SECONDARY 2 Honors ~ UNIT 4

Triangles and Quadrilaterals

4.1 ~ Isosceles and Equilateral Triangles and Midsegments Worksheet 4.1

4.2 ~ Classifying quadrilaterals and Proofs Worksheet 4.2

4.3 ~ Properties of Parallelograms Worksheet 4.3

4.4 ~ Special Parallelograms Worksheet 4.4

4.5 ~ Trapeziods and Kites

Worksheet 4.5

ANSWERS FOR ASSIGNMENTS: Remember: this is only to check your answers ~ all work must be shown, or no credit will be

given!! (Also – we’re only human, so there may be mistakes!)

4.1 answers:

1. 𝑉𝑋̅̅ ̅̅ (explain) 2. ∠𝑉𝑈𝑌 (𝑒𝑥𝑝𝑙𝑎𝑖𝑛) 3. x = 80°, y = 40° 4. x = 38°, y = 4 5. 64°

6. 42° 7. x = 50° 8. x = 6 9. x = 64°, y = 71° 10. n = 30°, m = 60°

11. n = 45°, m = 20° 12a. x = 25 b. angles: 40°, 40°, 100°

c. isosceles (explain why) 13. x = 9 14. x = 14 15. x = 11

16a. 𝑆𝑇̅̅̅̅ ∥ 𝑃𝑅̅̅ ̅̅ , 𝑆𝑈̅̅ ̅̅ ∥ 𝑄𝑅̅̅ ̅̅ , 𝑈𝑇̅̅ ̅̅ ∥ 𝑃𝑄̅̅ ̅̅ b. 40° 17. 𝑈𝑊̅̅ ̅̅ ̅ ∥ 𝑇𝑋̿̿ ̿̿ , 𝑈𝑌̅̅ ̅̅ ∥ 𝑉𝑋̅̅ ̅̅ , 𝑌𝑊̅̅ ̅̅ ̅ ∥ 𝑌𝑉̅̅ ̅̅

18a. 1050 ft. b. 437.5 ft. 19. 60° 20. 55 21. 37

2

22. x = 60 23. x = 50 24. x = 10 25. x = 6, y = 13

2

26. x = 3, DF = 24 27. x = 9, EC = 26

4.2 answers

1. Parallelogram, rectangle, rhombus, square 2. trapeziod 3. kite

4. rhombus 5. rhombus 6. kite

7 – 10 are proofs – you must have the proof

7. rhombus 8. Trapezoid 9. Quadrilateral 10. Kite

11. x = 11, y = 29 BC = 13, CD = 23, AD = 23 12. x = 2, y = 6, sides 2, 7, 7, 2

13. x = 3, y = 5, all sides 15 14. x = 58, r = 5, b = 9, all sides 6

15. x = 1, sides 2, 4, 7, 4 16. Check picture with teacher

4.3 answers:

1. x = 127° 2. x = 100° 3. x = 3

4 4. x = 22, BC = 18.5, CD =23.6, AB = 23.6

5. a = 18 6. a = 12, m∠𝑄 = 36°, 𝑚∠𝑅 = 144°, 𝑚∠𝑆 = 36°, 𝑚∠𝑃 = 144°

7. x = 45, y = 60, 𝑚∠𝐴 = 100°, 𝑚∠𝐷 = 80°, 𝑚∠𝐶 = 100°, 𝑚∠𝐵 = 80°

8. x = 6, y = 8 9. x = 7, y = 10 10. x = 5 11. x = 1.6, y = 1

12. x = 5 13. x = 3, y = 11 14. m = 23.4, k = 9

15. It must be a parallelogram because both pair of opposite sides are congruent

16. It must be a parallelogram because both pair of opposite angles are congruent

17. m∠1 = 38°, 𝑚∠2 = 32°, 𝑚∠3 = 110° 18. BC = AD = 29

2, AB = CD =

19

2

19. AB = CD = 13, BC = AD = 33

4.4 answers:

1. 𝑚∠1 = 38°, 𝑚∠2 = 38°, 𝑚∠3 = 38°, 𝑚∠4 = 38° 2. 𝑚∠1 = 118°, 𝑚∠2 = 31°, 𝑚∠3 = 31°

3. 𝑚∠1 = 32°, 𝑚∠2 = 90°, 𝑚∠3 = 58°, 𝑚∠4 = 32°

4. 𝑚∠1 = 55°, 𝑚∠2 = 35°, 𝑚∠3 = 55°, 𝑚∠4 = 90° 5. 𝑚∠1 = 90°, 𝑚∠2 = 55°, 𝑚∠3 = 90°

6. x = 3, LN = MP = 7 7. x = 9, LN = MP = 67 8. 𝑥 = 5

2, 𝐿𝑁 = 𝑀𝑃 =

25

2

9. Can be a parallelogram (you explain why) 10. Impossible (you explain why)

11. Can be a parallelogram (you explain why)

12. Measure sides and diagonals (you tell why this helps)

13. x = 30 14. 𝑥 = 15

2, 𝑦 = 3 15. 𝑥 = 5, 𝑦 = 32, 𝑧 =

15

2

16. AC = BD = 16 17. AC = BD = 1

4.5 answers:

