7.1 Oblique Triangles and the Law of Sines

 

Extending “solving triangles” from right triangles to all triangles

 

Congruence Axioms

SAS           Side-Angle-Side 

If two sides and the included angle of one triangle are equal, respectively, to two sides and the included angle of a second triangle, then the triangles are congruent.

 

ASA            Angle-Side Angle 

If two angles and the included side of one triangle are equal, respectively, to two angles and the included side of a second triangle, then the triangles are congruent.

 

SSS            Side-Side-Side 

If three sides of one triangle are equal, respectively, to three sides of a second triangle, then the tri angles are congruent.

 

Whenever SAS, ASA, or SSS is given, the triangle is unique.

 

Oblique Triangle   A triangle that is not a right triangle

 

Data Required for Solving Oblique Triangles

Case 1       One side and two angles are know (SAA or ASA)

Case 2       Two sides and one angle not included between the two sides are

                   known (SSA).  This case may lead to more than one triangle.

Case 3       Two sides and the angle included between the two sides are

                   known (SAS).

Case 4       Three sides are known (SSS).

 

Note – Three angles of a triangle (AAA) does not result in a unique triangle – just similarity.

 

Law of Sines

In any triangle ABC, with sides a, b, and c,

Hint Writing the equation so that the unknown variable is in the numerator and all other variables are known is sometimes useful when using the law of Sines.

 

Solving SAA and ASA Triangles (Case 1)

Given two angles and one side – solve the triangle

 

Example

Solve triangle ABC if

Always begin with a labeled sketch of the triangle!!

(Ans:  a=12.5, B=48.6o, b=19.4)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Application

Jerry Keefe wishes to measure the distance across the Big Muddy River.  He determines that   The distance across the river is a.              (Ans 65.1 ft)

 

 

 

 

 

 

 

 

 

 

Application

The bearing of a lighthouse from a ship was found to be .  After the ship sailed 5.8 km due south, the new bearing was .  Find the distance between the ship and the lighthouse at each location.

(Ans: 1st  4.7 km;   2nd  9.4 km)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Area of a Triangle

In any triangle ABC, the area A is given by the following formulas.

 

These formulae use two sides and the angle included between them.

 

Example

Find the area of Triangle DEF if  (Ans: 43 ft2)

 

 

 

 

 

 

 

 

Example

Find the area of triangle ABC if

                                      (Ans:  576 cm2)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Reminder   If possible, use given values in solving triangles to avoid any rounding errors from using intermediate values.