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6.1 The Polygon Angle-Sum Theorems
Polygon: Closed figure with at least 3 sides
Convex: Concave:
Ex: What is the sum of the interior angles of a heptagon?
(7 – 2) 180
(9) 180 = 900
Ex: What is the measure of
n = 5
(5 – 2) 180 = 542
m
m
Ex: What is the measure of each interior angle of a regular nonagon?
Ex: What is the value of x, y, and z?
2x + 20 + 112 + 96 = 360
x = 66
y = 84
z = 94
Ex: What is the value of the <1 in the regular pentagon?
*All interior angles are congruent, so all exterior angles are congruent
*Exterior angles of regular polygon =
Ex: The measure of the exterior angle of a polygon is 40. Find the measure of the exterior angle and the number of sides.
Interior angle: 180-40 = 140Number of sides:
360 = 40n
n = 9
6.2 Properties of Parallelograms
Parallelogram: A quadrilateral (4 sides) with both pairs of opposite sides parallel. ll and ll
-Can name as
-Opposite sides are congruent.
and
-Opposite angles are congruent.
∠Q ∠S and ∠P ∠R
-Consecutive angles (angles that share a side) are supplementary
m
m
-Diagonals bisect each other.
and
Find the values of x and y.
2y + 9 = 273x + 6 = 12
2y = 183x = 6
y = 9 x = 2
Find the values of x and y.
Write the 2 equations.Use substitution.
2x = y + 42x = x + 2 +4 y = 8
x + 2 = yx = 6
with diagonals and
Prove:
StatementsReasons
1. ABCD is a parallelogram with
diagonals and 1. Given
2. and 2. Definition of bisect
3. 3. Opposite sides of parallelograms are
4. 4. SSS
EH = 6.75
6.3 Proving that a Quadrilateral is a Parallelogram
Ex: For what values of x and y make PQRS a parallelogram?
3x – 5 = 2x + 1
X = 6
Y = 8
Ex: For what values of x and y make EFGH a parallelogram?
3y – 2 + y + 10 =1804x + 13 + 12x + 7 = 180
4y + 8 = 18016x + 20 = 180
4y = 17616x = 160
Y = 43 x = 10
Ex: For what values of x and y make ACBD a parallelogram?
2x = 4y -1 = 2y -7
X = 2y = 6
Ex: Can you prove that the quadrilateral is a parallelogram based on the given information?
No, not enough infoYes, alternate interior angles are congruent, so both sets of lines are Parallel.
6.4 Properties of Rhombuses, Rectangles, and Squares
Rhombus: a parallelogram with four congruent sides.
Rectangle: a parallelogram with four right angles.
Square: a parallelogram with four congruent sides and four right angles. (A square is a rhombus and a rectangle.)
1. If a parallelogram is a rhombus, then its diagonals are perpendicular.
2. If a parallelogram is a rhombus, then its diagonals bisect a pair of opposite angles.
3. A parallelogram is a rectangle if and only if its diagonals are congruent.
Rhombus ParallelogramRhombusRectangle
Ex. Find the measures of the angles.
m<1 = 26 m< 1 = 32
m<2 = 128 m<2 = 90
m< 3 = 128 m< 3 = 58
m < 4 = 32
Ex: LMNP is a rectangle. Find the value of x and the length of each diagonal.
LN = 3x + 1 and MP = 8x - 4
3x + 1 = 8x – 4
5 = 5x
1 = x
LN and MP = 4
Ex: Find the variables and the side lengths.
3y = 155x = 15
Y = 5 x =3
Sides = 15
6.5 Conditions for Rhombuses, Rectangles, and Squares
6.6 Trapezoids and Kites
Trapezoid: quadrilateral with exactly ONE pair of parallel sides.
Bases: Parallel sides (BC and AD)
Base angles: ∠A and ∠D and ∠B and ∠C
Legs: Nonparallel sides (AB and CD)
Isosceles Trapezoid: A trapezoid that has congruent legs
-Base angles are congruent
∠A ∠D and ∠B ∠C
-Diagonals are congruent
Ex: Find the measures of the numbered angles.
m< 1 = 49
m<2 = 131
m<3 = 131
Ex: Find EF.
EF = (AD + BC)
3x = (x +3 + 12)3x + 5 = (4 + 7x + 4)
6x = x + 156x + 10 = 7x + 8
5x = 15 2 = x
X = 3 EF = 11
EF = 9
Kite: A quadrilateral with two pairs of consecutive congruent sides and opposite sides are not congruent.
-Diagonals are perpendicular
-One pair of congruent opposite angles
Ex: Find the measures of the numbered angles.
m<1 = 108
m<2 = 108
m<1= 90
m<2= 52
m<3= 38
m<4= 37
m<5= 53
6.7 Polygons in the Coordinate Plane
Ex: Classify the triangle as scalene, isosceles, or equilateral.
A (1,3) B(3,1) C(-2, -2)
AB = = = = 2
BC = = =
AC = = =
Isosceles Triangle
How to classify a parallelogram as just a parallelogram, a rhombus, a rectangle, or a square:
Ex:
Step 1. Sketch graph.
Step 2. Find the slope of 2 consecutive sides.
SP = = = PA = = = 3
Not opposite reciprocals so either a parallelogram or rhombus
Step 3. Find the lengths of the 2 consecutive sides.
SP = PA =
= =
= =
Same side lengths so either a square or rhombus
Answer: Rhombus
Ex:
Step 1. Sketch graph.
Step 2. Find the slope of 2 consecutive sides.
HI = = = IJ = = = -2
Opposite reciprocals so either a square or rectangle
Step 3. Find the lengths.
HI = IJ =
= =
= =
Different side lengths so either a rectangle or parallelogram
Answer: Rectangle
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