To find the measure of \( m∠KGH \), we start by analyzing the information given in the problem:
- The segments \( HI \) and \( IJ \) are congruent (\( HI \cong IJ \)), which implies that triangle \( HIJ \) is isosceles with \( HI = IJ \).
- We are given that \( m∠KIJ = 35^\circ \).
In an isosceles triangle, the angles opposite the equal sides are equal. Therefore, we have:
- Let \( m∠HJI \) be the angle opposite side \( HI \), and \( m∠HIK \) be the angle opposite side \( IJ \). Since both sides are congruent, we have:
\[ m∠HJI = m∠HIK \]
Let \( x \) be the measure of these angles. Since the sum of the angles in triangle \( HIJ \) is \( 180^\circ \), we can write the equation:
\[ x + x + 35^\circ = 180^\circ \]
This simplifies to:
\[ 2x + 35^\circ = 180^\circ \]
Now, subtract \( 35^\circ \) from both sides:
\[ 2x = 180^\circ - 35^\circ = 145^\circ \]
Now, divide by 2:
\[ x = \frac{145^\circ}{2} = 72.5^\circ \]
So we have \( m∠HJI = m∠HIK = 72.5^\circ \).
Next, we want to find \( m∠KGH \). Since \( G \) is the center of the circle, \( m∠KGH \) is equal to the measure of the inscribed angle \( m∠KJI \), which subtends the same arc \( KJI \).
The inscribed angle theorem states that the measure of an inscribed angle is half the measure of the central angle that subtends the same arc. In this case:
\[ m∠KJI = m∠KHG \]
Since \( m∠HJI = 72.5^\circ \) is the angle we calculated before, \( m∠KJI \) will equal \( 2 \times 72.5^\circ \).
Now, we calculate:
\[ m∠KGH = m∠KJI = 2 \times 72.5^\circ = 145^\circ \]
However, we should note that the angle \( m∠KGH\) refers to the external angle \( m∠KGH = 180^\circ - m∠KJI \):
Thus \( m∠KGH = 180^\circ - 145^\circ = 35^\circ\)
So, my final answer is:
70 degrees.