Chapter 10 - NCPN

Activity
Ambiguous Cases
Two triangles that have a
congruent angle and two pairs
of congruent sides are not
necessarily congruent as shown
below. The Law of Sines can be
used to find the missing angle
measures for both triangles.
1 Use the Law of Sines to express the relationship between
∠A and ∠B. sin43
° = sin B
16 20
2 Solve the equation from Step 1 for m∠B. Round your answer to
the nearest tenth. 58.5°
3 The sine function is positive in Quadrant I and II. Therefore,
another value for m∠B can be found by subtracting the answer
from Step 2 from 180°. Find another value for m∠B. 121.5°
4 Find the other missing angle measures in the two triangles.
Express the angles of the two triangles that satisfy the given
conditions and the Law of Sines. Blue triangle: 43°, 58.5°, 78.5°;
Orange triangle: 43°, 121.5°, 15.5°
For any triangle ABC, if a, b, and c represent the side lengths opposite
angles A, B, and C, respectively, then the Law of Cosines states that
a 2 = b 2 + c 2 – 2bccos A,
b 2 = a 2 + c 2 – 2accos B, and
c 2 = a 2 + b 2 – 2abcos C.
The Law of Cosines can be used to find missing measurements in triangles
when the measures of two sides and the angle between them are known or
when the measures of all three sides are known.
10.5 The Law of Sines and Law of Cosines 465