Question

2 sketch the unique triangles that can be made with angle measures 40° and 100° and side length 3 units. how many unique triangles are there? how do you know you have sketched all possibilities? 3 here is triangle jkl. what is the least amount of information that you need to construct a triangle congruent to this one?

Explanation:

Question 2

Step1: Find the third - angle

The sum of angles in a triangle is \(180^{\circ}\). Let the third angle be \(x\). Then \(x + 40^{\circ}+100^{\circ}=180^{\circ}\), so \(x = 40^{\circ}\).

Step2: Analyze the cases based on the side - angle relationship

By the Law of Sines \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). Since two angles are \(40^{\circ}\) and one is \(100^{\circ}\), and we have a side of length \(3\) units.
Case 1: If the side of length \(3\) is opposite the \(40^{\circ}\) angle.
Case 2: If the side of length \(3\) is opposite the \(100^{\circ}\) angle. But since the angles are fixed (\(40^{\circ},40^{\circ},100^{\circ}\)), and using the Law of Sines (the ratio of side lengths is determined by the ratio of the sines of their opposite angles), there is only \(1\) unique triangle.
We know we have sketched all possibilities because the angles of the triangle are fixed (\(40^{\circ},40^{\circ},100^{\circ}\)) and by the Law of Sines (for a given set of angles and a side length, the other side lengths are uniquely determined).

According to the triangle congruence criteria (SSS - Side - Side - Side, SAS - Side - Angle - Side, ASA - Angle - Side - Angle, AAS - Angle - Angle - Side, HL - Hypotenuse - Leg for right - angled triangles). The least amount of information is \(3\) pieces of information. This can be either three side lengths (SSS), two sides and the included angle (SAS), two angles and the included side (ASA), or two angles and a non - included side (AAS).

Answer:

There is \(1\) unique triangle.

Question 3