Question

1. find the missing measures given that \\( \triangle a b c \cong \triangle d f e \\)

Explanation:

Step1: Corresponding sides and angles

Since \(\triangle ABC\cong\triangle DFE\), corresponding parts are equal.

• \(AB = DF = 8\)
• \(BC = FE = 3\)
• \(AC = DE = 5\)
• \(\angle B=\angle F = 60^{\circ}\)
• \(\angle E=\angle C = 84^{\circ}\)

Step2: Find the third angle

Using the angle - sum property of a triangle (\(\angle A+\angle B+\angle C = 180^{\circ}\)).
\(\angle A=180^{\circ}-\angle B - \angle C\)
Substitute \(\angle B = 60^{\circ}\) and \(\angle C = 84^{\circ}\)
\(\angle A=180^{\circ}-60^{\circ}-84^{\circ}=36^{\circ}\)
Since \(\angle A=\angle D\), \(\angle D = 36^{\circ}\)

Answer:

\(AC = 5\), \(BC = 3\), \(\angle A=\angle D = 36^{\circ}\), \(\angle C=\angle E = 84^{\circ}\), \(\angle B=\angle F = 60^{\circ}\)