Question

classify each set of conditions by the possible number of triangles, if any, that can be created with those conditions. drag each set of conditions to the table. more than one triangle can be formed exactly one triangle can be formed no triangles can be formed m∠a = 35° m∠b = 45° ab = 3 m∠a = 90° m∠b = 30° m∠c = 60° ab = 5 bc = 8 ca = 3 m∠a = 50° m∠b = 80° bc = 5 m∠c = 70° ca = 5 ab = 7

Explanation:

Step1: Recall triangle - forming rules

The sum of interior angles of a triangle is 180°. Also, for side - length conditions, we can use the triangle inequality theorem (the sum of the lengths of any two sides of a triangle must be greater than the length of the third side).

Step2: Analyze the first set of conditions

Given \(m\angle A = 35^{\circ}\), \(m\angle B=45^{\circ}\), \(AB = 3\). Since the sum of the given angles \(35^{\circ}+45^{\circ}=80^{\circ}\lt180^{\circ}\), and we have one side - length, exactly one triangle can be formed (ASA - Angle - Side - Angle criterion).

Step3: Analyze the second set of conditions

Given \(m\angle A = 90^{\circ}\), \(m\angle B = 30^{\circ}\), \(m\angle C=60^{\circ}\). The sum of the angles is \(90^{\circ}+30^{\circ}+60^{\circ}=180^{\circ}\), but no side - length is given. So, more than one triangle can be formed (similar triangles with different side - lengths).

Step4: Analyze the third set of conditions

Given \(AB = 5\), \(BC = 8\), \(CA = 3\). Since \(3 + 5=8\), which does not satisfy the triangle inequality theorem (\(a + b>c\) for all sides \(a\), \(b\), \(c\) of a triangle). So, no triangles can be formed.

Step5: Analyze the fourth set of conditions

Given \(m\angle A = 50^{\circ}\), \(m\angle B = 80^{\circ}\), \(BC = 5\). The sum of the given angles \(50^{\circ}+80^{\circ}=130^{\circ}\lt180^{\circ}\), and we have one side - length. Exactly one triangle can be formed (ASA criterion).

Step6: Analyze the fifth set of conditions

Given \(m\angle C = 70^{\circ}\), \(CA = 5\), \(AB = 7\). Using the Law of Sines \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\), we can find the possible values of the other angles and sides. Since we have two sides and a non - included angle, more than one triangle can be formed (the ambiguous case of the Law of Sines).

Answer:

• More than one triangle can be formed: \(m\angle A = 90^{\circ},m\angle B = 30^{\circ},m\angle C = 60^{\circ}\); \(m\angle C = 70^{\circ},CA = 5,AB = 7\)
• Exactly one triangle can be formed: \(m\angle A = 35^{\circ},m\angle B = 45^{\circ},AB = 3\); \(m\angle A = 50^{\circ},m\angle B = 80^{\circ},BC = 5\)
• No triangles can be formed: \(AB = 5,BC = 8,CA = 3\)