day 76 bellringer name - highschoolmathteachers.com fileday 76 activity name _____...
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Day 76 Bellringer Name ____________________________________
HighSchoolMathTeachers@2018 Page 1
1. Use the triangle below to answer the questions that follow. ππ =1
2π΄π΅
a) What is the ratio of AQ:QC?
b) What is the ratio of BR:RC?
c) Find the length of QR
2. In the diagram below, βπππ is similar to βπΎπΏπ. ππ = 12ππ, πΎπ = 4ππ and πΏπ = 6ππ.
a) Find the length of KL
b) Find the length of LN
A B
C
π π
2.6ππ
M N
O
K L
9ππ
Day 76 Bellringer Name ____________________________________
HighSchoolMathTeachers@2018 Page 2
Answer Key Day 76:
1. a) 1:1
b) 1:1
c) 1.3ππ
2. a) 3ππ
b) 12ππ
Day 76 Activity Name ____________________________________
HighSchoolMathTeachers@2018 Page 3
1. Draw a 4ππ long line in the middle of a plane paper.
2. Label this line AB.
3. Using the method of your choice construct a line parallel to and above line AB.
4. Make a mark anywhere above the line you have constructed and label it as C.
5. Using a ruler and a pencil join point C and end A.
6. Using a ruler and a pencil join point C and end B such that you have βπ΄π΅πΆ and a line parallel
to side AB passing through it.
7. Label the points where the parallel line intersects side AC and BC as Q and R respectively.
8. Using a ruler measure the lengths of AQ, QC, BR and RC and record them in the table below.
Line
AQ QC BR RC
Length
9. The ratios π΄π
ππΆ and
π΅π
π πΆ. Is π΄π
ππΆ=
π΅π
π πΆ?
Day 76 Activity Name ____________________________________
HighSchoolMathTeachers@2018 Page 4
In this activity, students will draw a triangle of their choice and a line parallel to one of the sides
then establish the proportionality of the parts of the sides divided by the parallel line.
Students will work in groups of at least three and each group is required to have a pencil, a ruler
a compass and a plain paper.
Answer Keys
Day 76:
1-8. No response
9. Yes
Day 76 Practice Name ____________________________________
HighSchoolMathTeachers@2018 Page 5
Use the figure below to answer questions 1 and 4.
π΄πΆ = 11ππ, π΄π· = 5ππ, π΄πΉ = 4ππ and πΆπΈ = 5ππ.
1. Find the length of FB
2. Find the length of AB
3. Find the length of EB?
4. Find the length of BC
Use the diagram below to answer questions 5 and 8.
5. Find the length of AK
A B
C
D E
F
12ππ
4ππ
J 6ππ A K
B
L
C
5in
Day 76 Practice Name ____________________________________
HighSchoolMathTeachers@2018 Page 6
6. Find the length of JK
7. Find the length of CL
8. Find the length of KL
Use the figure below to answer questions 9 and 11.
9. In which ratio does D divide SU?
10. Find the value of x
11. What is the length of side SU?
Use the diagram below to answer questions 12-17. π΄πΊ = 12ππ, πΊπΉ = 8ππ, π΄π΅ = 8ππ, πΆπ· =9ππ, π·πΈ = 10ππ, ππ΅ = 3ππ, ππΆ = 2ππ and ππΈ = 8ππ
12. Find the length of AC
S 2ππ πΈ 4ππ T
D
U
5ππ
π₯
π΄ π΅ πΆ π·
G E
F
O
Day 76 Practice Name ____________________________________
HighSchoolMathTeachers@2018 Page 7
13. Find the length of BC
14. Find the length of AB
15. Find the length of FE?
16. Find the length of DF?
17. Find the length of CG?
Use the diagram below to questions 18 - 19
18. Find the value of π₯
19. What is the length of MN?
20. Find the value of y in the figure below.
π π₯ π 15ππ π
π
O
24ππ
18ππ
y
9ππ 15ππ
3ππ
Day 76 Practice Name ____________________________________
HighSchoolMathTeachers@2018 Page 8
Answer Key
1. 4.8ππ
2. 8.8ππ
3. 4.17ππ
4. 9.17ππ
5. 2ππ
6. 8ππ
7. 15ππ
8. 20ππ
9. 1:2
10. 2.5ππ
11. 7.5 ππ
12. 13.5ππ
13. 3.375ππ
14. 10.125
15. 8.18ππ
16. 18.18ππ
17. 18ππ
18. 11.25ππ
19.26.25ππ
20.1.8ππ
Day 76 Exit Slip Name ____________________________________
HighSchoolMathTeachers@2018 Page 9
Use the diagram below to answer the question that follows.
