prove the triangles congruent by asa?

options:
∠adc ≅ ∠cab
∠adc ≅ ∠cba
∠dca ≅ ∠bca

12 fill in the blank 4 points
state how the triangles are congruent using sss, sas, asa, aas, or hl. if they are not congruent, type not.

Explanation:

First Sub - Question (Multiple Choice)

Step 1: Recall ASA Congruence Criterion

ASA (Angle - Side - Angle) congruence criterion states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. In the given figure, we have a common side \(AC\) for triangles \(ADC\) and \(BCA\) (or relevant triangles). We already have one pair of angles marked as congruent. For ASA, we need another pair of angles such that the side is included between the two angles.

Step 2: Analyze the Options

• Option 1: \(\angle ADC\cong\angle CAB\) - These angles are not related to the included side in a way that would satisfy ASA.
• Option 2: \(\angle ADC\cong\angle CBA\) - These angles do not help in forming the ASA condition with the existing elements.
• Option 3: \(\angle DCA\cong\angle BCA\) - If \(\angle DCA\cong\angle BCA\), and we have the common side \(AC\) and the other marked angle, this would satisfy the ASA criterion as the side \(AC\) is included between \(\angle DCA\) (or the other angle) and the angle related to it.

Step 1: Identify Corresponding Parts

In the figure, we can see that \(FT = TS\) (marked with 1), \(LT=TQ\) (marked with 2), and \(\angle LTF\) and \(\angle QTS\) are vertical angles, so \(\angle LTF\cong\angle QTS\).

Step 2: Apply Congruence Criterion

We have two sides and the included angle. The SAS (Side - Angle - Side) congruence criterion 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 triangles are congruent. Here, in triangles \(LFT\) and \(QST\), \(FT = TS\), \(LT=TQ\), and \(\angle LTF=\angle QTS\) (vertical angles), so by SAS, the triangles are congruent.

Answer:

\(\boldsymbol{\angle DCA\cong\angle BCA}\) (the option with \(\angle DCA\cong\angle BCA\))

Second Sub - Question (Fill in the Blank)