Video: Converse to the Hinge Theorem

Consider the two triangles in the diagram. Length π‘Ž is equal to length 𝑑 and length 𝑏 is equal to length 𝑒. Given that the perimeter of triangle 1 is less than the perimeter of triangle 2, what does the converse of the hinge theorem tell us about the measures of angles πœƒ and πœ™?

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Video Transcript

Consider the two triangles in the diagram. Length π‘Ž is equal to length 𝑑. And length 𝑏 is equal to length 𝑒. Given that the perimeter of triangle 1 is less than the perimeter of triangle 2, what does the converse of the hinge theorem tell us about the measures of angles πœƒ and πœ™?

Let’s just read through the information that we’re given in the question again. Firstly, we’re told that lengths π‘Ž and 𝑑 are equal. Secondly, we’re told that length 𝑏 and length 𝑒 are equal. We’re told that the perimeter of triangle 1 is less than the perimeter of triangle 2. We can express this as an inequality.

The perimeter of a triangle is found by adding its three sides together. So we have that the perimeter of triangle 1, π‘Ž plus 𝑏 plus 𝑐, is less than the perimeter of triangle 2, 𝑑 plus 𝑒 plus 𝑓. This can be simplified, as certain pairs of sides in the two triangles are equal. π‘Ž and 𝑑 are equal. So they cancel each other out. 𝑏 and 𝑒 are also equal. So they cancel each other out. And we’re left with the inequality 𝑐 is less than 𝑓, meaning that the third side of triangle 1 is shorter than the third side of triangle 2.

We’re told that we need to answer this question using the converse of the hinge theorem. So let’s recall what this tells us. The hinge theorem itself tells us this: if two triangles have two equal sides, then the triangle with the larger included angle has the longer third side. The included angle is the angle between the two sides that are equal. This is the setup that we have in this question, as triangles 1 and 2 have two equal sides and the angles that we’re given, πœƒ and πœ™, are the included angles.

To find the converse of the hinge theorem, we need to swap two parts around. We need to swap larger included angle with longer third side. The converse of the hinge theorem is that if two triangles have two equal sides, then the triangle with the longer third side has the larger included angle.

Remember, we already know the relationship that exists between the third sides of these two triangles. We know that 𝑐 is less than 𝑓. This means that triangle 2 has the longer third side. And therefore, by the converse of the hinge theorem, it has the larger included angle. The included angle in triangle 2 is angle πœ™. And so our statement about the measures of the two angles is that the measure of angle πœ™ is greater than the measure of angle πœƒ.

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