1. Quadrilaterals
Concept 27

2. What is a quadrilateral?
A quadrilateral is a 4 sided polygon (all straight sides and encloses space).

3. Angles of a Quadrilateral
The sum of the interior angles of a quadrilateral is 360 degrees.
180
𝑚∠𝐴+𝑚∠𝐵+𝑚∠𝐶+𝑚∠𝐷=360°
180
= 360

4. Find the measure of x and each interior angle.
𝑚∠𝐽=2(50)+15
=100+15
=115
𝑚∠𝐾=3 50 −20
=150−20
=130
𝑚∠𝐽+𝑚∠𝐾+𝑚∠𝑀+𝑚∠𝑁=360°
𝑚∠𝐽=50
2𝑥+15+3𝑥−20+𝑥+𝑥+15=360°
7𝑥+10=360°
𝑚∠𝑁=50+15
=65
7𝑥=350
𝑥=50

5. Find the measure of x and each interior angle.
𝑚∠𝑅+𝑚∠𝑆+𝑚∠𝑇+𝑚∠𝑈=360°
6𝑥 −4+2𝑥+8+6𝑥 −4+2𝑥+8=360
16𝑥+8=360
16𝑥=352
𝑥=22
∠𝑅=6 22 −4=128°
∠𝑆= =52°
∠𝑇=128°
∠𝑈=52°

6. Find the measure of x and each interior angle.
𝑚∠𝐴+𝑚∠𝐵+𝑚∠𝐶+𝑚∠𝐷=360°
2𝑥 −15+𝑥+2𝑥 −15+𝑥=360
6𝑥 −30=360
∠𝐴=2 65 −15=115°
6𝑥=390
𝑥=65
∠𝐶=115°
∠𝐵=65°
∠𝐷=65°

7. Find the measure of x and each interior angle.
𝑚∠𝐿+𝑚∠𝑀+𝑚∠𝑁+𝑚∠𝑃=360°
2𝑥+20+3𝑥−10+2𝑥 −10+2𝑥=360
9𝑥 =360
∠𝐿= =100°
𝑥=40
∠𝑀=3 40 −10=110°
∠𝑁=2 40 −10=70°
∠𝑃=2 40 =80°

8. Find the measure of x and each interior angle.
𝑚∠𝑆+𝑚∠𝑇+𝑚∠𝑈+𝑚∠𝑊=360°
2𝑥+16+2𝑥+16+𝑥+14+𝑥+14=360
6𝑥+60=360
6𝑥=300
∠𝑆= =116°
𝑥=50
∠𝑇=116°
∠𝑈=50+14=64°
∠𝑊=64°

9. Parallelograms
Concept 29

10. Parallelogram
A quadrilateral with two pairs of opposite sides ___________
parallel
If a quadrilateral is a parallelogram, then ________ pairs of opposite ___________ are _______________.
both
sides
𝐴𝐵=𝐶𝐷
congruent
𝐴𝐶=𝐵𝐷
If a quadrilateral is a parallelogram, then ________ pairs of opposite ___________ are _______________.
both
angles
𝑚∠𝐴=𝑚∠𝐷
congruent
𝑚∠𝐵=𝑚∠𝐶

11. If a quadrilateral is a parallelogram, then ________ diagonals __________ each ____________. E
both
bisect
𝐴𝐸=𝐷𝐸
congruent
𝐵𝐸=𝐶𝐸
If a quadrilateral is a parallelogram, then ________ pairs of consecutive angles are _______________.

12. Find the value of each variable. Name which characteristic fits.

15. Special Parallelograms
Concepts

16. Rhombus
A parallelogram with all sides congruent.

17. Theorem
A parallelogram is a rhombus if and only if its diagonals are perpendicular.

18. Theorem
A parallelogram is a rhombus if and only if the diagonals bisect a pair of opposite angles.

19. Quadrilateral ABCD is a rhombus. Find each value or measure.
1. If m∠ABD = 60, find m∠BDC.   
2. If AE = 8, find AC. 60
m∠BDC = 60 x
3. If AB = 26 and BD = 20, find AE.
𝑏 2 = 26 2
BE = 10
AE = 8 and EC = 8,
so AC = 16
100+ 𝑏 2 =676
𝑏 2 =576
Now use the Pythagorean theorem.
𝑎 2 + 𝑏 2 = 𝑐 2
𝑏 2 = 576
𝑏=24