1. Find the value of π₯
π₯ 2.5ππ
10ππ 8ππ
Day 76 Exit Slip Name ____________________________________
HighSchoolMathTeachers@2018 Page 10
Answer Keys
Day 76:
1. 2ππ
Day 77 Bellringer Name ____________________________________
HighSchoolMathTeachers@2018 Page 11
Use the figure below to answer the following questions. All length measurements are given in
inches.
1. (a) Identify two triangles similar to βABC.
(b) Identify two triangles that share πΌ as one of their angles.
(c) Identify two triangles that share π½ as one of their angles.
2. Using the similar triangles you have identified in the figure above and the ratio of
corresponding sides for proportionality, calculate to two decimal places, the length of the
following sides:
(a) BC
(b) BD
A
B C
D
πΌ
π½
16
8
24
Day 77 Bellringer Name ____________________________________
HighSchoolMathTeachers@2018 Page 12
Answer keys Day 77:
1. (a) βBDC and βADB
(b) βABC and βADB
(c) βABC and βBDC
2. (a) BC = 27.71 in.
(b) BD = 13.86 in.
Day 77 Activity Name ____________________________________
HighSchoolMathTeachers@2018 Page 13
1. Use suitable measurements to construct right ΞABC using a ruler and a protractor on the blank
paper such that β BAC = 60Β°, β ABC = 90Β° and β ACB = 30Β°. ΞABC should appear as shown
below.
2. Drop a perpendicular from point B to intersect ACΜ Μ Μ Μ at point D as shown below.
3. Identify triangles ΞBDC and ΞADB from ΞABC and sketch them on the blank paper. They
should appear as shown below.
A
B C
A
B C
D
A
D B
B
D C
Day 77 Activity Name ____________________________________
HighSchoolMathTeachers@2018 Page 14
4. Measure β ADB and β BDC and compare their measures to β ABC. What do you notice?
5. Identify two triangles that have β A as one of their angles.
6. Identify two triangles that have β C as one of their angles.
7. Considering the shapes and sizes of the angles of the triangles above, give the major
relationship between the three triangles above?
8. Measure the lengths ABΜ Μ Μ Μ , BCΜ Μ Μ Μ , ADΜ Μ Μ Μ , DCΜ Μ Μ Μ , BDΜ Μ Μ Μ and ACΜ Μ Μ Μ in inches.
9. Find the sum of the squares of the lengths ABΜ Μ Μ Μ and BCΜ Μ Μ Μ and compare it to the square of the
length ACΜ Μ Μ Μ on β ABC . Write down an identity to show the relationship between the three sides.
10. Find the sum of the squares of the lengths ADΜ Μ Μ Μ and BDΜ Μ Μ Μ and compare it to the square of the
length ABΜ Μ Μ Μ on β ADB . Write down an identity to show the relationship between the three sides.
11. Find the sum of the squares of the lengths BDΜ Μ Μ Μ and CDΜ Μ Μ Μ and compare it to the square of the
length BCΜ Μ Μ Μ on β BDC . Write down an identity to show the relationship between the three sides.
Day 77 Activity Name ____________________________________
HighSchoolMathTeachers@2018 Page 15
In this activity, students will work in groups of four to verify the Pythagoras theorem from
similar right triangles. The students in the respective groups will require a ruler, blank paper, and
a protractor.
