We will check each option by finding the sum of all three angles. Interior angle + Exterior angle = 180° Exterior angle = 180°-144° Therefore, the exterior angle is 36° The formula to find the number of sides of a regular polygon is as follows: Number of Sides of a Regular Polygon = 360° / Magnitude of each exterior angle. 2 Using the Polygon Angle-Sum Theorem As I said before, the main application of the polygon angle-sum theorem is for angle chasing problems. Polygon: Interior and Exterior Angles. That is, Interior angle + Exterior Angle = 180 ° Then, we have. Since the 65 degrees angle, the angle x, and the 30 degrees angle make a straight line together, the sum must be 180 degrees Since, 65 + angle x + 30 = 180, angle x must be 85 This is not a proof yet. By using the triangle inequality theorem and the exterior angle theorem, you should have no trouble completing the inequality proof in the following practice question. Find the nmnbar of sides for each, a) 72° b) 40° 2) Find the measure of an interior and an exterior angle of a regular 46-gon. Theorem: The sum of the interior angles of a polygon with sides is degrees. If the sides of the convex polygon are increased or decreased, the sum of all of the exterior angle is still 360 degrees. Polygon: Interior and Exterior Angles. $$\angle D$$ is an exterior angle for the given triangle.. Find the sum of the measure of the angles of a 15-gon. The sum is $$35^{\circ}+45^{\circ}+90^{\circ}=170^{\circ}<180^{\circ}$$. You crawl to A 2 and turn an exterior angle, shown in red, and face A 3. 2. According to the Polygon Interior Angles Sum Theorem, the sum of the measures of interior angles of an n-sided convex polygon is (n−2)180. Here lies the magic with Cuemath. Sum of the measures of exterior angles = Sum of the measures of linear pairs − Sum of the measures of interior angles. $$\therefore$$ The fourth option is correct. right A corollary of the Triangle Sum Theorem states that a triangle can contain no more than one _____ angle or obtuse angle. A pentagon has 5 interior angles, so it has 5 interior-exterior angle pairs. This mini-lesson targeted the fascinating concept of the Angle Sum Theorem. In several high school treatments of geometry, the term "exterior angle theorem" has been applied to a different result, namely the portion of Proposition 1.32 which states that the m $$a=65^{\circ}, b=115^{\circ}$$ and $$c=25^{\circ}$$. First, use the Polygon Angle Sum Theorem to find the sum of the interior angles: n = 9 What Is the Definition of Angle Sum Theorem? According to the Polygon Exterior Angles Sum Theorem, the sum of the measures of exterior angles of convex polygon, having one angle at each vertex is 360. Then there are non-adjacent vertices to vertex . Exterior Angle Theorem : The exterior angle theorem states that the measure of an exterior angle is equal to the sum of the measures of the two remote interior angles of the triangle. Click to see full answer let EA = external angle of that polygon polygon exterior angle sum theorem states that the sum of the exterior angles of any polygon is 360 degrees. Determine the sum of the exterior angles for each of the figures. Did you notice that all three angles constitute one straight angle? A More Formal Proof. The sum of all angles of a triangle is $$180^{\circ}$$. = 180 n − 180 ( n − 2) = 180 n − 180 n + 360 = 360. We can find the value of $$b$$ by using the definition of a linear pair. Create Class; Polygon: Interior and Exterior Angles. 354) Now, let’s consider exterior angles of a polygon. Before we carry on with our proof, let us mention that the sum of the exterior angles of an n-sided convex polygon = 360 ° I would like to call this the Spider Theorem. You can derive the exterior angle theorem with the help of the information that. The sum of all exterior angles of a convex polygon is equal to $$360^{\circ}$$. If we observe a convex polygon, then the sum of the exterior angle present at each vertex will be 360°. Draw three copies of one triangle on a piece of paper. Here are three proofs for the sum of angles of triangles. We know that the sum of the angles of a triangle adds up to 180°. Polygon Angles 1. Theorem. Inscribed angle theorem proof. Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic. The goal of the Polygon Interior Angle Sum Conjecture activity is for students to conjecture about the interior angle sum of any n-gon. The exterior angle of a given triangle is formed when a side is extended outwards. Be it problems, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in. So, we all know that a triangle is a 3-sided figure with three interior angles. The math journey around Angle Sum Theorem starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Here is the proof of the Exterior Angle Theorem. right A corollary of the Triangle Sum Theorem states that a triangle can contain no more than one _____ angle or obtuse angle. In the figures below, you will notice that exterior angles have been drawn from each vertex of the polygon. Alternate Interior Angles Draw Letter Z Alternate Interior Angles Interior And Exterior Angles Math Help . 2.1 MATHCOUNTS 2015 Chapter Sprint Problem 30 For positive integers n and m, each exterior angle of a regular n-sided polygon is 45 degrees larger than each exterior angle of a regular m-sided polygon. The exterior angle theorem states that the exterior angle of the given triangle is equal to the sum total of its opposite interior angles. Proof: Assume a polygon has sides. Since two angles measure the same, it is an isosceles triangle. The exterior angle of a given triangle is formed when a side is extended outwards. \begin{align}\angle PSR+\angle PRS+\angle SPR&=180^{\circ}\\115^{\circ}+40^{\circ}+c&=180^{\circ}\\155^{\circ}+c&=180^{\circ}\\c&=25^{\circ}\end{align}. Then, by exterior angle sum theorem, we have $$\angle 1+\angle 2=\angle 4$$. 2. Let us consider a polygon which has n number of sides. Sum of Interior Angles of Polygons. From the proof, you can see that this theorem is a combination of the Triangle Sum Theorem and the Linear Pair Postulate. Measure of Each Interior Angle: the measure of each interior angle of a regular n-gon. Find the nmnbar of sides for each, a) 72° b) 40° 2) Find the measure of an interior and an exterior angle of a regular 46-gon. Each exterior angle is paired with a corresponding interior angle, and each of these pairs sums to 180° (they are supplementary). Alternate Interior Angles Draw Letter Z Alternate Interior Angles Interior And Exterior Angles Math Help . This is the Corollary to the Polygon Angle-Sum Theorem. The sum is always 360. Inscribed angles. Proof 2 uses the exterior angle theorem. WORKSHEET: Angles of Polygons - Review PERIOD: DATE: USING THE INTERIOR & EXTERIOR ANGLE SUM THEOREMS 1) The measure of one exterior angle of a regular polygon is given. Polygon Exterior Angle Sum Theorem If we consider that a polygon is a convex polygon, the summation of its exterior angles at each vertex is equal to 360 degrees. An exterior angle of a triangle is formed when any side of a triangle is extended. Here are three proofs for the sum of angles of triangles. This concept teaches students the sum of exterior angles for any polygon and the relationship between exterior angles and remote interior angles in a triangle. The sum of the exterior angles is N. Sum of the measures of exterior angles = Sum of the measures of linear pairs − Sum of the measures of interior angles. Exterior Angle-Sum Theorem: sum of the exterior angles, one at each vertex, is 360⁰ EX 1: What is the sum of the interior angle measures of a pentagon? The sum is $$50^{\circ}+55^{\circ}+120^{\circ}=225^{\circ}>180^{\circ}$$. Next, we can figure out the sum of interior angles of any polygon by dividing the polygon into triangles. Theorem 3-9 Polygon Angle Sum Theorem. Therefore, the number of sides = 360° / 36° = 10 sides. Definition same side interior. The sum of the measures of the angles of a given polygon is 720. The Exterior Angle Theorem states that the sum of the remote interior angles is equal to the non-adjacent exterior angle. Here, $$\angle ACD$$ is an exterior angle of $$\Delta ABC$$. C. Angle 2 = 40 and Angle 3 = 20 D. Angle 2 = 140 and Angle 3 = 20 Use less than, equal to, or greater than to complete this statement: The sum of the measures of the exterior angles of a regular 9-gon, one at each vertex, is ____ the sum of the measures of the exterior angles of a … The sum is $$112^{\circ}+90^{\circ}+15^{\circ}=217^{\circ}>180^{\circ}$$. Author: Megan Milano. The remote interior angles are also termed as opposite interior … Author: pchou, Megan Milano. So, substituting in the preceding equation, we have. Rearrange these angles as shown below. Observe that in this 5-sided polygon, the sum of all exterior angles is 360∘ 360 ∘ by polygon angle sum theorem. So, $$\angle 1 + \angle 2+ \angle 3=180^{\circ}$$. Therefore, there the angle sum of a polygon with sides is given by the formula. The sum is $$95^{\circ}+45^{\circ}+40^{\circ}=180^{\circ}$$. Can you set up the proof based on the figure above? Proving that an inscribed angle is half of a central angle that subtends the same arc. 1) Exterior Angle Theorem: The measure of an The number of diagonals of any n-sided polygon is 1/2(n - 3)n. The sum of the exterior angles of a polygon is 360 degrees. exterior angle + interior angle = 180° So, for polygon with 'n' sides Let sum of all exterior angles be 'E', and sum of all interior angles be 'I'. Proof 1 uses the fact that the alternate interior angles formed by a transversal with two parallel lines are congruent. 