Video: The Type of the Exterior Angle of an Isosceles Triangle

In a triangle π‘‹π‘Œπ‘, if π‘‹π‘Œ = 𝑋𝑍, what type of angle is the exterior angle at vertex 𝑍? [A] Acute [B] Right [C] Obtuse [D] Reflex

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Video Transcript

In a triangle π‘‹π‘Œπ‘, if π‘‹π‘Œ is equal to 𝑋𝑍, what type of angle is the exterior angle at vertex 𝑍? Is it (A) acute, (B) right, (C) obtuse, or (D) reflex?

We begin by recalling the properties of our four angles. An acute angle is less than 90 degrees. A right angle is equal to 90 degrees. An obtuse angle lies between 90 degrees and 180 degrees. Finally, a reflex angle lies between 180 degrees and 360 degrees. In this question, we are told that in the triangle π‘‹π‘Œπ‘, the length of π‘‹π‘Œ is equal to the length of 𝑋𝑍. This means that we have an isosceles triangle. In any isosceles triangle, two interior angles are also equal. In this case, angle π‘‹π‘Œπ‘ is equal to angle π‘‹π‘π‘Œ.

We know that angles in a triangle sum to 180 degrees. This means that the interior angles at vertex π‘Œ and vertex 𝑍 must both be acute. They must both be less than 90 degrees. Otherwise the sum would be greater than 180. The exterior angle at vertex 𝑍 is shown on the diagram. It is the angle between the side length 𝑋𝑍 and the extension of side length π‘Œπ‘. We know that angles on a straight line sum to 180 degrees. This means that the interior and exterior angles at vertex 𝑍 must sum to 180. As the interior angle is acute, it is less than 90, the exterior angle must be greater than 90 and obtuse. In the isosceles triangle drawn, the exterior angle at vertex 𝑍 is obtuse.

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