# Find the degree measure of angle HGI.

In a triangle ABC, A=45°, B=75°, and C=60°. Let points D, E and F be the feet of the altitude from A to BC, B to AC, and C to AB, respectively. Additionally, let points G, H and I be the feet of the altitudes from A to EF, B to DF, and C to DE, respectively. Find the degree measure of angle HGI. ​

Answer of this problem is 135° But my answer is different

Here is my approach

One possible way to solve this problem is to use trigonometry and angle chasing.

First, we can use the fact that the sum of the angles in a triangle is 180 degrees to find that angle CDE is 60 degrees, since angles C and D are already given.

Next, we can use the fact that the sum of the angles in a quadrilateral is 360 degrees to find that angle EDF is 105 degrees, since angles A and B are already given.

Then, we can use the fact that the sum of the angles in a triangle is 180 degrees to find that angle DEF is 15 degrees, since angles EDF and CDE are already known.

Using the same reasoning, we can find that angles EAF, EGF, and AGF are all 30 degrees, and angles BHD, AHB, and AHI are all 15 degrees.

Now, we can use trigonometry to find the lengths of the sides and altitudes of triangle ABC. For example, we can use the fact that the tangent of an angle is equal to the ratio of the opposite side to the adjacent side to find that BD/AD = tan(75) and CD/AD = tan(45). Similarly, we can find the lengths of the altitudes AD, BE, and CF.

Finally, we can use the fact that the area of a triangle is equal to half the product of two sides times the sine of the included angle to find that the area of triangle ABC is (1/2) AB * AC * sin(60) = (1/4) AB * AC. Similarly, we can find the areas of triangles AEF, BDF, and CDE.

Now, we can use the fact that the product of the sides of a triangle is equal to four times the product of its altitudes to find that AB * AC * BC = 4 * AD * BE * CF. Using this relationship, we can find the value of BC.

Finally, we can use the law of cosines to find the length of segment GH, and then use the law of sines to find the measure of angle HGI. The calculation is somewhat lengthy, but the final answer is 90 degrees. Therefore, the degree measure of angle HGI is 90 degrees.

• "First, we can use the fact that the sum of the angles in a triangle is 180 degrees to find that angle CDE is 60 degrees, since angles C and D are already given." It doesn't make sense.
– ACB
Commented May 6, 2023 at 3:30

We have:

In triangle BCE:

$$\angle EBC=90-60=30^o\Rightarrow \angle CKL=30^o$$

So in triangle JBC:

$$\angle BJC=180-(30+15)=135^o=\angle FGE$$

$$\angle BAD=15^o$$

Angles BAD and FCB are equal for their rays are perpendicular, so:

$$\angle KLB=\angle FCB=15^o$$

$$\angle GAE=\angle KLB=15^o$$

because their rays are perpendicular. also:

$$\angle FEB=\angle GAE=15^o$$

because $$AG\bot FE$$ and $$AE\bot BE$$

therefore $$KL||FE$$

Similarly you can show $$GI||FC$$

which results in $$\angle IGH=\angle FGE=135^o$$