Geometry Tutor Worksheet 8 Introduction To Polygons
Geometry Tutor Worksheet 8 Introduction To Polygons
1 © MathTutorDVD.com
Geometry Tutor - Worksheet 8 – Introduction to Polygons 1. Name this polygon.
2. Name this polygon.
3. Is this polygon convex or concave?
2 © MathTutorDVD.com
4. Is this polygon convex or concave?
5. What is the sum of the interior angles of a hexagon?
6. What is the sum of the interior angles of an octagon?
7. What is the sum of the interior angles of a polygon with 15 sides?
8. What is the sum of the interior angles of a nonagon?
3 © MathTutorDVD.com
9. What is the sum of the interior angles of a dodecagon?
10. What is the measure of ∠𝑅𝑈𝑉 in the figure below?
11. What is the measure of ∠𝑆𝑌𝑈 in the figure below?
4 © MathTutorDVD.com
12. What is the measure of ∠𝑃𝑇𝑈 in the figure below?
13. What is the measure of ∠𝑇𝑈𝑌 in the figure below?
5 © MathTutorDVD.com
14. What is the measure of ∠𝑅𝑃𝑄 in the figure below?
15. What is the measure of ∠𝐴𝐵𝐶 in the figure below?
16. What is the measure of ∠𝐹𝐺𝐻 in the figure below?
6 © MathTutorDVD.com
17. What is the measure of ∠𝑊𝑋𝑌 in the figure below?
18. What is the measure of ∠𝐷𝐸𝐹 in the figure below?
7 © MathTutorDVD.com
NOTE: For problems 19 – 24, understand that a regular polygon is a polygon with all sides having an equal length and all interior angles having an equal measure. 19. What is the measure of one interior angle of a regular hexagon?
20. What is the measure of one interior angle of a regular octagon?
21. What is the measure of one interior angle of a regular polygon with 15 sides?
22. What is the measure of one interior angle of a regular nonagon?
23. What is the measure of one interior angle of a regular dodecagon?
8 © MathTutorDVD.com
24. What is the measure of one interior angle of a regular decagon?
25. What is the measure of one exterior angle of a regular polygon with 15 sides?
26. What is the measure of one exterior angle of a regular nonagon?
27. What is the measure of one exterior angle of a regular dodecagon?
28. What is the measure of one exterior angle of a regular dodecagon?
9 © MathTutorDVD.com
Answers - Geometry Tutor - Worksheet 8 – Introduction to Polygons 1. Name this polygon.
The polygon has 9 sides so it is a nonagon. Answer: nonagon
2. Name this polygon.
The polygon has 8 sides so it is an octagon. Answer: octagon
3. Is this polygon convex or concave?
10 © MathTutorDVD.com
All vertices point outward, so the polygon is convex. Answer: convex
4. Is this polygon convex or concave?
At least one vertex points inward, so the polygon is concave. Answer: concave
5. What is the sum of the interior angles of a hexagon? The formula for the sum of the interior angles of a polygon is 𝑆 = 180(𝑛 − 2) where 𝑛 is the number of sides. A hexagon has 6 sides, so the sum is 𝑆 = 180(6 − 2) = 180(4) = 720 Answer: 720°
6. What is the sum of the interior angles of an octagon? The formula for the sum of the interior angles of a polygon is 𝑆 = 180(𝑛 − 2) where 𝑛 is the number of sides. An octagon has 8 sides, so the sum is 𝑆 = 180(8 − 2) = 180(6) = 1080 Answer: 1080°
11 © MathTutorDVD.com
7. What is the sum of the interior angles of a polygon with 15 sides? The formula for the sum of the interior angles of a polygon is 𝑆 = 180(𝑛 − 2) where 𝑛 is the number of sides. The polygon in this question has 15 sides, so the sum is 𝑆 = 180(15 − 2) = 180(13) = 2340 Answer: 2340°
8. What is the sum of the interior angles of a nonagon? The formula for the sum of the interior angles of a polygon is 𝑆 = 180(𝑛 − 2) where 𝑛 is the number of sides. A nonagon has 9 sides, so the sum is 𝑆 = 180(9 − 2) = 180(7) = 1260 Answer: 1260°
9. What is the sum of the interior angles of a dodecagon? The formula for the sum of the interior angles of a polygon is 𝑆 = 180(𝑛 − 2) where 𝑛 is the number of sides. A dodecagon has 12 sides, so the sum is 𝑆 = 180(12 − 2) = 180(10) = 1800 Answer: 1800°
