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Nội dung chính- Breaking Down the Exterior Angle Theorem
- Applying the Exterior Angle Theorem
- Understanding Exterior and Interior Angles
- More Math Homework Help:
- What theorem states that the exterior angle of a triangle is equal to the sum of two remote interior angles of the triangle Brainly?
- What theorem states that the measure of an exterior angle of a triangle?
- What theorem justifies the statement the measure of an exterior angle of a triangle is greater than the measure of either of its remote interior angle?
- What is the exterior angle sum theorem states?
The exterior angle theorem for triangles states that the sum of “The measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles that are not adjacent to it; this is the exterior angle theorem. The sum of the measures of the three exterior angles (one for each vertex) of any triangle is $360$ degrees.” How can we prove this theorem?
So, for the triangle above, we need to prove why $angle CBD = angle BAC + angle BCA$.
asked Apr 10, 2022 4:08
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In the triangle $triangle ABC$, we know that $ABD$ is a straight line. So $angle ABC = 180^circ - angle CBD$. From the angle sum property of triangles we can infer that $angle BAC + angle ABC + angle BCA = 180^circ$ or $angle ABC = 180^circ-(angle BAC + angle BCA)$. Therefore:
$$angle ABC = 180^circ - angle CBD = 180^circ - (angle BAC + angle BCA)$$ $$Rightarrow - angle CBD = -(angle BAC + angle BCA)$$ $$Rightarrow - angle CBD times -1 = -(angle BAC + angle BCA) times -1$$ $$Rightarrow angle CBD = angle BAC + angle BCA$$
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In Elements I, 32 Euclid gives a visually satisfying proof of the exterior angle theorem by drawing $BE$ parallel to $AC$, and observing that $angle CBE=angle ACB$ (alternate interior angles) and $angle EBD=angle CAB$ (corresponding angles), making$$angle CBD=angle ACB+angle CAB$$ This theorem includes the further important result that the three angles of a triangle sum to $180^o$, or "two right angles" as Euclid says.
But if, as I suspect, the true intent of OP's question is, assuming the truth of the exterior angle theorem, prove that the sum of the three exterior angles of a triangle is $360^o$, then we can argue as follows.
Since by the exterior angle theorem$$angle CBD=angle BCA+angle BAC$$and$$angle ACE=angle CBA+angle BAC$$and$$angle BAF=angle CBA+angle BCA$$then by addition$$angle CBD+angle ACE+angle BAF= 2angle BCA+2angle CBA+2angle BAC=2cdot 180^o=360^o$$answered Apr 15, 2022 6:34
Edward PorcellaEdward Porcella
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Image credit: Desmos
Before we cover the exterior angle theorem, let's review a few definitions.
- Adjacent angles: angles that share a side and a vertex (ex., BCA and DCA)
- Supplementary angles: two angles that add to 180°
- Interior angles: the angles inside a triangle
- Exterior angles: angles formed between a side of a shape and a line that extends from the next side
We'll use the above triangle to demonstrate the exterior angle theorem's principles:
Breaking Down the Exterior Angle Theorem
Let's look how the exterior angle theorem works. First, let’s review the angle sum theorem, which states that the interior angles of a triangle equal 180°.
Image credit: Desmos
In the above triangle ECD, the exterior angle of DEF and its adjacent interior angle CED are linear pairs. That means together, they form a straight line and equal 180°.
Because these two adjacent angles add to 180° and the interior measures of the angles of a triangle also equal 180°, the sum of the remote interior angles ECD and CDE must equal the measure of exterior angle DEF.
Next, we'll use this knowledge to find angle measurements.
Applying the Exterior Angle Theorem
Let's use the exterior angle theorem in the triangle below:
Image credit: Desmos
Since we know that the angle EST = 125° and the adjacent interior angle TSU is its supplementary angle, let's solve for the measure of this interior angle:
Now let's use the second part of the exterior angle theorem: The exterior angle equals the sum of the remote interior angles. We'll follow this logic and find the remote interior angle TUS by subtracting STU from EST:
Understanding Exterior and Interior Angles
The exterior angle theorem states that:
- The measure of an exterior angle of a triangle is supplementary to its adjacent interior angle.
- The sum of the remote interior angles must equal the measure of the exterior angle of the triangle.
This theorem can help you solve for missing angles and understand the relationship between exterior and interior angles within a triangle.
More Math Homework Help:
- Vertical Angle Theorem: What It Is and How to Use It
- How to Find the Measure of an Angle in a Triangle: 3 Methods
- How to Recognize Adjacent Supplementary Angles
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 states that the measure of an 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.What theorem justifies the statement the measure of an exterior angle of a triangle is greater than the measure of either of its remote interior angle?
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 is the exterior angle sum theorem states?
Exterior Angle Sum Theorem states that the exterior angles of any polygon will always add up to 360 degrees. Tải thêm tài liệu liên quan đến nội dung bài viết Which theorem states that the measure of the exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles of the triangle *?
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