Question

find the values of m and n.
the value of m is 34°.
the value of n is □°.

Explanation:

Step1: Find the base angle of the isosceles triangle

Since the triangle is isosceles (marked with equal sides), and the exterior angle is \(124^{\circ}\), the adjacent interior angle is \(180 - 124=56^{\circ}\).

Step2: Use the right - triangle angle sum

In a right - triangle (\(90^{\circ}\) angle), the sum of angles is \(180^{\circ}\). We know one angle is \(90^{\circ}\) and another is \(56^{\circ}\) (from step 1). Let \(n\) be the unknown angle. Then \(n=90-(56 + 34)\) (but another way: using the property of the triangle with the exterior angle. The base angles of the isosceles part: the non - right angle adjacent to \(124^{\circ}\) is \(56^{\circ}\). In the right - triangle, we know \(m = 34^{\circ}\). Using the angle sum of \(180^{\circ}\) for the large right - triangle: \(90^{\circ}+(34^{\circ}+n)+(56^{\circ}-n)=180^{\circ}\). Also, using the property of the triangle where the exterior angle of the small triangle (the one with \(n\)): The base angle of the isosceles triangle (adjacent to \(124^{\circ}\)) is \(56^{\circ}\). In the right - triangle, \(n=90-(56 + 34)\) is wrong. Correctly, since the base angles of the isosceles triangle (the non - right - angled part) are equal. The angle adjacent to \(124^{\circ}\) is \(180 - 124=56^{\circ}\). In the right - triangle, we know \(m = 34^{\circ}\) (given or calculated as \(90 - 56=34\)). Now, consider the small triangle with \(n\). Since the two sides are equal (marked), the base angles are equal. The angle opposite to \(n\) (using the property of the straight line and the isosceles triangle) \(n=180-(124 + 34)\) is wrong. Another approach: The base angle of the isosceles triangle (adjacent to \(124^{\circ}\)) is \(180 - 124 = 56^{\circ}\). In the right - triangle, we know one non - right angle is \(m + 56^{\circ}\) (but wait, no. Wait, the correct way: Since the triangle with the \(124^{\circ}\) exterior angle is isosceles. The interior angle is \(180 - 124=56^{\circ}\). In the right - triangle (\(90^{\circ}\) angle), we know \(m = 34^{\circ}\) (because \(90-(56)=34\)). Now, for \(n\): Using the property of the triangle (the small one with \(n\)). Since the two sides are equal (marked), the base angles are equal. The angle adjacent to \(124^{\circ}\) is \(56^{\circ}\). In the right - triangle, we use the angle sum. Wait, another property: the exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. For the small triangle with \(n\): The exterior angle is \(56^{\circ}\) (the base angle of the isosceles part adjacent to \(124^{\circ}\)), and one non - adjacent interior angle is \(34^{\circ}\). So \(n=56 - 34\)

$$n=22^{\circ}$$

Answer:

\(22\)