Question

Q2) Given P is the Midpoint of QT and RS

prove:
Statement:
1. P is midpoint of QT
2. QP = TP
3. PR = PS
4. ???

5. triangle QPR = TPS

6.


Reason:
1. Given
2. Definition of midpoint
3. definition of midpoint
4. vertical angle congruence theorem
5. ???
6. ???

Answer 1

To prove that triangle QPR is congruent to triangle TPS, one must show that the sides and the included angle from one triangle are congruent to the corresponding parts of the other triangle, using the definition of midpoint and the vertical angle congruence theorem.

Step-by-step explanation:

The question asks to prove that triangle QPR is congruent to triangle TPS given that P is the midpoint of both QT and RS. The proof is constructed using the properties of a midpoint and congruences in triangles.

1. P is the midpoint of QT
2. QP = TP
3. PR = PS
4. ▵QPR ≈ ▵TPS

Since P is the midpoint of QT, by definition (statement 2), QP and TP are congruent. Similarly, since P is also the midpoint of RS, PR and PS are congruent (statement 3). By the vertical angle congruence theorem, angle QPR is congruent to angle TPS (statement 4). Seeing as we have two sides and the included angle congruent in both triangles, by the SAS Postulate (Side-Angle-Side), we can conclude that ▵QPR is congruent to ▵TPS (statement 5).

Congruence of triangles: Two triangles are said to be congruent if all three corresponding sides are equal and all the three corresponding angles are equal in measure. These triangles can be slides, rotated, flipped and turned to be looked identical. If repositioned, they coincide with each other. The symbol of congruence is’ ≅’.'A closed polygon made of three line segments forming three angles is known as a Triangle.

Two triangles are said to be congruent if their sides have the same length and angles have same measure. Thus, two triangles can be superimposed side to side and angle to angle.