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This project has received funding from the European

Union’s Horizon 2020 research and innovation programme

under grant agreement No. 710577.

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LINES AND PLANES II

JEOPARDY GAME

The goal is to gain maximum of the points by answering questions. The points forincorrect answers are subtracted. The game is designed either for a single player or

for two players (or two teams).

Created by P. Vondráková, P. Beremlijski, M. Litschmannová and R. Maříkfrom Department of Applied Mathematics, VŠB – Technical University of Ostrava.

Choose the number of players. For each player choose a face.

Single player Two players

1 1

Player 1Boy Girl

1

1

Player 2

Boy Girl

Verbal descriptionto the angle

Measureof the angle— two lines

Measureof the angle

— line and planeLength anddistance

The game finished.The gameboard on the previous page allows to access the questions again.

THE WINNER IS

1111

11

NO WINEREQUAL SCORE

11

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Wonderfull. That’s right!Sorry, this is not right. NEXTVerbal description to the angle for 100.

Give a verbal description to the angle shown in the picture.

ϕ

A B

CD

VA The angle between the edge and the

baseThe angle between the triangular faceand the square baseThe angle between two opposite edgesThe angle between the edge on a faceand the base edge

B The angle between the edge and thebaseThe angle between the triangular faceand the square baseThe angle between two opposite edgesThe angle between the edge on a faceand the base edge

C The angle between the edge and thebaseThe angle between the triangular faceand the square baseThe angle between two opposite edgesThe angle between the edge on a faceand the base edge

D The angle between the edge and thebaseThe angle between the triangular faceand the square baseThe angle between two opposite edgesThe angle between the edge on a faceand the base edge

Wonderfull. That’s right!Sorry, this is not right. NEXTVerbal description to the angle for 200.

Give a verbal description to the angle shown in the picture.

ϕ

A B

CD

VA The angle between two edges in a common trian-

gular faceThe angle between two opposite triangular facesThe angle between two opposite edgesThe angle between two triangular faces havingcommon edge

B The angle between two edges in a common trian-gular faceThe angle between two opposite triangular facesThe angle between two opposite edgesThe angle between two triangular faces havingcommon edge

C The angle between two edges in a common trian-gular faceThe angle between two opposite triangular facesThe angle between two opposite edgesThe angle between two triangular faces havingcommon edge

D The angle between two edges in a common trian-gular faceThe angle between two opposite triangular facesThe angle between two opposite edgesThe angle between two triangular faces havingcommon edge

Wonderfull. That’s right!Sorry, this is not right. NEXTVerbal description to the angle for 300.

Give a verbal description to the angle shown in the picture.

ϕ

A B

CD

VA The angle between the edge on triangular face

and the base edge from the same faceThe angle between the triangular face and a baseedge not in this faceThe angle between two triangular faces having acommon edgeThe angle between a triangular face and squarebase

B The angle between the edge on triangular faceand the base edge from the same faceThe angle between the triangular face and a baseedge not in this faceThe angle between two triangular faces having acommon edgeThe angle between a triangular face and squarebase

C The angle between the edge on triangular faceand the base edge from the same faceThe angle between the triangular face and a baseedge not in this faceThe angle between two triangular faces having acommon edgeThe angle between a triangular face and squarebase

D The angle between the edge on triangular faceand the base edge from the same faceThe angle between the triangular face and a baseedge not in this faceThe angle between two triangular faces having acommon edgeThe angle between a triangular face and squarebase

Wonderfull. That’s right!Sorry, this is not right. NEXTMeasure of the angle — two lines for 100.

Identify a valid relation involving the angle α defined as an angle between a solid diagonal and a facediagonal through the same vertex in a cube.

α

a

a

a

A tgα =√

22sinα =

√3

2cosα =√

53cotgα =√

3α = 45◦

B tgα =√

22sinα =√

32cosα =

√5

3cotgα =√

3α = 45◦

C tgα =√

22sinα =√

32cosα =

√5

3cotgα =√

3α = 45◦

D tgα =√

22sinα =√

32cosα =

√5

3cotgα =√

3α = 45◦

E tgα =√

22sinα =√

32cosα =

√5

3cotgα =√

3α = 45◦

Wonderfull. That’s right!Sorry, this is not right. NEXTMeasure of the angle — two lines for 200.

In the cube ABCDEFGH find the angle between the lines DB and AG.

D C

GH

A B

FE

A 90◦45◦35.26◦53.13◦

B 90◦45◦35.26◦53.13◦

C 90◦45◦35.26◦53.13◦

D 90◦45◦35.26◦53.13◦

Wonderfull. That’s right!Sorry, this is not right. NEXTMeasure of the angle — two lines for 300.

