Question

4 fill in the blank 3 points
complete the chart
hl theorem (hypotenuse - leg) answer postulate
ll theorem (leg - leg sss - (side - side - side) × postulate and answer postulate
ha theorem (hypotenuse - angle) aas - (angle - angle - side) × theorem
la theorem (leg - angle) answer theorem and answer postulate
sas (side - angle - side) asa (angle - side - angle) sss - (side - side - side) aas - (angle - angle - side)

Explanation:

HL theorem (hypotenuse - leg)

The HL (Hypotenuse - Leg) theorem is a special case of the SAS (Side - Angle - Side) postulate. In a right - triangle, if the hypotenuse and one leg of a right - triangle are equal to the hypotenuse and one leg of another right - triangle, we can consider the right - angle (included angle) between the leg and the hypotenuse. So, it is a special case of SAS.

LL theorem (leg - leg)

The LL (Leg - Leg) theorem for right - triangles. Since we have two legs (sides) equal. In non - right - triangles, SSS (Side - Side - Side) is a postulate for triangle congruence. For right - triangles, when we know two legs (right - angle is \(90^{\circ}\)), the right - angle is the included angle between the two legs. So, LL is a special case of SSS (because we can use the Pythagorean theorem to find the hypotenuse, but essentially for the two legs and the right - angle (which is a fixed angle of \(90^{\circ}\)), it also relates to SAS as the right - angle is the included angle between the two legs)

HA theorem (hypotenuse - angle)

The HA (Hypotenuse - Angle) theorem. In non - right - triangles, AAS (Angle - Angle - Side) is a theorem. For right - triangles, when we have a hypotenuse and an acute angle (non - right angle), since the right - angle is \(90^{\circ}\), we can use the fact that the sum of angles in a triangle is \(180^{\circ}\). But if we consider the right - angle, HA is also related to ASA. If we know the hypotenuse (a side), an acute angle, and the right - angle (\(90^{\circ}\)), and the side (hypotenuse) is between the right - angle and the acute angle.

LA theorem (leg - angle)

The LA (Leg - Angle) theorem. If we consider a leg (a side) and an angle (acute or non - right angle). If the angle is adjacent to the leg, and we know the right - angle (\(90^{\circ}\)), then it is a special case of AAS (because we have two angles: the given angle and the right - angle, and a non - included side (the leg)). Also, if the angle is opposite the leg, using the fact that the sum of angles in a triangle is \(180^{\circ}\) and the right - angle, it can also be related to ASA (if we consider the side (leg) and the two angles: the given angle and the right - angle, and the side is between the two angles in a sense (after using angle - sum property)).

Answer:

Theorem	Special Case of
LL theorem (leg - leg)	SSS (side - side - side) and SAS (side - angle - side)
HA theorem (hypotenuse - angle)	ASA (angle - side - angle)
LA theorem (leg - angle)	AAS (angle - angle - side) and ASA (angle - side - angle)