20. 6. If m∠CDB = 6y and m∠ACB = 2y + 10, find y. m∠CEB = 90°
4. Find m∠CEB.
5. If m∠CBD = 58, find m∠ACB.
6. If m∠CDB = 6y and m∠ACB = 2y + 10, find y.
m∠CEB = 90° 58
2y + 10 x
2y + 10
m∠ACB = 32° 6y
6y +2y = 180
8y = 180
8y = 80
y = 10

21. 8. If AD = 2x + 4 and CD = 4x – 4, find x.
7. If AE = 3x – 1 and AC = 16, find x.
8. If AD = 2x + 4 and CD = 4x – 4, find x.
3x – 1 + 3x – 1 = or 3x – 1 = 8
3x - 1
6x – 2 = 16
3x - 1
4x - 4
6x = 18
x = 3
2x + 4
4x – 4 = 2x + 4
2x = 8
x = 4

22. A parallelogram with 4 right angles.

23. Theorem
A parallelogram is a rectangle if and only if its diagonals are congruent.

24. Quadrilateral ABCD is a rectangle
9. If AE = 36 and CE = 2x – 4, find x.
10. If BE = 6y + 2 and CE = 4y + 6, find y.
6y + 2
4y+6
2x - 4
2x – 4 = 36 36
2x = 40
x = 20
6y + 2 = 4y + 6
2y = 4
y = 2

25. 11. If BC = 24 and AD = 5y – 1, find y. 5y - 1 = 24 5y = 25
12. If m∠BEA = 62, find m∠BAC. 24
5y - 1 = 24 x
5y = 25
62°
y = 5 x
5y - 1
180 – 62 = 118
118/2 = 59 °

26. 13. If m∠AED = 12x and m∠BEC = 10x + 20, find m∠AED.
14. If BD = 8y – 4 and AC = 7y + 3, find BD.
12x = 10x + 20
10x + 20
2x = 20
7y + 3
8y - 4
m∠AED = 12(10)
x = 10
12x
= 120
8y - 4 = 7y + 3
BD = 8(7) - 4
y = 7
BD = 52

27. 15. If m∠DBC = 10x and m∠ACB = 4x + 6, find m∠ ACB.
16. If AB = 6y and BC = 8y, find BD in terms of y. 8y
10x = 4x + 6
10x
4x + 6
6x = 6 6y 6y
x = 1
m∠ACB = 4(1) + 6
=10
(6𝑦) 2 + (8𝑦) 2 = 𝐵𝐷 2
Use the Pythagorean theorem.
𝑎 2 + 𝑏 2 = 𝑐 2
36𝑦 𝑦 2 = 𝐵𝐷 2
100𝑦 2 = 𝐵𝐷 2
100 𝑦 2 = 𝐵𝐷 2
10 y = BD

28. Rectangle
- A parallelogram (5 characteristics apply)
- Four congruent angles – right angles
- Diagonals are congruent

29. - a parallelogram with 4 congruent sides
and 4 congruent angles.

30. 18. If BD = 8y – 4 and AC = 7y + 3, find BD. 12x – 3 = 45
17. If m∠ABE = 12x – 3, find x.
18. If BD = 8y – 4 and AC = 7y + 3, find BD.
12x – 3
12x – 3
= 45
12x = 48
x = 4
8y – 4 = 7y + 3
BD = 8(7) - 4
y – 4 = 3
BD = 52
y = 7

31. 45°

32. Trapezoids and kites
Concepts

33. quadrilateral parallel sides bases legs base angles ∠A and ∠D
supplementary
leg
∠B and ∠C

34. trapezoid with one pair of opposite sides congruent (the legs).

35. base angles congruent ∠𝐴≅∠𝐵 and∠𝐶≅∠𝐷 congruent base angles ∠𝐴≅∠𝐵
or ∠𝐶≅∠𝐷 A B D C

36. it’s diagonals are congruent.
𝐵𝐷 ≅ 𝐴𝐶

37. Find each measure. 1. m∠C 2. m∠L and m∠J X + 125 = 180 X + 40 = 180
m∠K = m∠J
40 = m∠J

38. connects the midpoints of the legs.
parallel
the average 𝑴𝑵
𝑴𝑵 || 𝑨𝑩 || 𝑫𝑪
𝑴𝑵=𝟏/𝟐 (𝑨𝑩+𝑫𝑪)

