What theorem states that the exterior angle of a triangle is equal to the sum of two?
Exterior angle theoremThe exterior angle theorem tells us that any exterior angle of a triangle equals the sum of the opposite two interior angles and that the sum of all three interior angles of a triangle equals 180°, the sum of two right angles (triangle sum theorem). Show
Exterior angle theorem formulaThe exterior angle theorem is based on Euclid’s proposition 16 and proposition 32 of his “Elements.” Together they give us the exterior angle theorem that we can use to solve missing angle measurements of triangles. Exterior angles of a triangleTo understand the exterior angle theorem, you must know what an exterior angle of any polygon is. A triangle has three interior angles, but it also has six exterior angles, which are the angles between a side of a triangle and an extension of an adjacent side. Taking one exterior angle at each vertex, the sum of any polygon’s exterior three angles is always 360°. This works in either direction. Exterior Angle Theorem ProofLet's construct a triangle with an exterior angle and prove the exterior angle theorem. Here is △ABC, named for it's three angles, angle A, angle B, and angle C. We have extended one side, BC, far past the triangle: We add Point D on segment B C and now have segment BD. This gives us the exterior ∠ACD. Next, we construct a line segment parallel to segment AB: Since segments AB and CE are parallel, both line segments AC and BD are transversals of parallel lines. That means ∠BAC and ∠ACE are congruent because they are alternate interior angles of two parallel lines cut by a transversal. The two angles, ∠ECD and ∠ABC, are also congruent because they are corresponding angles. Therefore, ∠ACD is equal to the sum of the measures for ∠BAC + ∠ABC, the triangle’s two interior angles opposite the exterior ∠ACD. The last step, adding interior ∠ACB to ∠ACD to get the straight line segment BD, demonstrates that the three interior angles of the triangle sum to 180°. Exterior angle theorem exampleThe exterior angle theorem is useful for finding an unknown angle of any triangle. If you are given the measure of one exterior angle of the triangle, J, and one opposite angle, F, subtraction will give you the missing angle, G. The symbol, ∠ indicates a measured angle. Subtract the known interior angle from the exterior angle: ∠J - ∠F = ∠G Suppose an exterior angle measures 110° and you are told one of its opposite interior angles measures 47°. Plug in the knowns to find the unknown: 110° - 47° = 63° Now you know two of the three interior angles and can, if needed, easily find the third interior angle by subtracting them from 180°: 180° - 47° - 63° = 70° You can also use the theorem to find the angle adjacent to the exterior angle, simply by subtracting the exterior angle from 180°. Exterior angle theorem FAQDo you have this figured out? Check for understanding by answering these questions.
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Next Lesson:Corresponding angles Which theorem states that the exterior angle of a triangle is equal to the sum of its remote interior angles?The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles of the triangle.
What theorem states that the exterior angle of a triangle is equal to the sum of two remote interior angles of the triangle Brainly?This is Expert Verified Answer
The exterior angle theorem states that the measure of each exterior angle of a triangle is equal to the sum of the opposite and non-adjacent interior angles.
What theorem says that an exterior angle of a triangle is equal to the sum of the 2 nonDefinition: Exterior angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of two non-adjacent interior angles (opposite interior angles). An exterior angle of a triangle is formed by the extension of any one side of the triangle.
What theorem states that the exterior angle of a triangle?The exterior angle theorem is Proposition 1.16 in Euclid's Elements, which states that the measure of an exterior angle of a triangle is greater than either of the measures of the remote interior angles. This is a fundamental result in absolute geometry because its proof does not depend upon the parallel postulate.
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