5.07 Geometry The Triangle Sum Theorem 1 The sum of the interior angles of a triangle is 180 degrees. Cut out these two angles and place them together as shown below. Do these two angles cover $$\angle ACD$$ completely? Since the 65 degrees angle, the angle x, and the 30 degrees angle make a straight line together, the sum must be 180 degrees Since, 65 + angle x + 30 = 180, angle x must be 85 This is not a proof yet. Exterior angle sum theorem states that "an exterior angle of a triangle is equal to the sum of its two interior opposite angles.". USING THE INTERIOR & EXTERIOR ANGLE SUM THEOREMS 1) The measure of one exterior angle of a regular polygon is given. Ask subject matter experts 30 homework questions each month. A pentagon has 5 interior angles, so it has 5 interior-exterior angle pairs. The sum of all interior angles of a triangle is equal to $$180^{\circ}$$. This is the Corollary to the Polygon Angle-Sum Theorem. But the exterior angles sum to 360°. One of the acute angles of a right-angled triangle is $$45^{\circ}$$. Subscribe to bartleby learn! Theorem 6.2 POLYGON EXTERIOR ANGLES THEOREM - The sum of the measures of the exterior angles, one from each vertex, of a CONVEX polygon is 360 degrees. Exterior Angles of Polygons. The angles on the straight line add up to 180° Each exterior angle is paired with a corresponding interior angle, and each of these pairs sums to 180° (they are supplementary). The goal of the Polygon Interior Angle Sum Conjecture activity is for students to conjecture about the interior angle sum of any n-gon. The sum of the interior angles of any triangle is 180°. How many sides does the polygon have? 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. 3. The marked angles are called the exterior angles of the pentagon. He is trying to figure out the measurements of all angles of a roof which is in the form of a triangle. In $$\Delta ABC$$, $$\angle A + \angle B+ \angle C=180^{\circ}$$. 180(n – 2) + exterior angle sum = 180n. (Use n to represent the number of sides the polygon has.) Every angle in the interior of the polygon forms a linear pair with its exterior angle. The same side interior angles are also known as co interior angles. If a polygon does have an angle that points in, it is called concave, and this theorem does not apply. Determine the sum of the exterior angles for each of … The sum of all the internal angles of a simple polygon is 180 n 2 where n is the number of sides. Can you help him to figure out the measurement of the third angle? Leading to solving more challenging problems involving many relationships; straight, triangle, opposite and exterior angles. This is a fundamental result in absolute geometry because its proof does not depend upon the parallel postulate. Although you know that sum of the exterior angles is 360, you can only use formula to find a single exterior angle if the polygon is regular! Google Classroom Facebook Twitter. So, $$\angle 1+\angle 2+\angle 3=180^{\circ}$$. The Polygon Exterior Angle sum Theorem states that the sum of the measures of the exterior angles, one angle at each vertex, of a convex polygon is _____. The exterior angle of a regular n-sided polygon is 360°/n. The sum of 3 angles of a triangle is $$180^{\circ}$$. Exterior Angle Theorem – Explanation & Examples. Select/type your answer and click the "Check Answer" button to see the result. The same side interior angles are also known as co interior angles. From the proof, you can see that this theorem is a combination of the Triangle Sum Theorem and the Linear Pair Postulate. Click here if you need a proof of the Triangle Sum Theorem. Let $$\angle 1, \angle 2$$, and $$\angle 3$$ be the angles of $$\Delta ABC$$. Theorem for Exterior Angles Sum of a Polygon. Sum of exterior angles of a polygon. Sum of Interior Angles of Polygons. \begin{align}\angle PSR+\angle PSQ&=180^{\circ}\\b+65^{\circ}&=180^{\circ}\\b&=115^{\circ}\end{align}. If