10. What is the measure of ∠𝑅𝑈𝑉 in the figure below?
12 © MathTutorDVD.com
Begin with the fact that ∠𝑇𝑉𝑅 is supplementary to ∠𝑅𝑉𝑈 in the triangle, so 𝑚∠𝑅𝑉𝑈 = 60°. Now use the fact that the sum of the measures of the interior angles of a triangle is 180°, so 𝑚∠𝑅𝑉𝑈 + 𝑚∠𝑉𝑅𝑈 + 𝑚∠𝑅𝑈𝑉 = 180 60 + 50 + 𝑚∠𝑅𝑈𝑉 = 180 110 + 𝑚∠𝑅𝑈𝑉 = 180 𝑚∠𝑅𝑈𝑉 = 70 Answer: 70°
11. What is the measure of ∠𝑆𝑌𝑈 in the figure below?
Begin with the fact the sum of the measures of the interior angles of a triangle is 180°, so 𝑚∠𝑇𝑆𝑌 + 𝑚∠𝑆𝑇𝑌 + 𝑚∠𝑇𝑌𝑆 = 180 13 © MathTutorDVD.com
50 + 70 + 𝑚∠𝑇𝑌𝑆 = 180 120 + 𝑚∠𝑇𝑌𝑆 = 180 𝑚∠𝑇𝑌𝑆 = 60 Now, continue with the fact that ∠𝑇𝑌𝑆 is supplementary to ∠𝑆𝑌𝑈, so 𝑚∠𝑆𝑌𝑈 = 120°. Answer: 120°
12. What is the measure of ∠𝑃𝑇𝑈 in the figure below?
Begin with the fact that ∠𝑇𝑈𝑉 is supplementary to ∠𝑇𝑈𝑃 in the triangle, so 𝑚∠𝑇𝑈𝑃 = 65°. Now use the fact that the sum of the measures of the interior angles of a triangle is 180°, so 𝑚∠𝑇𝑈𝑃 + 𝑚∠𝑈𝑃𝑇 + 𝑚∠𝑃𝑇𝑈 = 180 65 + 50 + 𝑚∠𝑃𝑇𝑈 = 180 115 + 𝑚∠𝑃𝑇𝑈 = 180 𝑚∠𝑃𝑇𝑈 = 65 Answer: 65°
14 © MathTutorDVD.com
13. What is the measure of ∠𝑇𝑈𝑌 in the figure below?
Begin with the fact the sum of the measures of the interior angles of a triangle is 180°, so 𝑚∠𝑆𝑇𝑈 + 𝑚∠𝑇𝑆𝑈 + 𝑚∠𝑆𝑈𝑇 = 180 50 + 70 + 𝑚∠𝑆𝑈𝑇 = 180 120 + 𝑚∠𝑆𝑈𝑇 = 180 𝑚∠𝑆𝑈𝑇 = 60 Now, continue with the fact that ∠𝑆𝑈𝑇 is supplementary to ∠𝑇𝑈𝑌, so 𝑚∠𝑇𝑈𝑌 = 120°. Answer: 120°
14. What is the measure of ∠𝑅𝑃𝑄 in the figure below?
15 © MathTutorDVD.com
Begin with the fact the sum of the measures of the interior angles of a triangle is 180°, so 𝑚∠𝑃𝑅𝐺 + 𝑚∠𝑅𝐺𝑃 + 𝑚∠𝐺𝑃𝑅 = 180 95 + 35 + 𝑚∠𝐺𝑃𝑅 = 180 130 + 𝑚∠𝐺𝑃𝑅 = 180 𝑚∠𝐺𝑃𝑅 = 50 Now, continue with the fact that ∠𝐺𝑃𝑅 is supplementary to ∠𝑅𝑃𝑄, so 𝑚∠𝑅𝑃𝑄 = 130°. Answer: 130°
15. What is the measure of ∠𝐴𝐵𝐶 in the figure below?
Begin with the fact the sum of the measures of the interior angles of a triangle is 180°, so 𝑚∠𝐶𝐷𝐸 + 𝑚∠𝐷𝐸𝐶 + 𝑚∠𝐷𝐶𝐸 = 180 20 + 39 + 𝑚∠𝐷𝐶𝐸 = 180 59 + 𝑚∠𝐷𝐶𝐸 = 180 𝑚∠𝐷𝐶𝐸 = 121 Now, continue with the fact that ∠𝐷𝐶𝐸 is vertical to ∠𝐴𝐶𝐵, so 𝑚∠𝐴𝐶𝐵 = 121°. Continue again with the fact the sum of the measures of the interior angles of a triangle is 180°, so 16 © MathTutorDVD.com
𝑚∠𝐴𝐶𝐵 + 𝑚∠𝐶𝐴𝐵 + 𝑚∠𝐴𝐵𝐶 = 180 121 + 40 + 𝑚∠𝐴𝐵𝐶 = 180 161 + 𝑚∠𝐴𝐵𝐶 = 180 𝑚∠𝐴𝐵𝐶 = 19 Answer: 19°
