Question

given that δabc ≅ δdef, m<e = x = y = z =

Explanation:

Step1: Use angle - sum property of a triangle

In \(\triangle ABC\), we know that the sum of interior angles of a triangle is \(180^{\circ}\). But we first use the congruence of \(\triangle ABC\) and \(\triangle DEF\). Since \(\triangle ABC\cong\triangle DEF\), corresponding angles are equal.

Step2: Find \(m\angle E\)

In \(\triangle DEF\), we know two angles: \(m\angle D = 60^{\circ}\) and \(m\angle F=70^{\circ}\). Using the angle - sum property of a triangle (\(m\angle D + m\angle E+m\angle F = 180^{\circ}\)), we substitute the known values: \(60^{\circ}+m\angle E + 70^{\circ}=180^{\circ}\). Then \(m\angle E=180^{\circ}-(60^{\circ} + 70^{\circ})=50^{\circ}\).

Step3: Solve for \(x\)

Since \(m\angle E=x\), then \(x = 50\).

Step4: Solve for \(y\)

Corresponding sides are equal. Let's assume \(AB = DE\) and \(AC = DF\). If we consider the side - length relationships, we need more information about which sides are corresponding. But if we assume \(BC\) and \(EF\) are corresponding. Since \(\triangle ABC\cong\triangle DEF\), we can set up equations based on side - length equalities. Let's assume \(AC = DF\), so \(20=4y - 4\). Add 4 to both sides: \(20 + 4=4y\), i.e., \(24 = 4y\). Divide both sides by 4, we get \(y = 6\).

Step5: Solve for \(z\)

Assume \(AB=DE\), so \(22=3z + 1\). Subtract 1 from both sides: \(22-1=3z\), i.e., \(21 = 3z\). Divide both sides by 3, we get \(z = 7\).

Answer:

\(x = 50\), \(y = 6\), \(z = 7\)