Geometry Angles of Polygons
polygon
- a closed, two-dimensional shape with straight sides
Number of Sides Name
3 triangle 4 quadrilateral 5 pentagon 6 hexagon 7 heptagon 8 octagon 9 nonagon 10 decagon 12 dodecagon n n-gon
Solution
An octagon has eight sides. This means it can be divided into 8 - 2 = 6 triangles, with the sum of the interior angles of each triangle equaling 180°. Sum of angles====(n−2) * 180°(8−2) * 180°6 * 180°1080°Sum of angles=(n-2) * 180°=(8-2) * 180°=6 * 180°=1080° The sum of the measures of the interior angles of a convex octagon is 1080°. Narrative: Explain
Angles in a triangle
the sum of the measures of the interior angles of any triangle is always the same: 180°.
Check: To check your answers, make sure that the sum of the interior angle measures equals 720°.
150° + 130° + 135° + 90° + 125° + 90° 720° == 720° 720°
A diagonal
A diagonal can be used to divide a convex quadrilateral (a four-sided polygon) into two triangles. The sum of the interior angles of each triangle is 180°, so the sum of the measures of the interior angles of the quadrilateral is 2 × 180° = 360°2 × 180° = 360°.
what is a polygon?
A polygon is a closed, two-dimensional shape with straight sides. Polygons are named according to the number of sides they have. As noted in the following table, a polygon with 3 sides is a triangle. A polygon with 4 sides is a quadrilateral, and so on.
Example 2 Solution
A quadrilateral has four sides. The sum of these angle measures is 360°, so x + (2x + 3) + 127 + 125 = 360 Sum of the measures of the interior angles of a quadrilateral is 360°. 3x + 255 = 360 3x + 255 = 360 Combine like terms. 3x = 1053x = 105 Subtract 255 from both sides. x = 35 Divide both sides by 3.
Example 1
Find the sum of the measures of the interior angles of a convex octagon.
Example 1
Given △ABC as shown in the following figure, find m∠4m m∠6m and m∠8m if m∠A = 50 m∠B = 42m and m∠C = 88m
Linear Pair Theorem
If two angles form a linear pair, then the angles are supplementary.
a diagonal of a polygon.
Keep in mind that a segment joining two nonconsecutive vertices of a convex polygon is called a diagonal of a polygon.
Solution
Since each interior angle creates a linear pair with an adjacent exterior angle, you know their sum will be 180°. Set each linear pair equal to 180 and solve for the unknown angle measure.
The measure of an interior angle of a regular polygon is 144°144°. Find the number of sides of this polygon.
Since this is a regular polygon, all angles have the same measure. You will use the corollary you used in Example 4 and set it equal to 144 (for 144°144°).
Example 3
Solve for the value of x in the figure. Then use x to find m∠Am∠A and m∠Bm∠B.
Example 2
Solve for the value of x in the following figure. Then use that value of x to find m∠C and m<D.
hexagon
The hexagon is successfully divided into four triangles, so the sum of the measures of the interior angles of the hexagon is 4 * 180° = 720°4 * 180° = 720°.
Regular Polygon Interior Angle Corollary
The measure of each interior angle of a regular n-gon equals (n − 2) × 180°/n
Polygon Exterior Angle Sum Theorem
The sum of the measures of the exterior angles of a convex polygon, one exterior angle at each vertex, is 360°.
Polygon Interior Angle-Sum Theorem
The sum of the measures of the interior angles of a convex n-gon is (n− 2) * 180°n- 2 * 180°.
Interior Angle Sum of a Convex Quadrilateral Theorem
The sum of the measures of the interior angles of any quadrilateral is 360°. m∠1 + m∠2 + m∠3 + m∠4 = 360°
Triangle Angle-Sum Theorem
The sum of the measures of the interior angles of any triangle is 180° m∠1 + m∠2 + m∠3 = 180°
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There are always two fewer triangles than the number of sides. As the number of sides increases by one, the number of interior triangles increases by one.
n-gon
This pattern can be generalized to any number of sides: An n-gon (a polygon with n sides) can be divided into n - 2 triangles. The sum of the interior angles is (n − 2) * 180°n - 2 * 180°.
Use x = 35 to find m∠Cm∠C and m∠Dm∠D.
m∠D = x° = 35° m∠C = (2x + 3)° = (2 * 35 + 3)° = 73° Check: To check your answers, make sure that m∠A + m∠B + m∠C + m∠D = 360° 127° + 125° +73° + 35° 360° == 360° 360°127° + 125° +73° + 35° = 360° 360° = 360°
Interior Angles of Polygons
the sum of the measures of the interior angles of a convex quadrliteral is always 2 × 180° = 360°2 × 180° = 360°
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(n − 2) * 180° n = 144 Set the formula equal to 144 n * (n − 2) * 180° n = n * 144 Multiply both sides by n. (n − 2) * 180 = 144n Simplify. 180n − 360 = 144n Use the Distributive Property. 36n = 360 Subtract 144n and then add 360 to both sides. n = 10 Divide both sides by 36. The regular polygon has 10 sides. To check, let n =10n =10 in the Corollary formula. The result will be 144 for 144°144°.
explanation
(x + 20) + x + 135 + 90 + 125 + 90 = 720 Sum of the measures of the interior angles is 720°. 2x + 460 = 720 Combine like terms. x = 260 Subtract 460 from both sides. x = 130 Divide both sides by 2 Use x = 130, then m∠B = x° = 130°m∠B = x° = 130°, and m∠A = (x + 20)° = (130 + 20)° = 150°m∠A = x + 20° = 130 + 20° = 150°
regular polygon
- a polygon whose sides are all the same length and angles are all the same measure
diagonal
- a segment joining two nonconsecutive vertices of a convex polygon
vertices
- noncollinear points connected by sides of a polygon
convex polygon
-a polygon is convex if no line containing a side contains a point within the interior of the polygon
interior angle
-an angle on the inside of a polygon that is formed by two sides of the polygon
Solution
This polygon has six sides, so it is a hexagon. The sum of the measures of the interior angles of any convex hexagon is the sum of angles = (6 − 2) * 180° = 4 * 180° = 720°= 6 - 2 * 180° = 4 * 180° = 720°. To find x, add the interior angle measures of the polygon and set them equal to 720°. Remember that an angle with a right angle symbol has a measure of 90°.
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Using the same method, how many triangles would a convex hexagon (six-sided polygon) be divided into? There are two fewer triangles than the number of sides, so there are 6 - 2 = 4 triangles. Now determine the sum of the measures of the interior angles.
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You can use diagonals to divide a convex pentagon (five-sided polygon) into triangles. This will create three triangles. The sum of the angles of each triangle is 180°. The pentagon has been divided into three triangles, so the sum of the measures of the interior angles of the pentagon is 3 * 180° = 540°.
regular polygon
a polygon whose sides are all the same length and angles are all the same measure.