16. What is the measure of ∠𝐹𝐺𝐻 in the figure below?
Begin with the fact the sum of the measures of the interior angles of a triangle is 180°, so 𝑚∠𝐹𝐷𝐸 + 𝑚∠𝐷𝐸𝐹 + 𝑚∠𝐸𝐹𝐷 = 180 60 + 65 + 𝑚∠𝐸𝐹𝐷 = 180 125 + 𝑚∠𝐸𝐹𝐷 = 180 𝑚∠𝐸𝐹𝐷 = 55 Now, continue with the fact that 𝑚∠𝐸𝐹𝐷 + 𝑚∠𝐸𝐹𝐻 + 𝑚∠𝐻𝐹𝐺 = 180 because together they make a straight angle, so 55 + 50 + 𝑚∠𝐻𝐹𝐺 = 180 105 + 𝑚∠𝐻𝐹𝐺 = 180 17 © MathTutorDVD.com
𝑚∠𝐻𝐹𝐺 = 75 Continue again with the fact the sum of the measures of the interior angles of a triangle is 180°, so 𝑚∠𝐻𝐹𝐺 + 𝑚∠𝐹𝐻𝐺 + 𝑚∠𝐻𝐺𝐹 = 180 75 + 35 + 𝑚∠𝐻𝐺𝐹 = 180 110 + 𝑚∠𝐻𝐺𝐹 = 180 𝑚∠𝐻𝐺𝐹 = 70 Answer: 70°
17. What is the measure of ∠𝑊𝑋𝑌 in the figure below?
Begin with the fact the sum of the measures of the interior angles of a triangle is 180°, so 𝑚∠𝑈𝑉𝑊 + 𝑚∠𝑉𝑈𝑊 + 𝑚∠𝑈𝑊𝑉 = 180 36 + 84 + 𝑚∠𝑈𝑊𝑉 = 180 120 + 𝑚∠𝑈𝑊𝑉 = 180 𝑚∠𝑈𝑊𝑉 = 60
18 © MathTutorDVD.com
Now, continue with the fact that 𝑚∠𝑈𝑊𝑉 + 𝑚∠𝑌𝑊𝑋 = 90 because together they make a right angle, so 60 + 𝑚∠𝑌𝑊𝑋 = 90 𝑚∠𝑌𝑊𝑋 = 30 Continue again with the fact the sum of the measures of the interior angles of a triangle is 180°, so 𝑚∠𝑊𝑌𝑋 + 𝑚∠𝑌𝑊𝑋 + 𝑚∠𝑊𝑋𝑌 = 180 86 + 30 + 𝑚∠𝑊𝑋𝑌 = 180 116 + 𝑚∠𝑊𝑋𝑌 = 180 𝑚∠𝑊𝑋𝑌 = 64 Answer: 64°
18. What is the measure of ∠𝐷𝐸𝐹 in the figure below?
Begin with the fact the sum of the measures of the interior angles of a triangle is 180°, so 𝑚∠𝐹𝐵𝐶 + 𝑚∠𝐵𝐶𝐹 + 𝑚∠𝐵𝐹𝐶 = 180 20 + 35 + 𝑚∠𝐵𝐹𝐶 = 180 55 + 𝑚∠𝐵𝐹𝐶 = 180 19 © MathTutorDVD.com
𝑚∠𝐵𝐹𝐶 = 125 Now, continue with the fact that ∠𝐵𝐹𝐶 is vertical to ∠𝐷𝐹𝐸, so 𝑚∠𝐷𝐹𝐸 = 125°. Continue with the fact that ∠𝐴𝐷𝐹 and ∠𝐸𝐷𝐹 are supplementary because together they make a straight angle, so 𝑚∠𝐸𝐷𝐹 = 24°. Continue again with the fact the sum of the measures of the interior angles of a triangle is 180°, so 𝑚∠𝐸𝐷𝐹 + 𝑚∠𝐷𝐹𝐸 + 𝑚∠𝐷𝐸𝐹 = 180 24 + 125 + 𝑚∠𝐷𝐸𝐹 = 180 149 + 𝑚∠𝐷𝐸𝐹 = 180 𝑚∠𝐷𝐸𝐹 = 31 Answer: 31°
NOTE: For problems 19 – 24, understand that a regular polygon is a polygon with all sides having an equal length and all interior angles having an equal measure. 19. What is the measure of one interior angle of a regular hexagon? The formula for the sum of the interior angles of a polygon is 𝑆 = 180(𝑛 − 2) where 𝑛 is the number of sides. It follows that in a regular polygon with 𝑛 sides that the measure of each side is 𝑀 =
