Question

question 9 of 10
which of the following are right triangle congruence theorems?
check all that apply.
a. leg - angle (la)
b. leg - leg (ll)
c. hypotenuse - leg (hl)
d. hypotenuse - angle (ha)

Explanation:

• Leg - angle (LA): If a leg and an acute angle of one right - triangle are congruent to the corresponding leg and acute angle of another right - triangle, then the two right - triangles are congruent. This is based on the Angle - Side - Angle (ASA) or Angle - Angle - Side (AAS) congruence criteria (since in a right - triangle, if we know one acute angle and a non - hypotenuse side (leg), we can use the fact that the right angles are equal (\(90^{\circ}\)) to apply ASA or AAS).
• Leg - leg (LL): If the two legs of one right - triangle are congruent to the two legs of another right - triangle, then the two right - triangles are congruent. This is based on the Side - Side - Side (SSS) congruence criterion. In right - triangles, if \(a = a'\) and \(b = b'\) (where \(a,b\) are legs of the first right - triangle and \(a',b'\) are legs of the second right - triangle), and by the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\) and \(c'=\sqrt{a'^{2}+b'^{2}}\), so \(c = c'\) (since \(a = a'\) and \(b = b'\)), and then \(SSS\) ( \(a=a'\), \(b = b'\), \(c=c'\)) holds.
• Hypotenuse - leg (HL): If the hypotenuse and a leg of one right - triangle are congruent to the hypotenuse and a leg of another right - triangle, then the two right - triangles are congruent. This is a special - case congruence criterion for right - triangles. Let the hypotenuses be \(c\) and \(c'\) and the legs be \(a\) and \(a'\). By the Pythagorean theorem \(b=\sqrt{c^{2}-a^{2}}\) and \(b'=\sqrt{c'^{2}-a'^{2}}\). Since \(c = c'\) and \(a=a'\), then \(b = b'\), and then we can use \(SSS\) (\(a=a'\), \(b = b'\), \(c=c'\)) to show congruence.
• Hypotenuse - angle (HA): There is no such standard right - triangle congruence theorem named Hypotenuse - angle. The valid congruence theorems for right - triangles are LA, LL, and HL.

Answer:

A. Leg - angle (LA), B. Leg - leg (LL), C. Hypotenuse - leg (HL)