Question

in the diagram below, $overline{pq}congoverline{ut}$ and $overline{qr}congoverline{ts}$. note: figures are not drawn to scale. which statement must be true?

Explanation:

Step1: Recall the hinge - theorem

If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first triangle is larger than the included angle of the second triangle, then the third side of the first triangle is longer than the third side of the second triangle. Conversely, if two sides of one triangle are congruent to two sides of another triangle, and the third - side of the first triangle is longer than the third - side of the second triangle, then the included angle of the first triangle is larger than the included angle of the second triangle.
In \(\triangle PQR\) and \(\triangle UTS\), we know that \(PQ\cong UT\) and \(QR\cong TS\). Also, \(PR = 8\mathrm{cm}\) and \(US=7\mathrm{cm}\), so \(PR>US\).

Step2: Apply the hinge - theorem

By the hinge - theorem, since \(PQ = UT\), \(QR = TS\) and \(PR>US\), the included angle of the sides \(PQ\) and \(QR\) (which is \(\angle Q\)) is larger than the included angle of the sides \(UT\) and \(TS\) (which is \(\angle T\)). That is \(m\angle Q>m\angle T\).

Answer:

The statement \(m\angle Q > m\angle T\) must be true.