180(𝑛−2) 𝑛
. A hexagon has 6 sides, so the
measure of each interior angle is 𝑀=
180(6 − 2) 180(4) 720 = = = 120 6 6 6
Answer: 120°
20. What is the measure of one interior angle of a regular octagon?
20 © MathTutorDVD.com
The formula for the sum of the interior angles of a polygon is 𝑆 = 180(𝑛 − 2) where 𝑛 is the number of sides. It follows that in a regular polygon with 𝑛 sides that the measure of each side is 𝑀 =
180(𝑛−2) 𝑛
. An octagon has 8 sides, so the
measure of each interior angle is 𝑀=
180(8 − 2) 180(6) 1080 = = = 135 8 8 8
Answer: 135°
21. What is the measure of one interior angle of a regular polygon with 15 sides? The formula for the sum of the interior angles of a polygon is 𝑆 = 180(𝑛 − 2) where 𝑛 is the number of sides. It follows that in a regular polygon with 𝑛 sides that the measure of each side is 𝑀 =
180(𝑛−2) 𝑛
. The polygon in this question has
15 sides, so the measure of each interior angle is 𝑀=
180(15 − 2) 180(13) 2340 = = = 156 15 15 15
Answer: 156°
22. What is the measure of one interior angle of a regular nonagon? The formula for the sum of the interior angles of a polygon is 𝑆 = 180(𝑛 − 2) where 𝑛 is the number of sides. It follows that in a regular polygon with 𝑛 sides that the measure of each side is 𝑀 =
180(𝑛−2) 𝑛
. A nonagon has 9 sides, so the
measure of each interior angle is 𝑀=
180(9 − 2) 180(7) 1260 = = = 140 9 9 9
Answer: 140°
21 © MathTutorDVD.com
23. What is the measure of one interior angle of a regular dodecagon? The formula for the sum of the interior angles of a polygon is 𝑆 = 180(𝑛 − 2) where 𝑛 is the number of sides. It follows that in a regular polygon with 𝑛 sides that the measure of each side is 𝑀 =
180(𝑛−2) 𝑛
. A dodecagon has 12 sides, so the
measure of each interior angle is 𝑀=
180(12 − 2) 180(10) 1800 = = = 150 12 12 12
Answer: 150°
24. What is the measure of one interior angle of a regular decagon? The formula for the sum of the interior angles of a polygon is 𝑆 = 180(𝑛 − 2) where 𝑛 is the number of sides. It follows that in a regular polygon with 𝑛 sides that the measure of each side is 𝑀 =
180(𝑛−2) 𝑛
. A decagon has 10 sides, so the
measure of each angle is 𝑀=
180(10 − 2) 180(8) 1440 = = = 144 10 10 10
Answer: 144°
25. What is the measure of one exterior angle of a regular polygon with 15 sides? The sum of the exterior angles of any polygon is 360°. It follows that the measure of one exterior angle of a regular polygon with 𝑛 sides is 𝑀 = this question has 15 sides, so 𝑀=
360 = 24 15
Answer: 24°
22 © MathTutorDVD.com
360 𝑛
. The polygon in
26. What is the measure of one exterior angle of a regular nonagon? The sum of the exterior angles of any polygon is 360°. It follows that the measure of one exterior angle of a regular polygon with 𝑛 sides is 𝑀 =
360 𝑛
. A nonagon has
9 sides, so 𝑀=
360 = 40 9
Answer: 40°
27. What is the measure of one exterior angle of a regular dodecagon? The sum of the exterior angles of any polygon is 360°. It follows that the measure of one exterior angle of a regular polygon with 𝑛 sides is 𝑀 =
360 𝑛
. A dodecagon
has 12 sides, so 𝑀=
360 = 30 12
Answer: 30°
28. What is the measure of one exterior angle of a regular decagon? The sum of the exterior angles of any polygon is 360°. It follows that the measure of one exterior angle of a regular polygon with 𝑛 sides is 𝑀 = 10 sides, so 𝑀=
360 = 36 10
Answer: 36°
23 © MathTutorDVD.com
360 𝑛
. A decagon has
1 © MathTutorDVD.com
Geometry Tutor - Worksheet 8 – Introduction to Polygons 1. Name this polygon.
