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Write an informal proof for the following conjecture. use complete sentences and complete thoughts to give a reasoned explanation for how to go from the given information (including the diagram) to the prove statement using geometric reasoning. hint: you should start by restating the given, then using your math knowledge with each sentence building from the previous thought, until you can write the prove statement at the end. given: line segment qs bisects angle pqr and the measure of angle pqs is equal to 45 degrees. prove: triangle pqr is a right triangle. given: bisects. . prove: triangle pqr is a right triangle.

Answer :

Given that angle PQS is 45 degrees and QS bisects angle PQR, we can apply the angle bisector theorem to deduce that angle PQR is 90 degrees. Thus, triangle PQR is a right triangle.

Given: Line segment QS bisects angle PQR, and the measure of angle PQS is equal to 45 degrees.

To prove: Triangle PQR is a right triangle.

We start by noting that angle PQS is given as 45 degrees. This means that angle PQR, which is bisected by QS, must have twice the measure of angle PQS due to the angle bisector theorem. Therefore, angle PQR measures 2 * 45 degrees, which is 90 degrees.

Now, we have established that angle PQR is 90 degrees. To prove that triangle PQR is a right triangle, we need to show that the sum of its angles equals 180 degrees, as that is a property of triangles in Euclidean geometry.

We know that angles PQS and QSR form a straight line with angle PQR because QS bisects angle PQR. A straight line measures 180 degrees. So, angle PQS (45 degrees) + angle PQR (90 degrees) + angle QSR must also equal 180 degrees.

Therefore, triangle PQR is indeed a right triangle because it has a 90-degree angle, which fulfills the definition of a right triangle.

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