Question

find the values of m and n. ans: m, n 27, 36 36, 27 27, 27 54, 27

Explanation:

Step1: Find the value of \(x\)

The angle \(x\) and \(126^{\circ}\) are supplementary. So \(x = 180^{\circ}- 126^{\circ}=54^{\circ}\).

Step2: Find the value of \(m\)

Since the two right - angled triangles are congruent (by Hypotenuse - Leg congruence as the hypotenuse and one leg are equal), and the sum of angles in a right - angled triangle is \(180^{\circ}\). In the right - angled triangle with angle \(x = 54^{\circ}\), \(m=\frac{90^{\circ}-54^{\circ}}{2}=18^{\circ}\) (This step is wrong, correct as follows:
Since the two right - angled triangles are congruent (by Hypotenuse - Leg congruence as the hypotenuse and one leg are equal). Let's use the property of angle - sum in a triangle.
We know that in a triangle, the sum of angles is \(180^{\circ}\). For the non - right angle in the larger isosceles - like part:
The angle adjacent to \(126^{\circ}\) is \(x = 180 - 126=54^{\circ}\).
In the right - angled triangle (where one angle is \(90^{\circ}\) and one is \(x = 54^{\circ}\)), and since the two right - angled triangles are congruent.
We use the property that \(m=\frac{90 - (180 - 126)}{2}\).
Another way:
We know that the exterior angle property. But more simply, since the two right - angled triangles are congruent (by Hypotenuse - Leg, assume the equal sides are marked as equal in the figure).
The angle adjacent to \(126^{\circ}\) is \(x=180 - 126 = 54^{\circ}\).
In a right - angled triangle (sum of angles \(=180^{\circ}\)), if one angle is \(90^{\circ}\) and another is \(x = 54^{\circ}\) (for the combined non - right part of the two right - angled triangles). Then for each of the congruent right - angled triangles, using the angle - sum property of a triangle (\(90 + m+( \frac{180 - 126}{2})=180\)).
\(90+m + 27=180\), so \(m = 27^{\circ}\)

Step3: Find the value of \(n\)

We know that \(n=\frac{180 - 126}{2}=27^{\circ}\) (using the property of isosceles triangle formed by the two congruent right - angled triangles, the base angles of the non - right part of the combined figure (the part with angle \(126^{\circ}\)) are equal. So \(n = 27^{\circ}\))

Answer:

\(27,27\)