2. Name this polygon.
3. Is this polygon convex or concave?
2 © MathTutorDVD.com
4. Is this polygon convex or concave?
5. What is the sum of the interior angles of a hexagon?
6. What is the sum of the interior angles of an octagon?
7. What is the sum of the interior angles of a polygon with 15 sides?
8. What is the sum of the interior angles of a nonagon?
3 © MathTutorDVD.com
9. What is the sum of the interior angles of a dodecagon?
10. What is the measure of ∠𝑅𝑈𝑉 in the figure below?
11. What is the measure of ∠𝑆𝑌𝑈 in the figure below?
4 © MathTutorDVD.com
12. What is the measure of ∠𝑃𝑇𝑈 in the figure below?
13. What is the measure of ∠𝑇𝑈𝑌 in the figure below?
5 © MathTutorDVD.com
14. What is the measure of ∠𝑅𝑃𝑄 in the figure below?
15. What is the measure of ∠𝐴𝐵𝐶 in the figure below?
16. What is the measure of ∠𝐹𝐺𝐻 in the figure below?
6 © MathTutorDVD.com
17. What is the measure of ∠𝑊𝑋𝑌 in the figure below?
18. What is the measure of ∠𝐷𝐸𝐹 in the figure below?
7 © MathTutorDVD.com
NOTE: For problems 19 – 24, understand that a regular polygon is a polygon with all sides having an equal length and all interior angles having an equal measure. 19. What is the measure of one interior angle of a regular hexagon?
20. What is the measure of one interior angle of a regular octagon?
21. What is the measure of one interior angle of a regular polygon with 15 sides?
22. What is the measure of one interior angle of a regular nonagon?
23. What is the measure of one interior angle of a regular dodecagon?
8 © MathTutorDVD.com
24. What is the measure of one interior angle of a regular decagon?
25. What is the measure of one exterior angle of a regular polygon with 15 sides?
26. What is the measure of one exterior angle of a regular nonagon?
27. What is the measure of one exterior angle of a regular dodecagon?
28. What is the measure of one exterior angle of a regular dodecagon?
9 © MathTutorDVD.com
Answers - Geometry Tutor - Worksheet 8 – Introduction to Polygons 1. Name this polygon.
The polygon has 9 sides so it is a nonagon. Answer: nonagon
2. Name this polygon.
The polygon has 8 sides so it is an octagon. Answer: octagon
3. Is this polygon convex or concave?
10 © MathTutorDVD.com
All vertices point outward, so the polygon is convex. Answer: convex
4. Is this polygon convex or concave?
At least one vertex points inward, so the polygon is concave. Answer: concave
5. What is the sum of the interior angles of a hexagon? The formula for the sum of the interior angles of a polygon is 𝑆 = 180(𝑛 − 2) where 𝑛 is the number of sides. A hexagon has 6 sides, so the sum is 𝑆 = 180(6 − 2) = 180(4) = 720 Answer: 720°
6. What is the sum of the interior angles of an octagon? The formula for the sum of the interior angles of a polygon is 𝑆 = 180(𝑛 − 2) where 𝑛 is the number of sides. An octagon has 8 sides, so the sum is 𝑆 = 180(8 − 2) = 180(6) = 1080 Answer: 1080°
11 © MathTutorDVD.com
7. What is the sum of the interior angles of a polygon with 15 sides? The formula for the sum of the interior angles of a polygon is 𝑆 = 180(𝑛 − 2) where 𝑛 is the number of sides. The polygon in this question has 15 sides, so the sum is 𝑆 = 180(15 − 2) = 180(13) = 2340 Answer: 2340°
8. What is the sum of the interior angles of a nonagon? The formula for the sum of the interior angles of a polygon is 𝑆 = 180(𝑛 − 2) where 𝑛 is the number of sides. A nonagon has 9 sides, so the sum is 𝑆 = 180(9 − 2) = 180(7) = 1260 Answer: 1260°
9. What is the sum of the interior angles of a dodecagon? The formula for the sum of the interior angles of a polygon is 𝑆 = 180(𝑛 − 2) where 𝑛 is the number of sides. A dodecagon has 12 sides, so the sum is 𝑆 = 180(12 − 2) = 180(10) = 1800 Answer: 1800°
