3-5 1.2. 3. (For help, go to Lesson 1-6 and 3-4.) GEOMETRY LESSON 3-5 The Polygon Angle-Sum Theorems...
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Transcript of 3-5 1.2. 3. (For help, go to Lesson 1-6 and 3-4.) GEOMETRY LESSON 3-5 The Polygon Angle-Sum Theorems...
![Page 1: 3-5 1.2. 3. (For help, go to Lesson 1-6 and 3-4.) GEOMETRY LESSON 3-5 The Polygon Angle-Sum Theorems Find the measure of each angle of quadrilateral ABCD.](https://reader036.fdocuments.net/reader036/viewer/2022070305/5514dcca550346b0338b566d/html5/thumbnails/1.jpg)
3-5
1. 2.
3.
(For help, go to Lesson 1-6 and 3-4.)
GEOMETRY LESSON 3-5GEOMETRY LESSON 3-5
The Polygon Angle-Sum TheoremsThe Polygon Angle-Sum Theorems
Find the measure of each angle of quadrilateral ABCD.
Check Skills You’ll Need
![Page 2: 3-5 1.2. 3. (For help, go to Lesson 1-6 and 3-4.) GEOMETRY LESSON 3-5 The Polygon Angle-Sum Theorems Find the measure of each angle of quadrilateral ABCD.](https://reader036.fdocuments.net/reader036/viewer/2022070305/5514dcca550346b0338b566d/html5/thumbnails/2.jpg)
Solutions
GEOMETRY LESSON 3-5GEOMETRY LESSON 3-5
1. m DAB = 32 + 45 = 77; m B = 65; m BCD = 70 + 61 = 131; m D = 87
2. m DAC = m ACD = m D and m CAB = m B = m BCA; by the Triangle
Angle-Sum Theorem, the sum of the measures of the angles is 180,
so each angle measures , or 60. So, m DAB = 60 + 60 = 120,
m B = 60, m BCD = 60 + 60 = 120, and m D = 60.
3. By the Triangle Angle-Sum Theorem m A + 55 + 55 = 180, so m A = 70. m ABC = 55 + 30 = 85; by the Triangle Angle-Sum Theorem, m C + 30 + 25 = 180, so m C = 125; m ADC = 55 + 25 = 80
180 3
The Polygon Angle-Sum TheoremsThe Polygon Angle-Sum Theorems
3-5
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1. A triangle with a 90° angle has sides that are 3 cm, 4 cm, and 5 cm long. Classify the triangle by its sides and angles.
Use the diagram for Exercises 2–6.
2. Find m 3 if m 2 = 70 and m 4 = 42.
3. Find m 5 if m 2 = 76 and m 3 = 90.
4. Find x if m 1 = 4x, m 3 = 2x + 28, and m 4 = 32.
5. Find x if m 2 = 10x, m 3 = 5x + 40, and m 4 = 3x – 4.
6. Find m 3 if m 1 = 125 and m 5 = 160.
GEOMETRY LESSON 3-4GEOMETRY LESSON 3-4
scalene right triangle
68
166
30
8
105
Parallel Lines and the Triangle Angle-Sum TheoremParallel Lines and the Triangle Angle-Sum Theorem
3-4
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GEOMETRY LESSON 3-5GEOMETRY LESSON 3-5
The Polygon Angle-Sum TheoremsThe Polygon Angle-Sum Theorems
3-5
A polygon is a closed plane figure with at least three sides that are line segments. The sides intersect only at their endpoints, and no adjacent sides are collinear.
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GEOMETRY LESSON 3-5GEOMETRY LESSON 3-5
The Polygon Angle-Sum TheoremsThe Polygon Angle-Sum Theorems
3-5
Each segment that forms a polygon is a side of the polygon. The common endpoint of two sides is a vertex of the polygon. A segment that connects any two nonconsecutive vertices is a diagonal.
![Page 8: 3-5 1.2. 3. (For help, go to Lesson 1-6 and 3-4.) GEOMETRY LESSON 3-5 The Polygon Angle-Sum Theorems Find the measure of each angle of quadrilateral ABCD.](https://reader036.fdocuments.net/reader036/viewer/2022070305/5514dcca550346b0338b566d/html5/thumbnails/8.jpg)
GEOMETRY LESSON 3-5GEOMETRY LESSON 3-5
The Polygon Angle-Sum TheoremsThe Polygon Angle-Sum Theorems
3-5
To name a polygon, start at any vertex and list the vertices consecutively in a clockwise or counterclockwise direction.
Two names for this polygon are ABCDE and CBAED.
vertices:
sides: , , , ,AB BC CD DE EAA, B, C, D, E
angles: , , , ,A B C D E
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GEOMETRY LESSON 3-5GEOMETRY LESSON 3-5
The Polygon Angle-Sum TheoremsThe Polygon Angle-Sum Theorems
3-5
A polygon is concave if any part of a diagonal contains points in the exterior of the polygon. If no diagonal contains points in the exterior, then the polygon is convex.