Answer keys Day 77:
1. No response
2. No response
3. No response
4. β ADB = β BDC = β ABC = 90Β°; the three angles are congruent
5. ΞABC and ΞADB
6. ΞABC and ΞBDC
7.The three triangles are similar
8. The lengths should be accurately measured
9. ABΜ Μ Μ Μ 2 + BCΜ Μ Μ Μ 2 = ACΜ Μ Μ Μ 2
10. ADΜ Μ Μ Μ 2 + BDΜ Μ Μ Μ 2 = ABΜ Μ Μ Μ 2
11. BDΜ Μ Μ Μ 2 + CDΜ Μ Μ Μ 2 = BCΜ Μ Μ Μ 2
Day 77 Practice Name ____________________________________
HighSchoolMathTeachers@2018 Page 16
In the figure ππ = π, ππ = π, ππ = π, ππ = π, π π = π and ππ β₯ ππ at π . Study it and use it
to answer questions 1-8 below.
1. Express π in terms of π and π.
Given that β QPS = 56Β° and β PRQ = 34Β°. Find the measures of the following angles:
2. β QSR
3. β SQR
4. β PSQ
5. β PQS
P
Q R
S
π
π π
π
π 56Β°
34Β°
Day 77 Practice Name ____________________________________
HighSchoolMathTeachers@2018 Page 17
6. Identify two triangles similar to ΞPQR and label them in such a way that the corresponding
parts match the parts of ΞPQR.
Complete the proportionality statements represented below:
7. π
=π
π
8. π
π=
π
Use the proportionality statements in questions 7 and 8 to complete the equations below:
9. π2 = π Γ ____
10. π2 = π Γ ____
11. Use the equations in questions 9 and 10 to show that π2 + π2 = π2 by substitution.
Day 77 Practice Name ____________________________________
HighSchoolMathTeachers@2018 Page 18
In the figure below, ΞABC is a right triangle and BD β₯ AC. Use it to answer questions 12-20.
Given that β CBD = 47Β° and β ABD = 43Β°. Calculate the measures of the following angles:
12. β BCD
13. β BAD
14. β ADB
15. Write π in terms of π₯ and π¦
Fill in the gaps to complete the proportionality statements below:
16. π
π₯=
π with reference to ΞABC and ΞBDC
A
C B
D
π
π¦
π₯
π
π
47Β°
43Β°
Day 77 Practice Name ____________________________________
HighSchoolMathTeachers@2018 Page 19
17. π
π¦=
π with reference to ΞABC and ΞADB
Use the proportionality statements in questions 16 and 17 to complete the equations below:
18. π2 = π₯ Γ ____
19. π2 = π¦ Γ ____
20. Use the equations in questions 18 and 19 prove the identity π2 + π2 = π2 by substitution.
Day 77 Practice Name ____________________________________
HighSchoolMathTeachers@2018 Page 20
Answer keys Day 77:
1. π = π + π
2. 90Β°
3. 56Β°
4. 90Β°
5. 34Β°
6. ΞPSQ and ΞQSR
7. π
π=
π
π
8. π
π=
π
π
9. π2 = π Γ π
10. π2 = π Γ π
11. π2 + π2 = ππ + ππ = π(π + π) but π + π = π hence π(π + π) = π Γ π = π2
β΄ π2 + π2 = π2
12. 43Β°
13. 47Β°
14. 90Β°
15. π = π₯ + π¦
16. π
π₯=
π
π
17. π
π¦=
π
π
18. π2 = π₯ Γ π
19. π2 = π¦ Γ π
20. π2 + π2 = ππ₯ + ππ¦ = π(π₯ + π¦) but π₯ + π¦ = π hence π(π₯ + π¦) = π Γ π = π2
β΄ π2 + π2 = π2
Day 77 Exit Slip Name ____________________________________
HighSchoolMathTeachers@2018 Page 21
In the figure π΄π΅ = π, π΅πΆ = π, π΄πΆ = π, π΄π· = π₯ and π·πΆ = π¦.
(a) Write π in terms of π₯ and π¦.
(b) Complete the proportionality statement below using the sides on ΞABC and ΞADB.
π=
π
π
(c) Complete the proportionality statement below using the sides on ΞABC and ΞBDC.