10. What is the measure of ∠𝑅𝑈𝑉 in the figure below?
12 © MathTutorDVD.com
Begin with the fact that ∠𝑇𝑉𝑅 is supplementary to ∠𝑅𝑉𝑈 in the triangle, so 𝑚∠𝑅𝑉𝑈 = 60°. Now use the fact that the sum of the measures of the interior angles of a triangle is 180°, so 𝑚∠𝑅𝑉𝑈 + 𝑚∠𝑉𝑅𝑈 + 𝑚∠𝑅𝑈𝑉 = 180 60 + 50 + 𝑚∠𝑅𝑈𝑉 = 180 110 + 𝑚∠𝑅𝑈𝑉 = 180 𝑚∠𝑅𝑈𝑉 = 70 Answer: 70°
11. What is the measure of ∠𝑆𝑌𝑈 in the figure below?
Begin with the fact the sum of the measures of the interior angles of a triangle is 180°, so 𝑚∠𝑇𝑆𝑌 + 𝑚∠𝑆𝑇𝑌 + 𝑚∠𝑇𝑌𝑆 = 180 13 © MathTutorDVD.com
50 + 70 + 𝑚∠𝑇𝑌𝑆 = 180 120 + 𝑚∠𝑇𝑌𝑆 = 180 𝑚∠𝑇𝑌𝑆 = 60 Now, continue with the fact that ∠𝑇𝑌𝑆 is supplementary to ∠𝑆𝑌𝑈, so 𝑚∠𝑆𝑌𝑈 = 120°. Answer: 120°
12. What is the measure of ∠𝑃𝑇𝑈 in the figure below?
Begin with the fact that ∠𝑇𝑈𝑉 is supplementary to ∠𝑇𝑈𝑃 in the triangle, so 𝑚∠𝑇𝑈𝑃 = 65°. Now use the fact that the sum of the measures of the interior angles of a triangle is 180°, so 𝑚∠𝑇𝑈𝑃 + 𝑚∠𝑈𝑃𝑇 + 𝑚∠𝑃𝑇𝑈 = 180 65 + 50 + 𝑚∠𝑃𝑇𝑈 = 180 115 + 𝑚∠𝑃𝑇𝑈 = 180 𝑚∠𝑃𝑇𝑈 = 65 Answer: 65°
14 © MathTutorDVD.com
13. What is the measure of ∠𝑇𝑈𝑌 in the figure below?
Begin with the fact the sum of the measures of the interior angles of a triangle is 180°, so 𝑚∠𝑆𝑇𝑈 + 𝑚∠𝑇𝑆𝑈 + 𝑚∠𝑆𝑈𝑇 = 180 50 + 70 + 𝑚∠𝑆𝑈𝑇 = 180 120 + 𝑚∠𝑆𝑈𝑇 = 180 𝑚∠𝑆𝑈𝑇 = 60 Now, continue with the fact that ∠𝑆𝑈𝑇 is supplementary to ∠𝑇𝑈𝑌, so 𝑚∠𝑇𝑈𝑌 = 120°. Answer: 120°
14. What is the measure of ∠𝑅𝑃𝑄 in the figure below?
15 © MathTutorDVD.com
Begin with the fact the sum of the measures of the interior angles of a triangle is 180°, so 𝑚∠𝑃𝑅𝐺 + 𝑚∠𝑅𝐺𝑃 + 𝑚∠𝐺𝑃𝑅 = 180 95 + 35 + 𝑚∠𝐺𝑃𝑅 = 180 130 + 𝑚∠𝐺𝑃𝑅 = 180 𝑚∠𝐺𝑃𝑅 = 50 Now, continue with the fact that ∠𝐺𝑃𝑅 is supplementary to ∠𝑅𝑃𝑄, so 𝑚∠𝑅𝑃𝑄 = 130°. Answer: 130°
15. What is the measure of ∠𝐴𝐵𝐶 in the figure below?
Begin with the fact the sum of the measures of the interior angles of a triangle is 180°, so 𝑚∠𝐶𝐷𝐸 + 𝑚∠𝐷𝐸𝐶 + 𝑚∠𝐷𝐶𝐸 = 180 20 + 39 + 𝑚∠𝐷𝐶𝐸 = 180 59 + 𝑚∠𝐷𝐶𝐸 = 180 𝑚∠𝐷𝐶𝐸 = 121 Now, continue with the fact that ∠𝐷𝐶𝐸 is vertical to ∠𝐴𝐶𝐵, so 𝑚∠𝐴𝐶𝐵 = 121°. Continue again with the fact the sum of the measures of the interior angles of a triangle is 180°, so 16 © MathTutorDVD.com
𝑚∠𝐴𝐶𝐵 + 𝑚∠𝐶𝐴𝐵 + 𝑚∠𝐴𝐵𝐶 = 180 121 + 40 + 𝑚∠𝐴𝐵𝐶 = 180 161 + 𝑚∠𝐴𝐵𝐶 = 180 𝑚∠𝐴𝐵𝐶 = 19 Answer: 19°
