Question

question 9 (multiple choice worth 1 points)
(07.02 mc)
point d is the incenter of triangle bca. if m∠fdg = 128°, what is the measure of ∠fhg?

Explanation:

Step1: Find the measure of ∠FCH and ∠HCF

Since \(D\) is the in - center, \(DF\perp BC\), \(DG\perp AC\), \(DH\perp AB\). The sum of angles in quadrilateral \(DFCG\) is \(360^{\circ}\). Given \(\angle FDG = 128^{\circ}\), \(\angle DFC=\angle DGC = 90^{\circ}\).
Using the formula for the sum of angles in a quadrilateral \(\angle FDG+\angle DFC+\angle FCG+\angle DGC=360^{\circ}\), we substitute the known values: \(128^{\circ}+90^{\circ}+\angle FCG + 90^{\circ}=360^{\circ}\).
Simplifying the left - hand side gives \(308^{\circ}+\angle FCG=360^{\circ}\), so \(\angle FCG=360^{\circ}- 308^{\circ}=52^{\circ}\).
Since \(D\) is the in - center, \(CH\) bisects \(\angle FCG\). Let \(\angle FCH=\angle HCG=\frac{1}{2}\angle FCG\). So \(\angle FCH = 26^{\circ}\).

Step2: Use the property of right - angled triangles

In right - angled triangle \(DFC\) and \(DHC\) (because \(DH\perp AB\), \(DF\perp BC\) and \(CH\) is an angle bisector), \(\angle DHC=\angle DFC = 90^{\circ}\), \(CH = CH\) (common side), \(DH = DF\) (radii of the incircle). So \(\triangle DHC\cong\triangle DFC\) (by Hypotenuse - Leg congruence in right - angled triangles).
In quadrilateral \(DFAH\), \(\angle DFH=\angle DHH = 90^{\circ}\). The sum of angles around point \(H\):
We know that the sum of angles in a quadrilateral formed by two right angles (\(\angle DFH\) and \(\angle DHH\)) and two angles related to the angle bisector.
Another way: The sum of angles in a circle - related quadrilateral (the quadrilateral formed by the points of tangency and the in - center).
We use the property that \(\angle FHG = 180^{\circ}-\angle FDG\) (the sum of the central - like angle related to the in - center and the angle formed by the non - parallel sides of the quadrilateral formed by the tangents).
Since \(\angle FDG = 128^{\circ}\), then \(\angle FHG=180^{\circ}- 52^{\circ}=128^{\circ}\) (using the angle - bisector and quadrilateral angle - sum properties).

Answer:

\(128^{\circ}\)