Triangle Geometry Problem 1

Consider triangle ABC. Angle A is 60°, and angle C is 78°. If BD bisects the angle at vertex B, what are the measures of angles x and y created at vertex B by this bisector?

Detailed Solution:

For triangle ABC, with angles at vertices A and C given as 60° and 78° respectively, we can calculate the angle at vertex B. Recall that the sum of the interior angles of any triangle is always 180°

$$ C \hat{A} B + A \hat{B} C + B \hat{C} A =180° $$

Plugging in the known values for angles A and C

$$ 60°+A \hat{B} C +78°=180° $$

$$ A \hat{B} C +138°=180° $$

This simplifies to:

$$ A \hat{B} C +138°-138°=180°-138° $$

$$ A \hat{B} C =180°−138° $$

From which we derive:

$$ A \hat{B} C =42° $$

Given that BD is the angle bisector for B, it divides angle B into two equal parts.

So, each of the angles formed by the bisector is:

$$ \frac{ A \hat{B} C }{2} = \frac{42°}{2} =21° $$

Consequently, both angles x and y at vertex B are 21°

$$ x = 21° $$

$$ y = 21° $$

This concludes our solution.