Question

Angle N = 40 degrees, side NP = 8, and side QS = 8. What additional information would you need to prove that triangle NOP is congruent to triangle QRS by ASA

Answer 1

Answer: You would need to know that angle P is congruent to angle S.



For the Angle-side-angle congruence theorem (ASA) to be applicable, three values are needed, one of which is a side. The other two values should be angles. Hence the name ASA has two letter A’s for “angle” and one letter S for “side”.

We already know that side NP is congruent to side QS as they both equal 8. It is also given that angle N is 40°. Angle N of triangle NOP should correspond to angle Q in triangle QRS.

The additional information needed to prove triangle NOP is congruent to triangle QRS by the ASA theorem is a second angle. We want two angles and the included side. (*Note. What “included side” means is that the endpoints are measured angles. For example, endpoints of NP would be angle N and angle P. Similarly, endpoints of QS would be angle Q and angle S.)

The side we are given (NP/QS) must be BETWEEN two congruent angles to prove by the angle-side-angle congruence theorem.
So, you would need to know that angle P is congruent to angle S.

Here’s a pic to help show this:

Answer 2

The angle 2) P is congruent to angle S would be necessary. Therefore, 2) Angle P is congruent to angle S is correct.

Since side NP and side QS both equal 8, we already know that side NP is congruent to side QS.

It is also known that angle N is 40°, so angle N of triangle NOP should correspond to angle Q in triangle QRS.

These three values are necessary for the Angle-side-angle congruence theorem (ASA) to be applicable. One of the values should be a side.

The other two values should be angles.

The additional information required to demonstrate that triangle NOP is congruent to triangle QRS by the ASA theorem is a second angle.

We are interested in two angles as well as the included side.

(*Note: The term "included side" refers to the fact that the endpoints are measured angles; for instance, endpoints of NP would be angle N and angle P, and endpoints of QS would be angle Q and angle S.)

The side we are given (NP/QS) must be BETWEEN two congruent angles in order to demonstrate the angle-side-angle congruence theorem.

The angle 2) P is congruent to angle S would be necessary.

Question

Angle N=40 degrees, side NP=8 , angle Q=40 degrees, and side QS=8 What additional information would you need to prove that △ NOP=△ QRS by ASA .

1)Angle O is congruent to angle R .

2)Angle P is congruent to angle S .

3)Side NO is congruent to side QR.

4)Side OP is congruent to side RS.