Question

1. explain if the given measurements will form a triangle? a) 30°, 90° and 60°. b) 5cm, 11cm and 3 cm. c) 7 cm, 9cm and 10 cm. d) 75°, 45° and 55°. 13) draw a triangle with a 50°, and a 70° degree angle. the side between them must measure 6 cm. how many unique triangles with the same measurements could you make?

Explanation:

Step1: Recall triangle - angle sum property and side - length rules

The sum of interior angles of a triangle is 180°. For side lengths \(a\), \(b\), and \(c\) of a triangle, the following inequalities must hold: \(a + b>c\), \(a + c>b\), and \(b + c>a\).

Step2: Check part (a) for angle - sum

Calculate the sum of the angles: \(30^{\circ}+90^{\circ}+60^{\circ}=180^{\circ}\). So, these angles can form a triangle.

Step3: Check part (b) for side - length inequalities

\(5 + 3=8<11\). So, these side lengths cannot form a triangle.

Step4: Check part (c) for side - length inequalities

\(7+9 = 16>10\), \(7 + 10=17>9\), \(9+10 = 19>7\). So, these side lengths can form a triangle.

Step5: Check part (d) for angle - sum

Calculate the sum of the angles: \(75^{\circ}+45^{\circ}+55^{\circ}=175^{\circ}
eq180^{\circ}\). So, these angles cannot form a triangle.

Step6: Analyze part (13) using the ASA (angle - side - angle) criterion

By the ASA congruence criterion, if two angles and the included side of a triangle are given, then exactly one unique triangle can be constructed.

Answer:

a) Can form a triangle.
b) Cannot form a triangle.
c) Can form a triangle.
d) Cannot form a triangle.

1. One unique triangle.