16. What is the measure of ∠𝐹𝐺𝐻 in the figure below?
Begin with the fact the sum of the measures of the interior angles of a triangle is 180°, so 𝑚∠𝐹𝐷𝐸 + 𝑚∠𝐷𝐸𝐹 + 𝑚∠𝐸𝐹𝐷 = 180 60 + 65 + 𝑚∠𝐸𝐹𝐷 = 180 125 + 𝑚∠𝐸𝐹𝐷 = 180 𝑚∠𝐸𝐹𝐷 = 55 Now, continue with the fact that 𝑚∠𝐸𝐹𝐷 + 𝑚∠𝐸𝐹𝐻 + 𝑚∠𝐻𝐹𝐺 = 180 because together they make a straight angle, so 55 + 50 + 𝑚∠𝐻𝐹𝐺 = 180 105 + 𝑚∠𝐻𝐹𝐺 = 180 17 © MathTutorDVD.com
𝑚∠𝐻𝐹𝐺 = 75 Continue again with the fact the sum of the measures of the interior angles of a triangle is 180°, so 𝑚∠𝐻𝐹𝐺 + 𝑚∠𝐹𝐻𝐺 + 𝑚∠𝐻𝐺𝐹 = 180 75 + 35 + 𝑚∠𝐻𝐺𝐹 = 180 110 + 𝑚∠𝐻𝐺𝐹 = 180 𝑚∠𝐻𝐺𝐹 = 70 Answer: 70°
17. What is the measure of ∠𝑊𝑋𝑌 in the figure below?
Begin with the fact the sum of the measures of the interior angles of a triangle is 180°, so 𝑚∠𝑈𝑉𝑊 + 𝑚∠𝑉𝑈𝑊 + 𝑚∠𝑈𝑊𝑉 = 180 36 + 84 + 𝑚∠𝑈𝑊𝑉 = 180 120 + 𝑚∠𝑈𝑊𝑉 = 180 𝑚∠𝑈𝑊𝑉 = 60
18 © MathTutorDVD.com
Now, continue with the fact that 𝑚∠𝑈𝑊𝑉 + 𝑚∠𝑌𝑊𝑋 = 90 because together they make a right angle, so 60 + 𝑚∠𝑌𝑊𝑋 = 90 𝑚∠𝑌𝑊𝑋 = 30 Continue again with the fact the sum of the measures of the interior angles of a triangle is 180°, so 𝑚∠𝑊𝑌𝑋 + 𝑚∠𝑌𝑊𝑋 + 𝑚∠𝑊𝑋𝑌 = 180 86 + 30 + 𝑚∠𝑊𝑋𝑌 = 180 116 + 𝑚∠𝑊𝑋𝑌 = 180 𝑚∠𝑊𝑋𝑌 = 64 Answer: 64°
18. What is the measure of ∠𝐷𝐸𝐹 in the figure below?
Begin with the fact the sum of the measures of the interior angles of a triangle is 180°, so 𝑚∠𝐹𝐵𝐶 + 𝑚∠𝐵𝐶𝐹 + 𝑚∠𝐵𝐹𝐶 = 180 20 + 35 + 𝑚∠𝐵𝐹𝐶 = 180 55 + 𝑚∠𝐵𝐹𝐶 = 180 19 © MathTutorDVD.com
𝑚∠𝐵𝐹𝐶 = 125 Now, continue with the fact that ∠𝐵𝐹𝐶 is vertical to ∠𝐷𝐹𝐸, so 𝑚∠𝐷𝐹𝐸 = 125°. Continue with the fact that ∠𝐴𝐷𝐹 and ∠𝐸𝐷𝐹 are supplementary because together they make a straight angle, so 𝑚∠𝐸𝐷𝐹 = 24°. Continue again with the fact the sum of the measures of the interior angles of a triangle is 180°, so 𝑚∠𝐸𝐷𝐹 + 𝑚∠𝐷𝐹𝐸 + 𝑚∠𝐷𝐸𝐹 = 180 24 + 125 + 𝑚∠𝐷𝐸𝐹 = 180 149 + 𝑚∠𝐷𝐸𝐹 = 180 𝑚∠𝐷𝐸𝐹 = 31 Answer: 31°
NOTE: For problems 19 – 24, understand that a regular polygon is a polygon with all sides having an equal length and all interior angles having an equal measure. 19. What is the measure of one interior angle of a regular hexagon? The formula for the sum of the interior angles of a polygon is 𝑆 = 180(𝑛 − 2) where 𝑛 is the number of sides. It follows that in a regular polygon with 𝑛 sides that the measure of each side is 𝑀 =