the sides of the convex polygon are increased or decreased, the sum of all of the exterior angle … ... All you have to remember is kind of cave in words And so, what we just did is applied to any exterior angle of any convex polygon. Theorem. Sum of exterior angles of a polygon. In the first option, we have angles $$50^{\circ},55^{\circ}$$, and $$120^{\circ}$$. $$\angle A$$ and $$\angle B$$ are the two opposite interior angles of $$\angle ACD$$. Polygon: Interior and Exterior Angles. Create Class; Polygon: Interior and Exterior Angles. The polygon exterior angle sum theorem states that "the sum of all exterior angles of a convex polygon is equal to 360∘ 360 ∘." The polygon exterior angle sum theorem states that "the sum of all exterior angles of a convex polygon is equal to $$360^{\circ}$$.". Here are a few activities for you to practice. Ms Amy asked her students which of the following can be the angles of a triangle? Create Class; Polygon: Interior and Exterior Angles. CCSS.Math: HSG.C.A.2. So, we can say that $$\angle ACD=\angle A+\angle B$$. This just shows that it works for one specific example Proof of the angle sum theorem: Topic: Angles. x + 50° = 92° (sum of opposite interior angles = exterior angle) x = 92° – 50° = 42° y + 92° = 180° (interior angle + adjacent exterior angle = 180°.) Formula for sum of exterior angles: The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360°. In other words, all of the interior angles of the polygon must have a measure of no more than 180° for this theorem … The angle sum of any n-sided polygon is 180(n - 2) degrees. So, only the fourth option gives the sum of $$180^{\circ}$$. Here is the proof of the Exterior Angle Theorem. The sum is always 360.Geometric proof: When all of the angles of a convex polygon converge, or pushed together, they form one angle called a perigon angle, which measures 360 degrees. Interior and exterior angles in regular polygons. Solution: x + 24° + 32° = 180° (sum of angles is 180°) x + 56° = 180° x = 180° – 56° = 124° Worksheet 1, Worksheet 2 using Triangle Sum Theorem Inscribed angles. Plus, you’ll have access to millions of step-by-step textbook answers. You will get to learn about the triangle angle sum theorem definition, exterior angle sum theorem, polygon exterior angle sum theorem, polygon angle sum theorem, and discover other interesting aspects of it. Use (n 2)180 . Now it's the time where we should see the sum of exterior angles of a polygon proof. At Cuemath, our team of math experts are dedicated to making learning fun for our favorite readers, the students! 6 Solving problems involving exterior angles. The sum of the exterior angles of a triangle is 360 degrees. The Exterior Angle Theorem states that the sum of the remote interior angles is equal to the non-adjacent exterior angle. To answer this, you need to understand the angle. Now it's the time where we should see the sum of exterior angles of a polygon proof. 11 Polygon Angle Sum. Proof 2 uses the exterior angle theorem. In general, this means that in a polygon with n sides. Measure of a Single Exterior Angle Formula to find 1 angle of a regular convex polygon of n sides = Exterior Angle Theorem states that in a triangle, the measure of an exterior angle is equal to the sum of the two remote interior angles. let EA = external angle of that polygon polygon exterior angle sum theorem states that the sum of the exterior angles of any polygon is 360 degrees. Again observe that these three angles constitute a straight angle. Polygon Angle-Sum Theorem: sum of the interior angles of an n-gon. This just shows that it works for one specific example Proof of the angle sum theorem: Polygon: Interior and Exterior Angles. But there exist other angles outside the triangle which we call exterior angles.. We know that in a triangle, the sum of all … Sum of exterior angles of a polygon. In any triangle, the sum of the three angles is $$180^{\circ}$$. You can use the exterior angle theorem to prove that the sum of the measures of the three angles of a triangle is 180 degrees. 