Question

question 4 of 10
in right triangle (\triangle efg), (mangle e = 25^{circ}). in right triangle (\triangle hjk), (mangle h = 25^{circ}).
which similarity postulate or theorem proves that (\triangle efg) and (\triangle hjk) are similar?
a. hl
b. sss
c. sas
d. aa

Explanation:

• AA (Angle - Angle) Similarity Theorem:
• In a right - triangle, one of the angles is \(90^{\circ}\).
• For \(\triangle EFG\), \(\angle E = 25^{\circ}\) and \(\angle F=90^{\circ}\). Using the angle - sum property of a triangle (\(\angle E+\angle F+\angle G = 180^{\circ}\)), we can find the third - angle. But for similarity, we don't need to calculate the third angle.
• For \(\triangle HJK\), \(\angle H = 25^{\circ}\) and \(\angle J = 90^{\circ}\).
• Since two angles of \(\triangle EFG\) (\(25^{\circ}\) and \(90^{\circ}\)) are equal to two angles of \(\triangle HJK\) (\(25^{\circ}\) and \(90^{\circ}\)), by the AA (Angle - Angle) similarity theorem, if two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.
• HL (Hypotenuse - Leg): This is a congruence criterion for right - triangles, not a similarity criterion. It requires information about the hypotenuse and a leg of the right - triangles.
• SSS (Side - Side - Side): This similarity criterion requires the ratio of the lengths of all three corresponding sides of the two triangles to be equal. No side - length information is given in the problem.
• SAS (Side - Angle - Side): This similarity criterion requires the ratio of the lengths of two corresponding sides to be equal and the included angle to be congruent. No side - length information is given in the problem.

Answer:

D. AA