π=
π
π
A
B C
D
π₯
π¦ π
π
π
Day 77 Exit Slip Name ____________________________________
HighSchoolMathTeachers@2018 Page 22
Answer keys Day 77:
(a) π = π₯ + π¦
(b) π
π₯=
π
π
(c) π
π¦=
π
π
Day 78 Bellringer Name ____________________________________
HighSchoolMathTeachers@2018 Page 23
1. Use triangles below to answer the questions that follow.
a) Which criterion makes the two triangles similar?
b) Find the value of y
c) Find the value of x
2. Use the diagram below to answer the questions that follow.
a) Which postulate makes the triangles above congruent?
b) Find the value of z
9 ππ 3 ππ
15 ππ 12 ππ
π₯ π¦
(3π§ β 2) ππ
7 ππ
Day 78 Bellringer Name ____________________________________
HighSchoolMathTeachers@2018 Page 24
Answer Key Day 78:
1. a) AA criterion
b) 5 ππ
c) 4 ππ
2 a) A.S.A postulate
b) 3
Day 78 Activity Name ____________________________________
HighSchoolMathTeachers@2018 Page 25
1. Draw a rectangle measuring 6 in by 4 in on a plain paper and label it ABCD as shown.
2. Mark the midpoint of AB and label it as O.
3. Draw a straight line joining point O and C.
4. Join points D and O with a straight line.
5. Measure the lengths of sides DO and OC.
Are they equal?
6. Measure the lengths of AO and OB.
Are they equal?
7. Measure the lengths of AD and CB.
Is βπ΄π·π β βπ΅πΆπ?
Explain your answer.
D C
A B
Day 78 Activity Name ____________________________________
HighSchoolMathTeachers@2018 Page 26
In this activity, students will draw different triangles and identify the ones that are congruent.
Students will work in groups of at least three and each group is required to have a ruler, a pencil,
and a plain paper.
Answer Keys
Day 78:
1-4. No response
5. Yes
6. Yes
7. Yes, S.S.S postulate
Day 78 Practice Name ____________________________________
HighSchoolMathTeachers@2018 Page 27
Use the diagram below to answer the questions 1-3.
1. Find the value of r
2. Find the value of p
3. Find the length of AC
4. A student who is 5ft has a shadow of length 15ft. At the same time, the length of the shadow
of a building was 450 ft. What is the height of the building?
Use the diagram below to answer the questions 5 and 6.
5. Find the value of y
2.4 ππ 0.8 ππ
π¦
1.1 ππ π§
1.8 ππ
π΄ 6 ππ π΅ π πΆ
3 ππ
12 ππ
π
9 ππ
E
D
Day 78 Practice Name ____________________________________
HighSchoolMathTeachers@2018 Page 28
6. Find the value of z
Use the diagram below to answer questions 7-11.
ππ = 12ππ.
7. Find the value of π₯.
8. Find the value of π‘
9. Find the value of π¦
10. Find the value of s
11. What is the value of w?
12. A mobile phone manufacturing company makes two rectangular models of mobile phones
such that they are similar. The first model has a width of 2 in and a length of 3 in. If the second
model has a width of 2.5 in, what is its length?
A B M N
D E
C O
12 ππ
8 ππ
10 ππ
π‘ 3 ππ
π
(5π₯ + 2) ππ
y π€
Day 78 Practice Name ____________________________________
HighSchoolMathTeachers@2018 Page 29
Use the figure below to answer questions 13-15.
13. What is the value of π?
14. Find the value of b
15. Find the value of c
Use the diagram below to answer questions 16 to 20.
The two triangles are image and pre-image of one another under glide reflection.
16. What is the value of π?
(2π β 2)
(4π) ππ
10 ππ
6 ππ
(25 + 2π)Β°
(3π β 15)Β°
2 ππ
8 ππ
π 1.5 ππ
π
9 ππ m
π
π
Day 78 Practice Name ____________________________________
HighSchoolMathTeachers@2018 Page 30
17. Find the value of π
18. Find the value of l
19. Find the value of the of m
20. Find the value of π
Day 78 Practice Name ____________________________________
HighSchoolMathTeachers@2018 Page 31
Answer Keys
Day 78:
1. 2 ππ
2. 3 ππ
3. 8 ππ
4. 150 ππ‘
5. 3.3 ππ
6. 0.6 ππ
7. π₯ = 2 ππ
8. π‘ = 5 ππ
9. 15 ππ
10. 6 ππ
11. 9 ππ
12. 3.75 ππ
13. 40
14. 6
15. 3
2
16. 8 ππ
17. 6 ππ
18. 3 ππ
19. 12 ππ
20. 7.5 ππ
Day 78 Exit Slip Name ____________________________________
HighSchoolMathTeachers@2018 Page 32
1. Find the value of π in the diagram below.
(π + 2) 7 ππ
Day 78 Exit Slip Name ____________________________________
HighSchoolMathTeachers@2018 Page 33
Answer Keys Day 78:
1. 5 ππ
Day 79 Bellringer Name ____________________________________
HighSchoolMathTeachers@2018 Page 34
Use the following diagram to answer the following questions if πΊπΜ Μ Μ Μ is parallel to π»πΎΜ Μ Μ Μ .
1. Identify two triangles from the diagram above
2. Are the triangles similar or not?
3. Explain your answer.
4. Which condition should the two parallel lines meet for WHKT to be a parallelogram?
5. State the condition that should be met for the two triangles to be congruent.
.
G H
J
K
T
Day 79 Bellringer Name ____________________________________
HighSchoolMathTeachers@2018 Page 35
Answer Keys
Day 79:
1. βππ½πΊ and βπΎπ½π»
2. Yes
3. Corresponding angles are equal
β ππΊπ½ = β πΎπ»π½(Corresponding angles)
β π½ππΊ = β π½πΎπ»(Corresponding angles)
β πΊπ½π = β π»π½πΎ(Common to both triangles)
4. πΊπΜ Μ Μ Μ = 2π»πΎΜ Μ Μ Μ .
5. πΊπΜ Μ Μ Μ = π»πΎΜ Μ Μ Μ .
G H
J
K
T
Day 79 Activity Name ____________________________________
HighSchoolMathTeachers@2018 Page 36
1. Draw a rectangle LMNO using a ruler, a pencil and a protractor.
2. Draw a diagonal from L to N.
3. Draw another diagonal from M to O.
4. Label the intersection of the diagonals as P.
5. Measure LM and MO. What do you realize?
6. Measure LP and MP
7. Write a relation between LP and LN.
8. Write a relation between MO and MP.
9. Make a conclusion based on your answer in 8 and 7 above.
Day 79 Activity Name ____________________________________
HighSchoolMathTeachers@2018 Page 37
In this activity, students will show that the diagonals of a rectangle bisect each other at the point
of intersection. This is taken as a verification of the proof in the presentation. They will work in
groups of at least 3. Each group will require a protractor, a ruler, a pencil and a plain paper.
Answer Keys
Day 79:
1-4. No response
5. Difference responses
They are approximately equal
6. Different responses
7. 2πΏπ = πΏπ
8. 2ππ = ππ
9. The diagonals are bisected at the intersection, P.
Day 79 Practice Name ____________________________________
HighSchoolMathTeachers@2018 Page 38
Use the following information to answer questions 1 β 8
Consider the rhombus below. We want to prove that the diagonals of a rhombus bisects the
angles at their endpoints and intersect at a right angle.
Statement Reason
πΊπΜ Μ Μ Μ = ππΜ Μ Μ Μ = ππ»Μ Μ Μ Μ = π»πΊΜ Μ Μ Μ 1.
In triangle GHT and GST, πΊπ = πΊπ 2.
ππ = π»π, πΊπ = πΊπ», 3.
Triangles GHT and GST are congruent 4.
β π»πΊπ = β ππΊπ, β π»ππΊ = β πππΊ 5.
β πππ» = β πππΊ + β πΊππ»; β ππΊπ» = β ππΊπ +
β ππΊπ»
6.
β πππ» = 2β πππΊ = 2β πΊππ»; β ππΊπ» =
2β ππΊπ = 2β ππΊπ»
From 5 and 6 above
Diagonals of a rhombus intersect each other 7.
G H
T S
O
Day 79 Practice Name ____________________________________
HighSchoolMathTeachers@2018 Page 39
Let β πΊππ» = πΌ
β π»ππΊ = β πππΊ = πΌ 8.
πΊπ» β₯ ππ and πΊπ β₯ π»π 9.
β π»πΊπ = β πππΊ 10.
β πππΊ = β π»πΊπ = πΌ; β π»ππΊ = β πππΊ = πΌ 11.