1. 𝑚∠1 = 77°, 𝑚∠2 = 103°, 𝑚∠3 = 130° 2. 𝑚∠1 = 49°, 𝑚∠2 = 131°, 𝑚∠3 = 131°

3. 𝑚∠𝑄 = 115°, 𝑚∠𝑅 = 115°, 𝑚∠𝑆 = 65° 4. 𝑚∠1 = 90°, 𝑚∠2 = 68°

5. 𝑚∠1 = 90°, 𝑚∠2 = 45°, 𝑚∠3 = 45° 6. 𝑚∠1 = 108°, 𝑚∠2 = 108°

7. 𝑚∠1 = 90°, 𝑚∠2 = 26°, 𝑚∠3 = 90° 8. 𝑚∠1 = 90°, 𝑚∠2 = 40°, 𝑚∠3 = 90°

9. 𝑚∠1 = 90°, 𝑚∠2 = 55°, 𝑚∠3 = 90°, 𝑚∠4 = 55°, 𝑚∠5 = 35°

10. 𝑚∠1 = 90°, 𝑚∠2 = 52°, 𝑚∠3 = 38°, 𝑚∠4 = 37°, 𝑚∠5 = 53°

11. x = 15 12. x = 3 13. x = 4 14. x = 28

15. x = 18, y = 106 16. The sides are 12, 12, 21, 21

REVIEW answers:

1. 𝑥 =75°

2 2. 21 3. AC = 46 4. x =7 5. 57

6. quadrilateral, parallelogram, rhombus 7. parallelogram (must have proof)

8. Rectangle (you must give proof) 9. x = 2, NM = 8, OL = 8

10. 111° 11. 𝑚∠𝑐 = 148° 12. x = 5, y = 8 13. x = 15, y =

14. x = 7, y = 2 15. x = 5, y = 8; sides are 40, 40, 45, & 45

16. 𝑚∠1 = 90°, 𝑚∠2 = 33°, 𝑚∠3 = 57° 17. x = 28, PR = 475, QS = 475 18. 10

19. Parallelogram ABCD is a rhombus, because it has four congruent sides. x = 76°, y = 4

20. Rhombus; the measure of all numbered angles is 57°

2 21.

18

11

22. Measure all 4 sides to show all are congruent. Measure both diagonals to show they are congruent.

23. B 24. 25. x = 1, y = 10; sides are 6, 6, 7, & 7

26. 𝑚∠𝑘 = 117° 27. 𝑚∠1 = 16°, 𝑚∠3 = 74° 28. y = 8

29. by the definition of kite. (same line), so by SSS.

17

5

Name: Date: Period:

4.1 Isosceles & Equilateral Triangles and Midsegments Complete each statement and explain why it is true. Use the following diagram as a reference.

1. 𝑉𝑇̅̅̅̅ ≅ _____

2. ∠𝑉𝑌𝑈 ≅ _____

Find the values of x and y 3.

4.

Find each value. Use the following diagram

5. If 𝑚∠𝐿 = 58°, then 𝑚∠𝐿𝐾𝐽 = _____

6. If 𝑚∠𝐽𝐾𝑀 = 48°, then 𝑚∠𝐽 = _____

Find the values of the variables 7.

8.

9.

10.

11.

For problem #12, a triangle has angle measures 𝒙 + 𝟏𝟓, 𝟑𝒙 − 𝟑𝟓, 𝒂𝒏𝒅 𝟒𝒙 12a. Find the value of 𝑥

12b. Find the measure of each angle

12c. What type of triangle is it? Why?

Find the value of x 13.

14.

15.

Identify pairs of parallel segments in each diagram 16a.

16b. If 𝑚∠𝑄𝑆𝑇 = 40, find 𝑚∠𝑄𝑃𝑅 (see diagram in 16a)

17.

18a. Kate wants to paddle her canoe across the lake. To determine how far she must paddle, she paced out a triangle, counting the number of steps as shown below. If Kate’s strides average 3.5 ft, what is the length of the longest side of the triangle?

b. What distance must Kate paddle across the lake?

X is the midpoint of 𝑼𝑽̅̅ ̅̅ . Y is the midpoint of 𝑼𝑾̅̅ ̅̅ ̅.

