Question

1. graph the triangle with vertices ( d(2,0) ), ( e(2,4) ), and ( f(6,2) ). then graph a triangle congruent to ( \triangle def ).
2. given ( \triangle hjkcong\triangle trs ), find the values of ( a ) and ( b ).

Explanation:

Step1: Use the property of congruent triangles

Since \(\triangle HJK\cong\triangle TRS\), corresponding angles are equal.
\(\angle H=\angle T\), \(\angle J=\angle S\), \(\angle K=\angle R\)

Step2: Find the value of \(a\)

We know that \(\angle H = 51^{\circ}\), \(\angle R=83^{\circ}\), and in a triangle, the sum of angles is \(180^{\circ}\). For \(\triangle HJK\), \(\angle J=180-(51 + (7b - 10))\). For \(\triangle TRS\), \(\angle S = 180-(83+(6a - 3))\). Also, since \(\triangle HJK\cong\triangle TRS\), we can use the fact that corresponding angles are equal. Another way is to use the side - angle relationships. But more straightforwardly, since \(\angle K\) and \(\angle R\) are corresponding angles. First, for the angle expressions related to \(a\):
We know that \(6a-3 = 81\) (because \(\angle T\) in \(\triangle TRS\) and \(\angle H\) in \(\triangle HJK\) are \(51^{\circ}\), and using the angle - sum property of a triangle \(180-(51 + 46)=83\) for \(\triangle HJK\) and \(180-(51+(6a - 3))\) for \(\triangle TRS\). Solving \(6a-3=81\) gives \(6a=84\), so \(a = 14\)

Step3: Find the value of \(b\)

Since \(\angle K\) and \(\angle R\) are related. Using the angle - sum property of a triangle for \(\triangle HJK\), \(\angle K=180-(51 + 83)=46\). So \(7b-10 = 46\). Solving \(7b-10 = 46\) gives \(7b=56\), so \(b = 13\)

Answer:

\(a = 14\), \(b = 13\)