354) Now, let’s consider exterior angles of a polygon. You can use the exterior angle theorem to prove that the sum of the measures of the three angles of a triangle is 180 degrees. You can check out the interactive simulations to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page. Angle sum theorem holds for all types of triangles. Sal demonstrates how the the sum of the exterior angles of a convex polygon is 360 degrees. In the figures below, you will notice that exterior angles have been drawn from each vertex of the polygon. In geometry, the triangle inequality theorem states that when you add the lengths of any two sides of a triangle, their sum will be greater that the length of the third side. He knows one angle is of $$45^{\circ}$$ and the other is a right angle. So, the angle sum theorem formula can be given as: Let us perform two activities to understand the angle sum theorem. To answer this, you need to understand the angle sum theorem, which is a remarkable property of a triangle and connects all its three angles. In $$\Delta PQS$$, we will apply the triangle angle sum theorem to find the value of $$c$$. For any polygon: 180-interior angle = exterior angle and the exterior angles of any polygon add up to 360 degrees. Discovery and investigation (through measuring) of Theorem 6: Each exterior angle of a triangle is equal to the sum of the interior opposite angles. Polygon Exterior Angle Sum Theorem If we consider that a polygon is a convex polygon, the summation of its exterior angles at each vertex is equal to 360 degrees. From the picture above, this means that . The sum of the measures of the angles in a polygon ; is (n 2)180. \begin{align} \text{angle}_3 &=180^{\circ}-(90^{\circ} +45^{\circ}) \\ &= 45^{\circ}\end{align}. The Exterior Angle Theorem (Euclid I.16), "In any triangle, if one of the sides is produced, then the exterior angle is greater than either of the interior and opposite angles," is one of the cornerstones of elementary geometry.In many contemporary high-school texts, the Exterior Angle Theorem appears as a corollary of the famous result (equivalent to … Exterior Angles of Polygons. Click Create Assignment to assign this modality to your LMS. Done in a way that is not only relatable and easy to grasp, but will also stay with them forever. The central angles of a regular polygon are congruent. Email. Practice: Inscribed angles. Draw any triangle on a piece of paper. Students will see that they can use diagonals to divide an n-sided polygon into (n-2) triangles and use the triangle sum theorem to justify why the interior angle sum is (n-2)(180).They will also make connections to an alternative way to determine the interior … Thus, the sum of the measures of exterior angles of a convex polygon is 360. We have moved all content for this concept to for better organization. The sum of measures of linear pair is 180. In any polygon, the sum of an interior angle and its corresponding exterior angle is 180 °. Polygon: Interior and Exterior Angles. In order to find the sum of interior angles of a polygon, we should multiply the number of triangles in the polygon by 180°. Definition same side interior. But the interior angle sum = 180(n – 2). 7.1 Interior and Exterior Angles Date: Triangle Sum Theorem: Proof: Given: ∆ , || Prove: ∠1+∠2+∠3=180° When you extend the sides of a polygon, the original angles may be called interior angles and the angles that form linear pairs with the interior angles are the exterior angles. Example 1 Determine the unknown angle measures. Thus, the sum of interior angles of a polygon can be calculated with the formula: S = ( n − 2) × 180°. You can derive the exterior angle theorem with the help of the information that. In this mini-lesson, we will explore the world of the angle sum theorem. Apply the Exterior Angles Theorems. Polygon Angle Sum Theorem The sum of the measures of the interior angles of a convex polygon with n sides is (n – 2) 180°. In the fourth option, we have angles $$95^{\circ}, 45^{\circ}$$, and $$40^{\circ}$$. The sum of the interior angles of any triangle is 180°. But the exterior angles sum to 360°. (pg. (pg. It should also be noted that the sum of exterior angles of a polygon is 360° 3. 'What Is The Polygon Exterior Angle Sum Theorem Quora May 8th, 2018 - The Sum Of The Exterior Angles Of A Polygon Is 360° You Can Find An Illustration Of It At Polygon Exterior Angle Sum Theorem' 'Polygon Angle Sum Theorem YouTube April 28th, 2018 - Polygon Angle Sum Theorem Regular Polygons Want music and videos with zero ads Get YouTube Red' The Triangle Sum Theorem is also called the Triangle Angle Sum Theorem or Angle Sum Theorem. $$\angle 4$$ and $$\angle 3$$ form a pair of supplementary angles because it is a linear pair. Consider, for instance, the pentagon pictured below. which means that the exterior angle sum = 180n – 180(n – 2) = 360 degrees. The exterior angle theorem states that