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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 <Y
n=5
(5 – 2) 180 = 542
m<Y = 540 – (110 + 10 + 120 + 150)
m<Y = 70
Ex: What is the measure of each interior angle of a regular nonagon?
(9−2)180
9
=
(7)180
9
= 140
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 =
360
5
360
𝑛
= 72
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 = 140
Number of sides:
360
𝑛
= 40
360 = 40n
n=9
6.2 Properties of Parallelograms
̅̅̅̅ and ̅̅̅̅
Parallelogram: A quadrilateral (4 sides) with both pairs of opposite sides parallel. ̅̅̅̅
𝑄𝑃 ll 𝑅𝑆
𝑄𝑅 ll ̅̅̅̅
𝑃𝑆
-Can name as
-Opposite sides are congruent.
̅̅̅̅ ≅ 𝑃𝑆
̅̅̅̅ and 𝑄𝑅
̅̅̅̅ ≅ 𝑃𝑆
̅̅̅̅
𝑄𝑃
-Opposite angles are congruent.
∠Q ≅ ∠S and ∠P ≅ ∠R
-Consecutive angles (angles that share a side) are supplementary
m<Q + m<R = 180
m<P + m<S = 180
m<Q + m<P = 180
m<S + m<R = 180
-Diagonals bisect each other.
̅̅̅̅̅
𝑄𝑀 ≅ ̅̅̅̅
𝑆𝑀 and ̅̅̅̅̅
𝑃𝑀 ≅ ̅̅̅̅̅
𝑅𝑀
Find the values of x and y.
2y + 9 = 27
3x + 6 = 12
2y = 18
3x = 6
y=9
x=2
Find the values of x and y.
Write the 2 equations.
Use substitution.
2x = y + 4
2x = x + 2 +4
x+2=y
x=6
with diagonals ̅̅̅̅
𝐴𝐶 and ̅̅̅̅
𝐵𝐷
Prove: ∆𝐴𝐸𝐷
≅ ∆𝐶𝐸𝐵
Statements
Reasons
1. ABCD is a parallelogram with
diagonals ̅̅̅̅
𝐴𝐶 and ̅̅̅̅
𝐵𝐷
1. Given
2. ̅̅̅̅
𝐴𝐸 ≅ ̅̅̅̅
𝐶𝐸 and ̅̅̅̅
𝐵𝐸 ≅ ̅̅̅̅
𝐷𝐸
2. Definition of bisect
3. ̅̅̅̅
𝐴𝐷 ≅ ̅̅̅̅
𝐶𝐵
3. Opposite sides of parallelograms are ≅
4. ∆𝐴𝐸𝐷 ≅ ∆𝐶𝐸𝐵
4. SSS
EH = 6.75
y=8
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 =180
4x + 13 + 12x + 7 = 180
4y + 8 = 180
16x + 20 = 180
4y = 176
16x = 160
Y = 43
x = 10
Ex: For what values of x and y make ACBD a parallelogram?
2x = 4
y -1 = 2y -7
X=2
y=6
Ex: Can you prove that the quadrilateral is a parallelogram based on the given information?
No, not enough info
Yes, 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
Parallelogram
Rhombus
Rectangle
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 = 15
5x = 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
̅̅̅̅ ≅ BC
̅̅̅̅
𝐴𝐷
Ex: Find the measures of the numbered angles.
m< 1 = 49
m<2 = 131
m<3 = 131
Ex: Find EF.
1
EF = 2 (AD + BC)
1
2
1
2
3x = (x +3 + 12)
3x + 5 = (4 + 7x + 4)
6x = x + 15
6x + 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
̅̅̅̅
𝐴𝐶 ⊥ ̅̅̅̅
𝐵𝐷
-One pair of congruent opposite angles
<B ≅ <D
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 = √(3 − 1)2 + (1 − 3)2 = √(2)2 + (−2)2
= √8 = 2√2
BC = √(−2 − 3)2 + (−2 − 1)2 = √(−5)2 + (−3)2 = √34
AC = √(−2 − 1)2 + (−2 − 3)2 = √(−3)2 + (−5)2 = √34
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.
3−4
−1
1
SP = 1−4 = −3 = 3
4−1
3
PA = 4−3 = 1 = 3
Not opposite reciprocals so either a parallelogram or rhombus
Step 3. Find the lengths of the 2 consecutive sides.
SP = √(1 − 4)2 + (3 − 4)2
PA = √(4 − 3)2 + (4 − 1)2
= √(−3)2 + (−1)2
= √(1)2 + (3)2
= √10
= √10
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.
−3−0
−3
1
0−2
HI = −2−4 = −6 = 2
IJ = 4−3 =
−2
1
= -2
Opposite reciprocals so either a square or rectangle
Step 3. Find the lengths.
HI = √(−2 − 4)2 + (−3 − 0)2
IJ = √(4 − 3)2 + (0 − 2)2
= √(−6)2 + (−3)2
= √(1)2 + (−2)2
= √45
= √5
Different side lengths so either a rectangle or parallelogram
Answer: Rectangle