The picture shows a square pyramid. The side of a base square is a = 4 cm and the height of thepyramid is v = 6 cm. Find the angle ϕ.

ϕ

A B

CD

V

A tgϕ = 62√

2=⇒ ϕ

.= 64◦46′tg ϕ2 = 26 =⇒ ϕ

.= 36◦52′tg ϕ2 = 22√

10=⇒ ϕ

.= 35◦6′tg ϕ2 = 2√

26 =⇒ ϕ

.= 50◦29′

B tgϕ = 62√

2=⇒ ϕ

.= 64◦46′tg ϕ2 = 26 =⇒ ϕ

.= 36◦52′tg ϕ2 = 22√

10=⇒ ϕ

.= 35◦6′tg ϕ2 = 2√

26 =⇒ ϕ

.= 50◦29′

C tgϕ = 62√

2=⇒ ϕ

.= 64◦46′tg ϕ2 = 26 =⇒ ϕ

.= 36◦52′tg ϕ2 = 22√

10=⇒ ϕ

.= 35◦6′tg ϕ2 = 2√

26 =⇒ ϕ

.= 50◦29′

D tgϕ = 62√

2=⇒ ϕ

.= 64◦46′tg ϕ2 = 26 =⇒ ϕ

.= 36◦52′tg ϕ2 = 22√

10=⇒ ϕ

.= 35◦6′tg ϕ2 = 2√

26 =⇒ ϕ

.= 50◦29′

Wonderfull. That’s right!Sorry, this is not right. NEXTMeasure of the angle — line and plane for 100.

Consider a cone of base radius r and a special shape: the shape is such that the volume of the cone isrelated to the base radius by the formula V = πr3. Find the angle between the side of the cone andthe base. Round your answer to two decimal places.

A 45◦63.43◦71.57◦ B 45◦63.43◦71.57◦ C 45◦63.43◦71.57◦

Wonderfull. That’s right!Sorry, this is not right. NEXTMeasure of the angle — line and plane for 200.

The side of a regular hexagonal prism ABCDEFA′B′C ′D′E′F ′ is a = 3 cm and the height v = 8 cm.Find the angle between the diagonal AD′ and the base plane ABC (round your result).

AB

C

DE

F

A′B′

C ′

D′

E′

F ′

a

v

A 53◦37◦45◦61◦72◦

B 53◦37◦45◦61◦72◦

C 53◦37◦45◦61◦72◦

D 53◦37◦45◦61◦72◦

E 53◦37◦45◦61◦72◦

Wonderfull. That’s right!Sorry, this is not right. NEXTMeasure of the angle — line and plane for 300.

The base ABCD of a square pyramid ABCDV has side 6 cm. The height of the pyramid is 4 cm.The point M is the middle of the side CV . Find the angle between the line AM and the plane ABC.Round to two decimal places.

V

A B

CD

MA 17.45◦34.50◦18.32◦

B 17.45◦34.50◦18.32◦

C 17.45◦34.50◦18.32◦

Wonderfull. That’s right!Sorry, this is not right. NEXTLength and distance for 100.

The side of a regular hexagonal prism ABCDEFA′B′C ′D′E′F ′ is a = 3 cm and the height v = 8 cm.Find the length of the diagonal AD′.

AB

C

DE

F

A′B′

C ′

D′

E′

F ′

a

v

A 10 cm√

73 cm√

82 cm2√

8 cm2√

6 cm

B 10 cm√

73 cm√

82 cm2√

8 cm2√

6 cm

C 10 cm√

73 cm√

82 cm2√

8 cm2√

6 cm

D 10 cm√

73 cm√

82 cm2√

8 cm2√

6 cm

E 10 cm√

73 cm√

82 cm2√

8 cm2√

6 cm

Wonderfull. That’s right!Sorry, this is not right. NEXTLength and distance for 200.

The base ABCD of a square pyramid ABCDV has side 6 cm. The height of the pyramid is 4 cm. Thepoint M is the middle of the side CV . Find the distance between the point M and the plane ABC.

V

A B

CD

MA 2 cm

√342 cm5

2 cm

B 2 cm√

342 cm5

2 cm

C 2 cm√

342 cm5

2 cm

Wonderfull. That’s right!Sorry, this is not right. NEXTLength and distance for 300.

The base ABCD of a square pyramid ABCDV has side 6 cm. The height of the pyramid is 4 cm.Find the distance between the line AD and the plane BCV .

V

A B

CD

A245 cm15√

345 cm6 cm

B245 cm15

√34

5 cm6 cm

C245 cm15

√34

5 cm6 cm

Wonderfull. That’s right!Sorry, this is not right. NEXT