the exterior angle of the given triangle is equal to the sum total of its opposite interior angles. Can you find the missing angles $$a$$, $$b$$, and $$c$$? Can you set up the proof based on the figure above? Observe that in this 5-sided polygon, the sum of all exterior angles is $$360^{\circ}$$ by polygon angle sum theorem. Thus, the sum of the measures of exterior angles of a convex polygon is 360. arrow_back. This is a fundamental result in absolute geometry because its proof does not depend upon the parallel postulate.. Adding $$\angle 3$$ on both sides of this equation, we get $$\angle 1+\angle 2+\angle 3=\angle 4+\angle 3$$. Interactive Questions on Angle Sum Theorem, $\angle A + \angle B+ \angle C=180^{\circ}$. Exterior Angle Theorem states that in a triangle, the measure of an exterior angle is equal to the sum of the two remote interior angles. You can visualize this activity using the simulation below. The radii of a regular polygon bisect the interior angles. The sum of the measures of the interior angles of a convex polygon with 'n' sides is (n - 2)180 degrees Polygon Exterior Angle Sum Theorem The sum of the measures of the exterior angles, one angle at each vertex, of a convex polygon What this means is just that the polygon cannot have angles that point in. E+I= n × 180° E =n×180° - I Sum of interior angles is (n-2)×180° E = n × 180° - (n -2) × 180°. x° + Exterior Angle = 180 ° 110 ° + Exterior angle = 180 ° Exterior angle = 70 ° So, the measure of each exterior angle corresponding to x ° in the above polygon is 70 °. In the second option, we have angles $$112^{\circ}, 90^{\circ}$$, and $$15^{\circ}$$. 3. For the nonagon shown, find the unknown angle measure x°. The angle sum theorem can be found using the statement "The sum of all interior angles of a triangle is equal to $$180^{\circ}$$.". 12 Using Polygon Angle-Sum Theorem \begin{align}\angle PQS+\angle QPS+\angle PSQ&=180^{\circ}\\60^{\circ}+55^{\circ}+a&=180^{\circ}\\115^{\circ}+a&=180^{\circ}\\a&=65^{\circ}\end{align}. The angles on the straight line add up to 180° Choose an arbitrary vertex, say vertex . The angle sum property of a triangle states that the sum of the three angles is $$180^{\circ}$$. Since two angles measure the same, it is an. One These pairs total 5*180=900°. The Polygon Exterior Angle sum Theorem states that the sum of the measures of the exterior angles, one angle at each vertex, of a convex polygon is _____. Triangle Angle Sum Theorem Proof. Imagine you are a spider and you are now in the point A 1 and facing A 2. The sum of all exterior angles of a triangle is equal to $$360^{\circ}$$. These pairs total 5*180=900°. Following Theorem will explain the exterior angle sum of a polygon: Proof. Triangle Angle Sum Theorem Proof. Topic: Angles, Polygons. The proof of the Polygon Exterior Angles Sum Theorem. Sum of Interior Angles of Polygons. Scott E. Brodie August 14, 2000. State a the Corollary to Theorem 6.2 - The Corollary to Theorem 6.2 - the measure of each exterior angle of a regular n-gon (n is the number of sides a polygon has) is 1/n(360 degrees). The angle sum theorem for quadrilaterals is that the sum of all measures of angles of the quadrilateral is $$360^{\circ}$$. The marked angles are called the exterior angles of the pentagon. Arrange these triangles as shown below. Exterior Angles of Polygons. In $$\Delta PQS$$, we will apply the triangle angle sum theorem to find the value of $$a$$. What is the formula for an exterior angle sum theorem? Proof 1 uses the fact that the alternate interior angles formed by a transversal with two parallel lines are congruent. Take a piece of paper and draw a triangle ABC on it. sum theorem, which is a remarkable property of a triangle and connects all its three angles. I Am a bit confused. The sum of all angles of a triangle is $$180^{\circ}$$ because one exterior angle of the triangle is equal to the sum of opposite interior angles of the triangle. A quick proof of the polygon exterior angle sum theorem using the linear pair postulate and the polygon interior angle sum theorem. 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. 1. The sum of all the internal angles of a simple polygon is 180 n 2 where n is the number of sides. interior angle sum* + exterior angle sum = 180n . Identify the type of triangle thus formed. Hence, the polygon has 10 sides. From the picture above, this means that. Example: Find the value of x in the following triangle. 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