Question

question 4 of 10
based only on the information given in the diagram, which congruence theorems or postulates could be given as reasons why \\( \triangle a b c \cong \triangle l m n \\)?
check all that apply.
\\( \square \\) a. asa
\\( \square \\) b. ll
\\( \square \\) c. aas
\\( \square \\) d. sas
\\( \square \\) e. hl
\\( \square \\) f. la

Explanation:

Step1: Analyze the given triangles

We have two right - angled triangles \(\triangle ABC\) and \(\triangle LMN\). Let's recall the congruence theorems for right - angled and non - right - angled triangles.
For right - angled triangles, the \(LL\) (Leg - Leg) theorem states that if the two legs of one right - triangle are congruent to the two legs of another right - triangle, then the triangles are congruent.
For non - right - angled triangles, \(AAS\) (Angle - Angle - Side) states that if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the triangles are congruent.

Step2: Check for \(LL\)

In right - angled triangles \(\triangle ABC\) and \(\triangle LMN\), the legs \(AC\cong LN\) (marked as equal) and \(BC\cong MN\) (marked as equal). So, by the \(LL\) (Leg - Leg) theorem for right - angled triangles, \(\triangle ABC\cong\triangle LMN\).

Step3: Check for \(AAS\)

We know that \(\angle C=\angle N = 90^{\circ}\). Also, from the side - angle relationships (since we have two sides and the right - angle, we can consider the non - included side situation). If we assume the right - angle as one of the angles, and we have two pairs of sides (which can be related to angles in the context of triangle congruence). In fact, for right - angled triangles, if we consider the right - angle (\(90^{\circ}\)), and we can show that another pair of angles (by the fact that in right - angled triangles \(\sin A=\frac{BC}{AB}\) and \(\sin L=\frac{MN}{LM}\), and since \(BC = MN\) and \(AB=LM\) (from the side - marking equivalence in the congruence sense)), and a non - included side (the hypotenuse or a leg), we can use \(AAS\).

Answer:

B. LL, C. AAS