Given:
[tex]\Delta MAT\cong \Delta MHT[/tex] and [tex]m\angle 2=82^\circ[/tex].
To find:
Whether [tex]m\angle 1=47^\circ[/tex] is possible or not.
Solution:
We have,
[tex]\Delta MAT\cong \Delta MHT[/tex]
[tex]\angle AMT\cong \angle HMT[/tex] (CPCTC)
[tex]m\angle 3=m\angle 8[/tex] ...(i)
And,
[tex]m\angle 1=m\angle 8[/tex] ...(ii) (Vertically opposite angles)
From (i) and (ii), we get
[tex]m\angle 1=m\angle 3[/tex] ...(iii)
Now,
[tex]m\angle 1+m\angle 2+m\angle 3=180^\circ[/tex] (Linear pair)
[tex]m\angle 1+82^\circ+m\angle 1=180^\circ[/tex] (Using (iii))
[tex]2m\angle 1=180^\circ-82^\circ[/tex]
[tex]2m\angle 1=98^\circ[/tex]
Divide both sides by 2.
[tex]m\angle 1=\dfrac{98^\circ}{2}[/tex]
[tex]m\angle 1=49^\circ[/tex]
The measure of angle 1 is 49 degrees so it cannot be equal to 47 degrees.
Therefore, the required answer is "no", the given statement [tex]m\angle 1=47^\circ[/tex] is not possible.
