The bisectors of the outer angles at the vertices B and A of the triangle ABC intersect at point D.

univerkov | August 8, 2021 | education

The bisectors of the outer angles at the vertices B and A of the triangle ABC intersect at point D. Find the angle BCA if the angle BDA = 70 °

1) Let ВD be the bisector of the outer angle of the angle <С1ВА, hence, <AED = <С1ВD. Since AD is the bisector of the external angle BAA1, then <A1AD = <DAB. But from the triangle DAB it is clear that the sum of the angles <DBA + <DAB = 180 – 70 = 110.

2) From the triangle ABC it can be seen that the angle <BCA = 180 ° – (<CBA + <CAB) = 180 ° – (180 ° – <АА1В) – (180 ° – <В1ВА) = 180 ° – (180 – 2 * <DBA) – 180 ° – 2 * <DAB) = 2 * (<DBA + <DAB) – 180 °.

3) But the sum of the angles (<DBA + <DAB) = 110 ° we found in action 1). Thus, <BCA = 2 * 110 ° – 180 ° = 220 ° – 180 ° = 40 °.

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