Solving the Triangle Angle Bisector Problem

The question asks us to find the measure of the angle formed by the angle bisectors of angles B and C in a triangle ABC, given the measure of angle A.

Understanding Angle Bisectors in a Triangle

An angle bisector is a line segment that divides an angle into two equal angles. In Δ ABC, BO is the angle bisector of ∠B, and CO is the angle bisector of ∠C. These bisectors meet at a point O inside the triangle. Point O is known as the incenter of the triangle.

Given Information

• In Δ ABC, ∠A = 50°
• BO bisects ∠B, so ∠OBC = ∠ABO = $\frac{1}{2}\ang;B$
• CO bisects ∠C, so ∠OCB = ∠ACO = $\frac{1}{2}\ang;C$

Finding ∠BOC

We can find ∠BOC using the properties of triangles.

Method 1: Using Sum of Angles in Triangles

In any triangle, the sum of interior angles is 180°. In Δ ABC:

$\ang;A + \ang;B + \ang;C = 180°$

Substitute the given value of ∠A:

$50° + \ang;B + \ang;C = 180°$

So, $\ang;B + \ang;C = 180° - 50°$

$\ang;B + \ang;C = 130°$

Now consider the triangle Δ OBC. The sum of angles in Δ OBC is also 180°:

$\ang;OBC + \ang;OCB + \ang;BOC = 180°$

Since BO and CO are angle bisectors:

$\ang;OBC = \frac{1}{2}\ang;B$

$\ang;OCB = \frac{1}{2}\ang;C$

Substitute these into the equation for Δ OBC:

$\frac{1}{2}\ang;B + \frac{1}{2}\ang;C + \ang;BOC = 180°$

Factor out $\frac{1}{2}$:

$\frac{1}{2}(\ang;B + \ang;C) + \ang;BOC = 180°$

We know that $\ang;B + \ang;C = 130°$, so substitute this value:

$\frac{1}{2}(130°) + \ang;BOC = 180°$

$65° + \ang;BOC = 180°$

Solve for ∠BOC:

$\ang;BOC = 180° - 65°$

$\ang;BOC = 115°$

Method 2: Using the Angle Bisector Formula

There is a specific formula relating the angle formed by the interior angle bisectors (∠BOC) and the angle at the opposite vertex (∠A):

$\ang;BOC = 90° + \frac{1}{2}\ang;A$

Substitute the given value ∠A = 50°:

$\ang;BOC = 90° + \frac{1}{2}(50°)$

$\ang;BOC = 90° + 25°$

$\ang;BOC = 115°$

Both methods yield the same result.

Summary of Calculation Steps

The angle ∠BOC is equal to 115°.

Revision Table: Triangle Angle Properties

Additional Information on Triangle Centers

The intersection point of angle bisectors (O) is called the incenter. The incenter is equidistant from the sides of the triangle and is the center of the inscribed circle (incircle).

Besides the incenter, triangles have other important centers:

• Centroid: The intersection of medians. Medians connect a vertex to the midpoint of the opposite side. The centroid divides each median in a 2:1 ratio.
• Orthocenter: The intersection of altitudes. Altitudes are perpendicular lines from a vertex to the opposite side.
• Circumcenter: The intersection of perpendicular bisectors of the sides. The circumcenter is equidistant from the vertices and is the center of the circumscribed circle (circumcircle).

Understanding these centers helps in solving various geometry problems related to triangles.