In this textbook, a polygon is convex unless stated otherwise.
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GEOMETRY LESSON 3-5GEOMETRY LESSON 3-5
The Polygon Angle-Sum TheoremsThe Polygon Angle-Sum Theorems
3-5
You can name a polygon by the number of its sides. The table shows the names of some common polygons.
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GEOMETRY LESSON 3-5GEOMETRY LESSON 3-5
The Polygon Angle-Sum TheoremsThe Polygon Angle-Sum Theorems
3-5
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GEOMETRY LESSON 3-5GEOMETRY LESSON 3-5
The Polygon Angle-Sum TheoremsThe Polygon Angle-Sum Theorems
3-5
All the sides are congruent in an equilateral polygon.
All the angles are congruent in an equiangular polygon.
A regular polygon is one that is both equilateral and equiangular. If a polygon is not regular, it is called irregular.
A regular polygon is always convex.
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Name the polygon. Then identify its vertices, sides,
and angles.
The polygon can be named clockwise or counterclockwise, starting at any vertex.
Possible names are ABCDE and EDCBA.
GEOMETRY LESSON 3-5GEOMETRY LESSON 3-5
Its vertices are A, B, C, D, and E.
Its angles are named by the vertices, A (or EAB or BAE), B (or ABC or CBA), C (or BCD or DCB), D (or CDE or EDC), and E (or DEA or AED).
The Polygon Angle-Sum TheoremsThe Polygon Angle-Sum Theorems
Its sides are AB or BA, BC or CB, CD or DC, DE or ED, and EA or AE.
3-5
Quick Check
Naming Polygons
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GEOMETRY LESSON 3-5GEOMETRY LESSON 3-5
Starting with any side, count the number of sides clockwise around the figure. Because the polygon has 12 sides, it is a dodecagon.
Classify the polygon below by its sides. Identify it as convex or
concave.
Think of the polygon as a star. If you draw a diagonal connecting two points of the star that are next to each other,that diagonal lies outside the polygon, so the dodecagon is concave.
The Polygon Angle-Sum TheoremsThe Polygon Angle-Sum Theorems
3-5
Quick Check
Classifying Polygons
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A decagon has 10 sides, so n = 10.
Sum = (n – 2)(180) Polygon Angle-Sum Theorem
= (10 – 2)(180) Substitute 10 for n.
= 8 • 180 Simplify.
= 1440
Find the sum of the measures of the angles of a decagon.
GEOMETRY LESSON 3-5GEOMETRY LESSON 3-5
The Polygon Angle-Sum TheoremsThe Polygon Angle-Sum Theorems
3-5
Quick Check
Finding a Polygon Angle Sum
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m X + m Y + m Z + m W = (4 – 2)(180) Polygon Angle-Sum Theorem
m X + m Y + 90 + 100 = 360 Substitute.
m X + m Y + 190 = 360 Simplify.
m X + m Y = 170 Subtract 190 from each side.
2m X = 170 Simplify.
m X = 85 Divide each side by 2.
m X + m X = 170 Substitute m X for m Y.
GEOMETRY LESSON 3-5GEOMETRY LESSON 3-5
The figure has 4 sides, so n = 4.
Find m X in quadrilateral XYZW.
The Polygon Angle-Sum TheoremsThe Polygon Angle-Sum Theorems
3-5
Quick CheckUsing the Polygon Angle-Sum Theorem
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Because supplements of congruent angles are congruent, all the angles marked 1 have equal measures.
Sample: The hexagon is regular, so all its angles are congruent.
An exterior angle is the supplement of a polygon’s angle because they are adjacent angles that form a straight angle.
GEOMETRY LESSON 3-5GEOMETRY LESSON 3-5
A regular hexagon is inscribed in a rectangle. Explain how you
know that all the angles labeled 1 have equal measures.
The Polygon Angle-Sum TheoremsThe Polygon Angle-Sum Theorems
3-5
Quick Check
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1. 2.
3. Find the sum of the measures of the angles in an octagon.
4. A pentagon has two right angles, a 100° angle and a 120° angle. What is the measure of its fifth angle?
5. Find m ABC.
6. XBC is an exterior angle at vertex B. Find m XBC.
quadrilateral ABCD;
AB, BC, CD, DA
not a polygon becausetwo sides intersect at a point other than endpoints
GEOMETRY LESSON 3-5GEOMETRY LESSON 3-5
1080
140
144
36
The Polygon Angle-Sum TheoremsThe Polygon Angle-Sum Theorems
For Exercises 1 and 2, if the figure is a polygon, name it by its vertices and identify its sides. If the figure is not a polygon, explain why not.
ABCDEFGHIJ is a regular decagon.
3-5