Final answer:
To prove two triangles congruent by the Side-Angle-Side Postulate, Brooke must show the included angles are congruent. This could be done if the angles are corresponding angles when parallel lines are cut by a transversal or if they have some angle relationships like vertical angles.
Step-by-step explanation:
The student's question revolves around proving that two triangles, APQT and ARQS, are congruent using the Side-Angle-Side Postulate. The Side-Angle-Side Postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. To justify that the included angles are congruent in Brooke’s proof, one would need a statement that confirms the equality of the two angles. Possible justifications could include angles being opposite to equal sides in an isosceles triangle, angles being alternate interior angles created by a transversal crossing parallel lines, or angles being corresponding angles where two lines are cut by a transversal.
For instance, if it was known that line segment PQ is parallel to line segment RS, then angle PQS and angle QRT would be corresponding angles and therefore congruent. A similar argument could be made if angle APQ is known to be congruent to angle AQR through some property such as vertical angles or supplementary angles that add up to 180 degrees. In geometry and trigonometry, the Pythagorean theorem is often used to relate the sides of right triangles, but it’s not directly applicable for the justification of included angles unless associated with right angles. In Brooke's case, if the proof includes a right triangle and the angle in question is the right angle, this could be a part of her justification.