39. For trapezoid HJKL, M and N are the midpoints of the legs.
3. If HJ = 32 and LK = 60, find MN.
4. If HJ = 8 and LK = 21, find MN.
5. If HJ = 18 and MN = 28, find LK.
6. If HJ = 15 and MN = 25, find LK.
𝑴𝑵= 𝟏 𝟐 (𝟑𝟐+𝟔𝟎)
𝑴𝑵= 𝟏 𝟐 (𝟗𝟐)
𝑴𝑵=𝟒𝟔
𝑴𝑵= 𝟏 𝟐 (𝟖+𝟐𝟏)
𝑴𝑵= 𝟏 𝟐 (𝟐𝟗)
𝑴𝑵=𝟏𝟒.𝟓
𝟐𝟖= 𝟏 𝟐 (𝟏𝟖+𝑳𝑲)
𝟓𝟔=𝟏𝟖+𝑳𝑲
𝟑𝟖=𝑳𝑲
𝟐𝟓= 𝟏 𝟐 (𝟏𝟓+𝑳𝑲)
𝟓𝟎=𝟏𝟓+𝑳𝑲
𝟑𝟓=𝑳𝑲

40. quadrilateral
two
consecutive
congruent
opposite
sides
congruent

41. its diagonals are perpendicular. 𝐴𝐶 ⊥ 𝐵𝐷
If quadrilateral ABCD is a kite, then
∠𝐴𝐵𝐷≅∠𝐶𝐵𝐷
and ∠𝐴𝐷𝐵≅∠𝐶𝐷𝐵

42. exactly one pair of
opposite angles are congruent.
𝐵𝐶 ≅ 𝐵𝐴
∠𝐴≅𝐶 𝑎𝑛𝑑 ∠𝐵 ≅∠𝐷

43. If GHJK is a kite, find each measure. 7. Find m∠JRK.
8. If RJ = 3 and RK = 10, find JK.
m∠JRK = 90
(3) 2 + (10) 2 = 𝐽𝐾 2
9+100= 𝐽𝐾 2
109= 𝐽𝐾 2
109 = 𝐽𝐾 2
109 = JK

44. 9. If m∠GHJ = 90 and m∠GKJ = 110, find m∠HGK.
m∠HGK m∠HJK = 360
x x = 360
2x = 360
2x = 160
10. If HJ = 7, find HG.   
11. If HG = 7 and GR = 5, find HR.
x = 80
HG = 7
110
(5) 2 + (𝐻𝑅) 2 = (7) 2
25+ 𝐻𝑅 2 =49
𝐻𝑅 2 =24
𝐻𝑅 2 = 24
𝐻𝑅= 24

45. 12. If m∠GHJ = 52 and m∠GKJ = 95, find m∠HGK.
13. If m∠GHJ = 120 and m∠HGK = 75, find m∠GKJ.
120 52
m∠HGK m∠HJK + 95 = 360
x x + 95 = 360 75 75
2x = 360
2x = 213
x = 106.5 95
m∠GKJ = 360
x = 360
x = 360
x = 90

46. Quadrilateral properties
Concept 34

47. Quadrilaterals
Trapezoid
Isosceles
Trapezoid
Quadrilaterals
Rectangle
Rhombus
Square
Kite

48. X X X X X X X X X X X X X X X X X X X X X X Property Parallelogram
Rectangle
Rhombus
Square
Trapezoid
Isosceles Trapezoid
Kite
1. Both pairs of opposite sides are parallel.
2. Both pairs of opposite angles are congruent.
3. Both pairs of opposite sides are congruent.
4. All pairs of angles are congruent.
5. Exactly one pair of opposite sides are parallel.
6. Exactly one pair of opposite sides are congruent.
7. Exactly one pair of opposite angles are congruent.
8. All sides are congruent
9. Only two pairs of consecutive sides are congruent.
10. Base angles are congruent. X X X X X X X X X X X X X X X X X X X X X X

49. X X X X X X X X X X X X X X X X X X X X Property Parallelogram
Rectangle
Rhombus
Square
Trapezoid
Isosceles Trapezoid
Kite
11. Only two pairs of consecutive angles are supplementary
12. All pairs of consecutive angles are supplementary.
13. Diagonals are congruent.
14. Diagonals are perpendicular.
15. Diagonal bisect each other.
16. Diagonals bisect opposite angles.
17. Diagonals bisect only one pair of opposite angels.
18. Only one diagonal is bisected. X X X X X X X X X X X X X X X X X X X X