Question

lesson 12 practice problems

1. a. show that the two pentagons are congruent.

b. find the side lengths of abcde and the angle measures of fghij.

Explanation:

Step1: Recall congruence criteria

Two polygons are congruent if their corresponding sides and angles are equal.

Step2: Match corresponding parts

By observing the figure, we can match the sides and angles of pentagon \(ABCDE\) and pentagon \(FGHIJ\). We see that \(\angle A = 162^{\circ}\), \(\angle B=108^{\circ}\), \(\angle C = 117^{\circ}\), \(\angle D=94^{\circ}\), \(\angle E = 59^{\circ}\) and the side - lengths of \(FGHIJ\) are given as \(FJ = 2.8\), \(JI=2.2\), \(IH = 1.4\), \(HG=3.2\), \(GF = 4.1\). Since the corresponding sides and angles match (assuming proper correspondence which can be visually verified from the grid - placement and labeling), the two pentagons are congruent.

Step3: Find side - lengths of \(ABCDE\)

Since the pentagons are congruent, \(AB = FJ=2.8\), \(BC = JI = 2.2\), \(CD=IH = 1.4\), \(DE=HG = 3.2\), \(EA=GF = 4.1\).

Step4: Find angle - measures of \(FGHIJ\)

Since the pentagons are congruent, \(\angle F=\angle A = 162^{\circ}\), \(\angle G=\angle B = 108^{\circ}\), \(\angle H=\angle C = 117^{\circ}\), \(\angle I=\angle D=94^{\circ}\), \(\angle J=\angle E = 59^{\circ}\)

Answer:

a. The two pentagons are congruent because their corresponding sides and angles are equal (by visual inspection and matching of parts).
b. Side - lengths of \(ABCDE\): \(AB = 2.8\), \(BC = 2.2\), \(CD=1.4\), \(DE=3.2\), \(EA = 4.1\). Angle - measures of \(FGHIJ\): \(\angle F = 162^{\circ}\), \(\angle G=108^{\circ}\), \(\angle H = 117^{\circ}\), \(\angle I=94^{\circ}\), \(\angle J = 59^{\circ}\)