Question

based only on the information given in the diagram, which congruence theorems or postulates could be given as reasons why $\triangle abc\cong \triangle lmn$?
check all that apply.
a. hl
b. la
c. asa
d. sas
e. aas

Explanation:

1. HL: HL applies to right triangles when the hypotenuse and one leg are congruent. Here, hypotenuse $AB \cong LM$ and leg $AC \cong LN$, so HL is valid.
2. LA: LA (Leg-Acute angle) applies to right triangles when one leg and an acute angle are congruent. We have leg $AC \cong LN$ and acute $\angle A \cong \angle L$, so LA is valid.
3. ASA: ASA requires two angles and the included side. We know $\angle A \cong \angle L$, $\angle C \cong \angle N$ (right angles), and included side $AC \cong LN$, so ASA is valid.
4. SAS: SAS requires two sides and the included angle. We have $AC \cong LN$, $\angle A \cong \angle L$, but the marked side on the hypotenuse is not the second side forming the included angle (the other side would be $BC$ and $MN$, which are not marked congruent), so SAS is invalid.
5. AAS: AAS requires two angles and a non-included side. We have $\angle A \cong \angle L$, $\angle C \cong \angle N$, and non-included side $AB \cong LM$, so AAS is valid.

Answer:

A. HL
B. LA
C. ASA
E. AAS