Understanding the Triangle Altitude and Angle Division

Given a triangle with base angles A and B (where B > A), and an altitude drawn to the base from the vertex angle C. This altitude divides the vertex angle C into two parts: C₁ and C₂, where C₂ is adjacent to side a. We are to find the correct relationship among the options.

Step 1: Visualize the Triangle and Altitude

Consider triangle ABC, where the base is BC. The base angles are A (at vertex A) and B (at vertex B), with B > A. The altitude from vertex A to the base BC divides the vertex angle C into C₁ and C₂. Let the foot of the altitude on BC be point D. Then, angle C is split such that in right triangles ADC and ADB, we can relate the angles.

Step 2: Express Angles in Terms of Known Quantities

In triangle ABC, the sum of angles is: $A+B+C=180°$ Since the altitude AD is perpendicular to BC, we have two right triangles: ABD and ADC.

In right triangle ABD: $A+C_1=90°$ (because angle at D is 90°, and the sum of angles in a triangle is 180°).

In right triangle ADC: $B+C_2=90°$ (similarly).

Step 3: Subtract the Two Equations

Subtract the second equation from the first: $(A+C_1)-(B+C_2)=90°-90°$
Which simplifies to: $A-B+C_1-C_2=0$
Rearranging terms: $C_1-C_2=B-A$

Step 4: Final Answer

Therefore, the correct relationship is: $C_1-C_2=B-A$
This matches the first option.

Related Topics and Formulae

Key Concepts:

• Triangle Angle Sum Theorem: The sum of the interior angles of a triangle is always 180°.
• Properties of Altitudes: An altitude in a triangle forms two right triangles, allowing the use of trigonometric ratios and angle relationships.
• Angle Chasing: Using known angles to find unknown ones by constructing equations based on geometric properties.

Important Formulae:

• Sum of angles in a triangle: $A+B+C=180°$
• In a right triangle, the two acute angles are complementary: $\alpha +\beta =90°$