Solving Triangles

Solving Triangles: Level 4 Challenges


In a triangle ABCABC, the measures of the angles are A=30,\angle A=30^\circ, B=80.\angle B=80^\circ. Point MM lies inside the triangle, such that MAC=10,\angle MAC = 10^\circ, MCA=30.\angle MCA = 30^\circ. What is the measure (in degrees) of BMC?\angle BMC?

Triangle ABCABC is isosceles with AC=BCAC = BC and ACB=106\angle ACB = 106^\circ Point MM is inside the triangle so that MAC=7\angle MAC = 7^\circ and MCA=23\angle MCA = 23^\circ. Find the measure of CMB\angle CMB in degrees.

In a triangle ABCABC A=84,\angle A =84^\circ, C=78.\angle C=78^\circ. Points DD and EE are taken on the sides ABAB and BC,BC, so that ACD=48,\angle ACD =48^\circ, CAE=63.\angle CAE =63^\circ. What is the measure (in degrees) of CDE\angle CDE ?

In triangle ABC ABC , cosA:cosB:cosC=2:9:12 \cos A : \cos B : \cos C = 2 : 9 : 12 .

If sinA:sinB:sinC=p:q:r \sin A : \sin B : \sin C = p : q : r , where p p , q q and r r are coprime positive integers, find pqr(p+q+r) pqr-(p+q+r) .

Triangle ABCABC has area equal to 9034 \dfrac {90 \sqrt{3}}{4} and perimeter equal to 30.30. Also, one of its angles is equal to 60. 60^\circ. What is the product of the sides of ABC?ABC?


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