β πππ» = 2πΌ; β ππΊπ» = 2πΌ 13.
β πΊππ + β ππΊπ» = 180Β° 14.
β πΊππ + 2πΌ = 180Β° 15.
β πΊππ = 180Β° β 2πΌ 16.
β πΊπ»π = β πΊππ = 180Β° β 2πΌ 17.
β πΊππ» = β πππ» = 90Β° β πΌ; β πΊπ»π = β ππ»π
= 90Β° β πΌ
18.
β π»ππ = β πππ = β πππΊ = β πΊππ»
= 180 β ((90 β πΌ) + πΌ)
= 90Β°
19.
Diagonals of a rhombus intersect at a right
angle
20.
Day 79 Practice Name ____________________________________
HighSchoolMathTeachers@2018 Page 40
Answer keys Day 79:
Statement Reason
πΊπΜ Μ Μ Μ = ππΜ Μ Μ Μ = ππ»Μ Μ Μ Μ = π»πΊΜ Μ Μ Μ 1. Properties of rhombus
In triangle GHT and GST, πΊπ = πΊπ 2. Common to both triangles
ππ = π»π, πΊπ = πΊπ», 3. By properties of a rhombus
Triangles GHT and GST are congruent 4.Corresponding sides are equal
β π»πΊπ = β ππΊπ, β π»ππΊ = β πππΊ 5.Corresponding angles of congruent triangles
β πππ» = β πππΊ + β πΊππ»; β ππΊπ» = β ππΊπ +
β ππΊπ»
6. Sum of Adjacent angles
β πππ» = 2β πππΊ = 2β πΊππ»; β ππΊπ» =
2β ππΊπ = 2β ππΊπ»
From 5 and 6 above
Diagonals of a rhombus intersect each other 7.β πππ» = 2β πππΊ = 2β πΊππ»; β ππΊπ» =
2β ππΊπ = 2β ππΊπ»
Day 79 Practice Name ____________________________________
HighSchoolMathTeachers@2018 Page 41
Let β πΊππ» = πΌ
β π»ππΊ = β πππΊ = πΌ 8.Since β π»ππΊ = β πππΊ
πΊπ» β₯ ππ and πΊπ β₯ π»π 9. Properties of rhombus
β π»πΊπ = β πππΊ 10. Alternate angles
β πππΊ = β π»πΊπ = πΌ; β π»ππΊ = β πππΊ = πΌ 11. Since β π»πΊπ = β πΊππ and β π»πΊπ = πΌ
Since β π»ππΊ = β πππΊ and β π»ππΊ = πΌ
β πππ» = 2πΌ; β ππΊπ» = 2πΌ 13. β πππ» = 2β πππΊ = 2β πΊππ»; β ππΊπ» =
2β ππΊπ = 2β ππΊπ»
β πΊππ + β ππΊπ» = 180Β° 14. Adjacent angles of a rhombus
β πΊππ + 2πΌ = 180Β° 15. Substitution; β ππΊπ» = 2πΌ
β πΊππ = 180Β° β 2πΌ 16. Algebraic equality of substitution
β πΊπ»π = β πΊππ = 180Β° β 2πΌ 17. Opposite angles of a rhombus
β πΊππ» = β πππ» = 90Β° β πΌ; β πΊπ»π = β ππ»π
= 90Β° β πΌ
18. Diagonals of a rhombus intersect each
other
β π»ππ = β πππ = β πππΊ = β πΊππ»
= 180 β ((90 β πΌ) + πΌ)
= 90Β°
19. Interior angles of a triangle
Diagonals of a rhombus intersect at a right
angle
20. β π»ππ = β πππ = β πππΊ = β πΊππ» =
90Β°
Day 79 Exit Slip Name ____________________________________
HighSchoolMathTeachers@2018 Page 42
Determine if the two triangles in the figure below are similar if angle DFH and HGE are equal.
Explain your answer.
D E
F
G
H
Day 79 Exit Slip Name ____________________________________
HighSchoolMathTeachers@2018 Page 43
Answer Keys
Day 79
β π·πΉπ» and β π»πΊπΈ are equal (Given)
β πΊπΈπ» = β πΉπΈπ· (Common to both triangles)
Thus AA criteria is satisfied showing that they are similar