1. CHAPTER III: POLYGONS
3.1 INTRODUCING POLYGONS.
Objectives To identify and name polygons and their parts
To identify and draw convex and concave polygons
To identify regular polygons
A polygon is the union of three or more coplanar segments such that:
• each endpoint is shared by exactly two segments;
• segments intersect only at their endpoints;
• intersecting segments are noncollinear.
Number of side Name Number of side Name
3 Triángulo 8 Octágono
4 Cuadrilátero 9 Nonagono
5 Pentágono 10 Decágono
6 Hexágono 11 Endecágono
7 Heptágono 12 Dodecágono
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2. A diagonal of a polygon is a segment joining two nonconsecutive vertices.
A polygon is convex if the segment XY joining any two interior points of the polygon is
in the interior of the polygon. If a polygon is not convex, then it concave.
Convex polygons
Concave Polygons
A regular polygon is a convex polygon that is both equilateral and equiangular.
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3. EXERCISES
Complete the statement for polygon PQRST.
S
RS and _____________ are adjacent sides. T R
TP and ______________ are nonadjacent sides
< S and ______________ are consecutive angles
< P and ______________ are nonconsecutive angles P Q
Name the polygon. State whether it is convex or concave.
Complete the table.
Number of side of polygon Number of diagonals from Number of diagonals
one vertex
3
4
5
6
7
50
n
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4. 3.2 INTERIOR ANGLES OF POLYGONS
Objectives To find the sum of the angle measures of a convex polygon
To find angle measures and the number of sides of polygons
To find angle measures and the number of sides of regular polygons.
Theorem
The sum of the measures of the interior angles of a convex polygon with n
sides is (n – 2) 180.
Corollary
The sum of the measures of the interior angles of a convex quadrilateral is
360°.
The measure o fan angle of a regular polygon with n sides is (n-2)180
n
Polygon Number of sides Number of Sum of Angle
Triangles Measures
Quadrilateral 4 2 2 . 180 = 360
Pentagon 5 3 3 . 180 = 540
Hexagon 6 4 4 . 180 = 720
Heptagon 7 5 5 . 180 = 900
Octagon 8 6 6 . 180 = 1,080
EXERCISES.
Find the sum of the measures of the angles of the convex polygon.
1. a pentagon
2. a hexagon
3. a decagon
4. a 30-gon
5. a 62-gon
6. a 100-gon
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5. The sum of the measures of the angles of a convex polygon is given. Find the number
of sides of the polygon.
1. 900
2. 1260
3. 1980
4. 3600
5. 4500
6. 7560
Find the measure of an angle of the regular polygon.
1. a square
2. a pentagon
3. a decagon
4. a 20-gon
5. a 30-gon
6. a 100-gon
Each of a regular polygon has the given measure. How many sides does the polygon
have?
1. 60
2. 135
3. 108
Find the measure of each angle of quadrilateral ABCD. Draw the figure.
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6. • m < A = 10x, m < B = 6x + 10, m < C = 12x – 10, m < D = 8x
• m < A = 8x + 5, m < B = 10x + 5, m < C = 10x – 8, m < D = 13x – 11.
Solve the problems.
1. the sum of the measures of four angle of a pentagon is 498. what is the measure
of the unknown angle?
2. the sum of the measures of nine angles of a decagon is 1320. What is the
measure of the missing angle?
3. Three angle of a hexagon are congruent. The other three angles are also
congruent. Each of the first three angles has a measure twice that one of the
second three angles. What is the measure of each angle of the hexagon?
4. In a pentagon, the measure of one angle is twice that of a second angle. The
remaining angles are congruent, each having a measure of three times that of the
second angle. What is the measure of each angle of the pentagon?
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7. 3.3 EXTERIOR ANGLES OF POLYGONS
Objective To identify the exterior angles of a polygon.
To find the measure of exterior angles of polygons
To explain why the exterior angles of regular polygon are congruent.
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8. The sumo f the measures of the exterior angles, one at each vertex, of any
convex polygon is 360°.
The measure of an exterior angle of a regular polygon with n sides is 360.
n
EXERCISES.
Find the measure of each exterior angle of the given regular polygon.
1. a pentagon
2. an octagon
3. a decagon
4. a 15-gon
Draw a triangle with its exterior angles.
3.4 QUADRILATERALS AND PARALLELOGRAMS
Objectives To identify parts an investigate properties of quadrilaterals
To prove an apply theorems about parallelograms
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9. A parallelogram is a quadrilateral with both pairs of opposite sides
parallel.( EFGH means parallelogram EFGH).
A diagonal of a parallelogram forms two congruent triangles.
The diagonals of parallelogram bisect each other.
Opposite sides of a parallelogram are congruent.
Opposite angles of a parallelogram are congruent.
Consecutive angles of a parallelogram are supplementary.
EXERCISES
Complete the statement for the parallelogram TUVW.
WV is opposite ___________ W T
< U and ________ are supplementary.
< T = ________
TU and ___________ are adjacent.
< W is opposite __________
< T and ________ are consecutive angles. V U
EFGH is a parallelogram. Find the unknown measure.
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10. 1. EF = 17, GH = _________ H G
2. m < EFG = 67, m < GHE = ___________
3. m < HEF = 119, m < GHE = __________
4. m < HGF = 125, m < EHG = __________
E F
5. EF = 12x, GH = 10x + 12, GH = _______
6. EF = 5x – 7, GH = 3x + 1, EF = ________
7. EH = 2x + 2, FG = 3x -5, FG = _________
8. EO = 3x + 2, GO = 5x – 8, EO = ________
9. FO = 4x + 13, HO = 5x + 1, FH = _______
10. m < EFG = 6x + 6, m < FGH = 3x + 3,
M < FGH = _____________
11. m < EFG = 12x -24, m < GHE = 9x + 12,
m < EFG = _____________
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