19. If 𝑚∠𝑈𝑋𝑌 = 60°, find 𝑚∠𝑉

20. If 𝑉𝑊 = 110, find 𝑋𝑌.

21. 𝐼�̅� is a midsegment of Δ𝐹𝐺𝐻. 𝐼𝐽 = 7, 𝐹𝐻 = 10, 𝑎𝑛𝑑 𝐺𝐻 = 13. Find the perimeter of Δ𝐼𝐽𝐻.

Find the value of each variable 22.

23.

24.

25.

Use the figure below for #26 and #27

26. If 𝐵𝐸 = 2𝑥 + 6 𝑎𝑛𝑑 𝐷𝐹 = 5𝑥 + 9, find the value of

x, then find 𝐷𝐹. 27. If 𝐸𝐶 = 3𝑥 − 1 𝑎𝑛𝑑 𝐴𝐷 = 5𝑥 + 7, find the value of

x, then find 𝐸𝐶.

Name: Date: Period:

4.2 Classifying Quadrilaterals Judging by appearance, classify each quadrilateral in as many ways as possible. 1.

2.

3.

Determine the most precise name for each quadrilateral. 4.

5.

6.

Graph and label each quadrilateral with the given vertices. Then determine and justify the most precise name for each quadrilateral. Use your own paper and attach it to this worksheet. 7. 𝐴(3,5), 𝐵(7,6), 𝐶(6,2), 𝐷(2,1)

8. 𝐽(2,1), 𝐾(5,4), 𝐿(7,2), 𝑀(2, −3)

9. 𝑁(−6, −4), 𝑃(−3,1), 𝑄(0,2), 𝑅(−3, 5)

10. W(-1, 1), X(0,2), Y(1,1), Z(0, -2)

Find the values of the variables. Then find the length of the sides. 11. Kite

12. Kite

13. Rhombus

14. Rhombus HKJI

15.

Isosceles Trapezoid

16. Draw the figure described below. If it is not possible, explain why. A parallelogram that is neither a rectangle nor a rhombus.

Name: Date: Period:

4.3 Properties of Quadrilaterals Find the value of x 1.

2.

3.

Find the value of the variables and the length of each side (#4 only) or the measure of each angle (#6, 7 only) for the following parallelograms 4.

5.

6.

7.

Find the values of x and y in the following parallelogram. Use the figure for #7 and #8.

Find the values of the variables for which ABCD must be a parallelogram 10.

11.

12.

13.

14.

Determine whether the quadrilateral must be a parallelogram. Explain. 15.

16.

17. Find the measure of the numbered angles for the parallelogram

Use the given information to find the lengths of all four sides of parallelogram ABCD 18. The perimeter is 48 in. AB is 5 in. less than BC 19. The perimeter is 92 cm. AD is 7 cm more than twice AB

Name: Date: Period:

4.4 Special Parallelograms Find the measures of the numbered angles in each rhombus 1.

2.

3.

4.

5.

LMNP is a rectangle. Find the value of x and the length of each diagonal. 6. 𝐿𝑁 = 5𝑥 − 8 𝑎𝑛𝑑 𝑀𝑃 = 2𝑥 + 1

7. 𝐿𝑁 = 9𝑥 − 14 𝑎𝑛𝑑 𝑀𝑃 = 7𝑥 + 4

8. 𝐿𝑁 = 3𝑥 + 5 𝑎𝑛𝑑 𝑀𝑃 = 9𝑥 − 10

Determine whether the quadrilateral can be a parallelogram. If not, write impossible. Explain. Diagrams might not be to scale. 9. Each diagonal is 3 cm long an two opposite sides are

2 cm long.

10.

11.

12. A carpenter is building a bookcase. How can they use a tape measure to check that the bookshelf is rectangular? Justify your answer and name any theorems used.

Find the value(s) of the variable(s) for each parallelogram 13.

14.

15.

ABCD is a rectangle. Find the length of each diagonal. 16. 𝐴𝐶 = 2(𝑥 − 3) 𝑎𝑛𝑑 𝐵𝐷 = 𝑥 + 5 17.

𝐴𝐶 =3𝑦

5 𝑎𝑛𝑑 𝐵𝐷 = 3𝑦 − 4

Name: Date: Period:

4.5 Trapezoids and Kites Each trapezoid is isosceles. Find the measure of each angle. 1.

2.

3.

Find the measures of the numbered angles in each kite. 4.

5.

6.

7.

8.

9.

10.

Find the value of the variable(s) in each isosceles trapezoid or kite 11.

12.

13.

14.

15.

16. The perimeter of a kite is 66 cm. The length of one of its sides is 3 cm less than twice the length of another. Find the length of each side of the kite.