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Difficulty: HardTriangle Properties and Angle Theorems

In acute triangle ABCABC, the measure of ∠A\angle A is 74∘74^\circ. The altitude from vertex BB to side ACAC and the altitude from vertex CC to side ABAB intersect at point HH inside the triangle. What is the measure, in degrees, of ∠BHC\angle BHC?

Answer: 106 degrees

Answer

The measure of ∠BHC\angle BHC is 106 degrees.
In quadrilateral AEHDAEHD, the sum of the angles is 360∘360^\circ. Since the angles at EE and DD are 90∘90^\circ because they are formed by altitudes, the sum of ∠A\angle A and ∠EHD\angle EHD must be 180∘180^\circ. Thus, ∠EHD=180∘−74∘=106∘\angle EHD = 180^\circ - 74^\circ = 106^\circ. Since ∠BHC\angle BHC and ∠EHD\angle EHD are vertical angles, ∠BHC=106∘\angle BHC = 106^\circ.

Step-by-Step Solution

1
Define the intersection points of the altitudes with the opposite sides.
Let the altitude from vertex BB intersect side ACAC at point DD, and let the altitude from vertex CC intersect side ABAB at point EE. Therefore, ∠ADH=90∘\angle ADH = 90^\circ and ∠AEH=90∘\angle AEH = 90^\circ.
Altitudes by definition are perpendicular to the sides they intersect, forming 90∘90^\circ angles.
2
Analyze the sum of interior angles in the quadrilateral AEHDAEHD.
The sum of the interior angles of a quadrilateral is 360∘360^\circ, so ∠A+∠AEH+∠EHD+∠ADH=360∘\angle A + \angle AEH + \angle EHD + \angle ADH = 360^\circ.
Any quadrilateral can be split into two triangles, making the sum of its interior angles 360∘360^\circ.
3
Substitute the known angle measures to find the measure of ∠EHD\angle EHD.
74∘+90∘+∠EHD+90∘=360∘  ⟹  254∘+∠EHD=360∘  ⟹  ∠EHD=106∘74^\circ + 90^\circ + \angle EHD + 90^\circ = 360^\circ \implies 254^\circ + \angle EHD = 360^\circ \implies \angle EHD = 106^\circ.
Solving the linear equation for the unknown angle ∠EHD\angle EHD.
4
Relate ∠EHD\angle EHD to the target angle ∠BHC\angle BHC.
Since line segments BDBD and CECE intersect at HH, the angles ∠BHC\angle BHC and ∠EHD\angle EHD are vertical angles, so ∠BHC=∠EHD=106∘\angle BHC = \angle EHD = 106^\circ.
Vertical angles are equal in measure.

Key Concept

The sum of angles in quadrilaterals, the definition of altitudes, and vertical angles within triangles.
Estimated Time:2m 0s
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Topics

Triangle Properties and Angle Theorems

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