180(𝑛−2) 𝑛
. A hexagon has 6 sides, so the
measure of each interior angle is 𝑀=
180(6 − 2) 180(4) 720 = = = 120 6 6 6
Answer: 120°
20. What is the measure of one interior angle of a regular octagon?
20 © MathTutorDVD.com
The formula for the sum of the interior angles of a polygon is 𝑆 = 180(𝑛 − 2) where 𝑛 is the number of sides. It follows that in a regular polygon with 𝑛 sides that the measure of each side is 𝑀 =
180(𝑛−2) 𝑛
. An octagon has 8 sides, so the
measure of each interior angle is 𝑀=
180(8 − 2) 180(6) 1080 = = = 135 8 8 8
Answer: 135°
21. What is the measure of one interior angle of a regular polygon with 15 sides? The formula for the sum of the interior angles of a polygon is 𝑆 = 180(𝑛 − 2) where 𝑛 is the number of sides. It follows that in a regular polygon with 𝑛 sides that the measure of each side is 𝑀 =
180(𝑛−2) 𝑛
. The polygon in this question has
15 sides, so the measure of each interior angle is 𝑀=
180(15 − 2) 180(13) 2340 = = = 156 15 15 15
Answer: 156°
22. What is the measure of one interior angle of a regular nonagon? The formula for the sum of the interior angles of a polygon is 𝑆 = 180(𝑛 − 2) where 𝑛 is the number of sides. It follows that in a regular polygon with 𝑛 sides that the measure of each side is 𝑀 =
180(𝑛−2) 𝑛
. A nonagon has 9 sides, so the
measure of each interior angle is 𝑀=
180(9 − 2) 180(7) 1260 = = = 140 9 9 9
Answer: 140°
21 © MathTutorDVD.com
23. What is the measure of one interior angle of a regular dodecagon? The formula for the sum of the interior angles of a polygon is 𝑆 = 180(𝑛 − 2) where 𝑛 is the number of sides. It follows that in a regular polygon with 𝑛 sides that the measure of each side is 𝑀 =
180(𝑛−2) 𝑛
. A dodecagon has 12 sides, so the
measure of each interior angle is 𝑀=
180(12 − 2) 180(10) 1800 = = = 150 12 12 12
Answer: 150°
24. What is the measure of one interior angle of a regular decagon? The formula for the sum of the interior angles of a polygon is 𝑆 = 180(𝑛 − 2) where 𝑛 is the number of sides. It follows that in a regular polygon with 𝑛 sides that the measure of each side is 𝑀 =
180(𝑛−2) 𝑛
. A decagon has 10 sides, so the
measure of each angle is 𝑀=
180(10 − 2) 180(8) 1440 = = = 144 10 10 10
Answer: 144°
25. What is the measure of one exterior angle of a regular polygon with 15 sides? The sum of the exterior angles of any polygon is 360°. It follows that the measure of one exterior angle of a regular polygon with 𝑛 sides is 𝑀 = this question has 15 sides, so 𝑀=
360 = 24 15
Answer: 24°
22 © MathTutorDVD.com
360 𝑛
. The polygon in
26. What is the measure of one exterior angle of a regular nonagon? The sum of the exterior angles of any polygon is 360°. It follows that the measure of one exterior angle of a regular polygon with 𝑛 sides is 𝑀 =
360 𝑛
. A nonagon has
9 sides, so 𝑀=
360 = 40 9
Answer: 40°
27. What is the measure of one exterior angle of a regular dodecagon? The sum of the exterior angles of any polygon is 360°. It follows that the measure of one exterior angle of a regular polygon with 𝑛 sides is 𝑀 =
360 𝑛
. A dodecagon
has 12 sides, so 𝑀=
360 = 30 12
Answer: 30°
28. What is the measure of one exterior angle of a regular decagon? The sum of the exterior angles of any polygon is 360°. It follows that the measure of one exterior angle of a regular polygon with 𝑛 sides is 𝑀 = 10 sides, so 𝑀=
360 = 36 10
Answer: 36°
23 © MathTutorDVD.com
360 𝑛
. A decagon has
