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How do you prove that the exterior angle of a triangle is equal to the sum of the two opposite interior angles?
Assume a triangle ABC with a line AB (containing the side AB) with external angle D which is formed when line AB and line segment AC intersect. We are asked to prove that the external angle D is equal to the sum of the two interior angles B and C. Angles A and D are supplementary angles (they sum to 180 degrees) because they are linear angles (both together make a straight line, or a 180 degree angle). This means:
m<A + m<D = 180 degrees.
m<A = 180 deg - m<D
Then because A, B, and C are the three angles in a triange:
m<A + m<B + m<C = 180 deg
m<A = 180 deg - m<B - m<C
By substituting 180 deg - m<D in for m<A in the above equation we get:
180 deg - m<D = 180 deg - m<B - m<C
Subtract 180 deg from each side:
-m<D = -m<B - m<C
Multiply both sides by -1
m<D = m<B + m<C
Which proves that the measure of the external angle D is equal to the sum of the two opposite interior angles B and C for any given triangle.
wow. that's a lot. lol.
What else can I help you with?
What is the relationship between exterior and opposite interior angles?
In a triangle, each exterior angle is equal to the sum of the two opposite interior angles.
What is relationship between exterior angles and opposite interior angles of a triangle?
The exterior angle of a triangle is equal to the sum of the two opposite interior angles. So if a triangle had points A, B and C: The exterior angle at B would equal the sum of interior angles at A and C. Similarly, the exterior angle at C would equal the sum of interior angles at A and B And the the exterior angle at A would equal the sum of interior angles at C and B.
What type of angles do the pairs of exterior angles of a triangle form?
The pairs of exterior angles of a triangle form adjacent angles with the interior angles of the triangle. Specifically, each exterior angle is supplementary to the interior angle at its corresponding vertex, meaning they add up to 180 degrees. Additionally, the exterior angles of a triangle are equal to the sum of the two opposite interior angles, establishing a relationship among them.
An exterior angle of a triangle is equal to the adjacent interior angle?
No. It is equal to the sum of the opposite interior angles.
What is the measure of each exterior angle of a triangle?
Any exterior angle of a triangle always equals the sum of the two interior opposite angles.
What is the relationship between exterior and opposite interior angles?
In a triangle, each exterior angle is equal to the sum of the two opposite interior angles.
What is relationship between exterior angles and opposite interior angles of a triangle?
The exterior angle of a triangle is equal to the sum of the two opposite interior angles. So if a triangle had points A, B and C: The exterior angle at B would equal the sum of interior angles at A and C. Similarly, the exterior angle at C would equal the sum of interior angles at A and B And the the exterior angle at A would equal the sum of interior angles at C and B.
What type of angles do the pairs of exterior angles of a triangle form?
The pairs of exterior angles of a triangle form adjacent angles with the interior angles of the triangle. Specifically, each exterior angle is supplementary to the interior angle at its corresponding vertex, meaning they add up to 180 degrees. Additionally, the exterior angles of a triangle are equal to the sum of the two opposite interior angles, establishing a relationship among them.
An exterior angle of a triangle is equal to the adjacent interior angle?
No. It is equal to the sum of the opposite interior angles.
The measure of an exterior angle of a triangle is equal to the sum of?
any other exterior angle of the triangle * * * * * No. The sum of the two opposite interior angles.
What is the measure of each exterior angle of a triangle?
Any exterior angle of a triangle always equals the sum of the two interior opposite angles.
Which polygon has the larger exterior angle triangle or heptagon?
Interior Angles of a Triangle: 60 Exterior Angles of a Triangle: 120 Interior Angles of a Heptagon 128 (approx) Exterior Angles of A Heptagon 52 (approx) Answer: Triangle
The measure of an exterior angle of a triangle is equal to the sum of the measures of its?
two opposite interior angles.
Which is the best definition of an exterior angle of a triangle?
an angle that forms a linear pair with one of the interior angles of the trinagle.(Apex)
Why does a remote exterior angle equal to the sum to the opposite exterior angles?
In a triangle, the remote exterior angle is formed by extending one side of the triangle, while the opposite interior angles are those that do not share a vertex with the exterior angle. According to the exterior angle theorem, the measure of the remote exterior angle is equal to the sum of the measures of the two opposite interior angles. This relationship holds because the angles in a triangle sum up to 180 degrees, and the exterior angle effectively "completes" the linear pair with the adjacent interior angle, reinforcing the equality. Thus, the theorem demonstrates a fundamental property of triangles and their angles.
Does exterior angle of a triangle equals the sum of the interior angles?
Actually, no. The sum of the exterior angles of a triangle is 360 degrees. The sum of the interior angles of a triangle is 180 degrees.
Can an acute triangle never have an acute exterior angle?
No, an acute triangle can have an acute exterior angle. An exterior angle is formed by extending one side of the triangle, and its measure is equal to the sum of the two opposite interior angles. Since all interior angles of an acute triangle are less than 90 degrees, the corresponding exterior angles can also be acute, depending on which side is extended.
