PLEASE HELPProve: Ray QS bisects Angle PQR

1. (REASON) Given
The statement in this item which states that "the segment SP is perpendicular to QP" can already be found in the problem statement that's why we don't need to come up with another reason to support it.

2. (REASON) Given
Likewise, the statement here can also be found in the problem statement. We can notice that the given "SR ⊥ QR" just translates to the statement "the segment SR is perpendicular to QR" so we don't need to support the statement any further.

3. (REASON) Given
For this item, we can also find the statement among the given like the previous two. Notice that the statement in the given which goes like SP ≡ SR simply just means that the segment SP is congruent to the segment SR.

4. (STATEMENT) The measure of angles QPS and QRS is equal to 90 degrees
Here our next statement is about the two angles being equal to 90 degrees. In making these statements, bear in mind that our goal is to prove that ray QS bisects angle PQR upon the 10th statement, so we need to think of the direction immediately.

4. (REASON) Definition of perpendicular lines
In this fourth statement we made use of the fact that the first and second statement just allowed us to declare that there are two pairs of perpendicular lines in the figure. Thus, the 4th statement just expands on the definition of a perpendicular line which states that two perpendicular lines intersect to form a 90-degree angle.

5. (STATEMENT) Triangle QPS is a right triangle
By this statement, we decided that we need to prove that the two triangles are congruent so that we can prove that the two new angles which resulted from the "bisecting" of ray QS is indeed equal. Now, we proceed to declare that QPS is a right triangle.

5. (REASON) Definition of a right triangle
For the fifth statement, we made use of the basic definition of a right triangle to advance our proof. A right triangle is simply defined as a triangle having one 90-degree interior angle. We have already stated in the fourth statement that the measure of angle QPS is equal to 90 degrees.

6. (STATEMENT) Triangle QRS is a right triangle
We declare the same thing for the sixth statement but this time we do it for the triangle QRS. Like the triangle QPS, QRS can also be classified as one. This will be crucial since we need to be proving that the two triangles are congruent.

6. (REASON) Definition of a right triangle
Like our reasoning for the fifth statement, we will also back up the sixth statement with the definition of a right triangle. We can recall that we have also stated that the measure of angle QRS is also equal to 90 degrees.

7. (STATEMENT) The segment QS is equal to QS.
In here we just declare that line segment QS is equal to itself. While seemingly pointless, this statement would be aligned with one of the triangle congruence postulates that we need to prove.

7. (REASON) Reflexive Property
Although it may sound like a worthless property, the reflexive property actually exists and it basically describes what we have just done for the seventh statement. According to this property, a number is equal to itself.

8. (STATEMENT) The triangle QPS is congruent to triangle QRS.
In this statement we have finally declared that the two triangles are equal. With only two statements left after this one, we can now easily prove that the ray absolutely bisects the two angles mentioned in the problem statement.

8. (REASON) HL Postulate
For the eighth statement, we have used the HL Postulate or the Hypotenuse-Leg Postulate. This postulate is a special one that is used to prove congruencies of right triangles exclusively. Since we have already stated that the two triangles are right triangles, we can validly apply this postulate.

9. (STATEMENT) The angle PQS is congruent to angle RQS
Now that we have successfully stated the congruency of the two triangles, we can proceed to stating the congruence of their corresponding parts. Stating this would be close to proving the bisecting ray since a ray bisecting essentially divides the angle into two congruent parts.

9. (REASON) Definition of congruency
In the ninth statement we just used the definition of congruency as a support. As stated before, once we have already proved the congruency of the two triangles, then we can also state the congruency of their corresponding parts.

10. (STATEMENT) Ray QS bisects angle PQR
For the last statement, we have successfully proved that ray QS indeed bisects angle PQR. Through the succession of statements, we were able to trace the logic required to reach this point in just ten statements.

10. (REASON) Definition of an angle bisector
We have just used the definition of an angle bisector to come up with the tenth statement. Since we know that angles PQS and RQS form the larger angle PQR, and we have already proven the congruency of these two angles, we can deduce that the ray that cut through the larger angle bisected it.