Question

multiple answer 5 points
which postulates and theorems may be used to determine if the triangles are congruent? check all that apply.
triangle diagram with labels a, b, c, d, markings for congruent sides and angles, right angle at c
options: not congruent, aas, asa, sss, hl, sas

Explanation:

Step1: Analyze triangle markings

From the diagram, \( AB = AD \) (marked with single ticks), \( BC = CD \) (marked with double ticks), and \( \angle ACB=\angle ACD = 90^\circ \) (right angle). Also, \( \angle B=\angle D \) (marked angles).

Step2: Check congruence postulates

• AAS: Two angles and a non - included side. We have \( \angle B=\angle D \), \( \angle ACB=\angle ACD \), and \( AC \) is common (non - included side), so AAS applies.
• ASA: Two angles and the included side. \( \angle B=\angle D \), \( BC = CD \), \( \angle ACB=\angle ACD \), so ASA applies.
• SSS: Three sides. \( AB = AD \), \( BC = CD \), \( AC = AC \) (common), so SSS applies.
• HL: Hypotenuse - leg for right triangles. \( AB = AD \) (hypotenuse), \( AC = AC \) (leg), so HL applies.
• SAS: Two sides and included angle. \( BC = CD \), \( \angle ACB=\angle ACD \), \( AC = AC \), so SAS applies.
• "not congruent" is incorrect as we can prove congruence.

Answer:

AAS, ASA, SSS, HL, SAS (the checkboxes for these options should be selected)