Question

directions: compare the sides and angles by filling in the blank with a < or > symbol.

1. qr____rs
2. ab____cd
3. m∠fjg____m∠hjg
4. m∠qsp____m∠qsr

Explanation:

21.

Step1: Apply the Hinge Theorem

The Hinge Theorem states that if two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is smaller than the included angle of the second triangle, then the third side of the first triangle is shorter than the third side of the second triangle. In \(\triangle QRT\) and \(\triangle SRT\), \(QT = ST\) (marked as equal), \(RT=RT\) (common side), and \(\angle QTR = 121^{\circ}\), \(\angle STR=123^{\circ}\). Since \(121^{\circ}<123^{\circ}\), by the Hinge Theorem, \(QR

22.

Step1: Use the Law of Sines in \(\triangle ABD\) and \(\triangle BCD\)

In \(\triangle ABD\), \(AD = BC\) (marked as equal), \(BD = BD\) (common side). In \(\triangle ABD\), \(\angle ADB = 68^{\circ}\), in \(\triangle BCD\), \(\angle BDC\) (let's find \(\angle BDC\) in \(\triangle BCD\). Since \(BC = AD\) (marked equal), \(BD = BD\). Using the Law of Sines in \(\triangle ABD\): \(\frac{AB}{\sin\angle ADB}=\frac{BD}{\sin\angle A}\), in \(\triangle BCD\): \(\frac{CD}{\sin\angle CBD}=\frac{BD}{\sin\angle C}\). Also, we can use the Hinge - like concept (comparing angles). The angle opposite \(AB\) (in \(\triangle ABD\)): \(\angle ADB = 68^{\circ}\), the angle opposite \(CD\) (in \(\triangle BCD\)): \(\angle CBD=65^{\circ}\). Since \(68^{\circ}>65^{\circ}\) and \(AD = BC\), \(BD = BD\), then \(AB > CD\).

23.

Step1: Apply the converse of the Hinge Theorem

In \(\triangle FJG\) and \(\triangle HJG\), \(FJ = HJ\) (marked as equal), \(JG = JG\) (common side). Since \(FG = 12\) and \(HG = 13\), and \(12<13\). By the converse of the Hinge Theorem (if two sides of one triangle are congruent to two sides of another triangle, and the third side of the first triangle is shorter than the third side of the second triangle, then the included angle of the first triangle is smaller than the included angle of the second triangle), \(m\angle FJG

24.

Step1: Use the Hinge Theorem

In \(\triangle QSP\) and \(\triangle QSR\), \(QP = 6\), \(QR = 5\) (\(QP>QR\)), \(QS = QS\) (common side), and the sides \(PS = RS\) (marked as equal). By the Hinge Theorem (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), \(m\angle QSP>m\angle QSR\).

Answer:

1. \(QR
2. \(AB > CD\)
3. \(m\angle FJG
4. \(m\angle QSP>m\angle QSR\)