Repetition ll
155
1the measure of angle a in the triangle shown
in the figure below is equal to 25 40
A. 35°
B. 30°,
C29.
D. 20º.
30
VM Angle a is four times smaller than angle B.
What is the measure of angle y?
A. 30° C. 45° a|b b
D. 72°
63 Assess the truthfulness of the given sentences. Select Tif the sentence is true
or F- if it is false.
There exists a triangle in which every angle is more than 50°. PF
There exists a triangle in which every angle is less than 60". P F
Repetition
6 4 Different points A, B, C, Dlie on one line and AB = 5 cm, AC =7 cm, BC=
12 cm, CD = 7 cm. Assess the truthfulness of the given sentences. Select Tif the
sentence is true or F- if it is false.
P F
BD= 5 cm
P F
DA= 14 cm
65 Points K, L, M, P are marked on the plane so that KM = 20 cm, KP =12 cm, KL = 9 cm,
LM =11 cm and LP =7 cm. Are these points the vertices of a quadrangle? Choose
the answer T (yes) or N(no) and its justification from the sentences A-C.
AND. point P does not lie on the line KL.
T
Because points K, L, Mlie on one straight line.
O. KM <KL+ LP +PK.
6 Draw six consecutive segments that do not lie on one
straight line in such a way that
one. Mark with
each subsequent one has an endpoint in common with the previous
arcs allnon-convex angBus dsned by any two consecutive segments.
4 Calculate the measure of angle a. Use the information given in the drawing.
and b) 300°
405
120/60o\10 46
315°
64
Repetition ll
8Calculate the remaining angle measures of the
triangle.
and)
b) c)
68
35°
125° 45
9 In a right triangle, one of the acute angles is 80" greater
than the other. Calcuian9
the measures of the angles in this
triangle. myh 8sand
10 An So olaa e go.
equilateral triangle is cut intotwo triangles. In one ofTthem, the smallest angle is 10. st dje
Determine the measures of the angles of the second
triangle.
11On side AB of triangle ABC,
point Dis
that BC =BD and DC = DA (figure defined such
a) Calculate the measure of angle B for
opposite).
the given value of
a.a = 10,
a=11°, a= 12° AND D B
b)Try to determine the measure of angle B
without performing calculations when a = 13*.
Check your guess using calculations.
12Angle ABC of the triangle in the figure below measures
60. Point D is located so that AB = BD = DC. Find the
measures of the remaining angles of triangle AB.. D
X60°, 60° 30°,90°
(B40", 80° 70 AND
3Ina certain rectangle, the acute angle tethe diagonals is 66 . A
bisector of
the right angle of this rectangle is dra irem one of the vertices. What is the
measure of the angle between this bisector and 1he diagonal starting from this vertex?
14 The vertices of the triangle ABC are locatas s
circle with center Oso that A0C=132, are: s
108° (figure opposite). Calculate the meusures
of the angles of triangle ABC.
M32108
15 How many 60° angles can a quadrangle have?
ANG
16 Does there exist a quadrilateral whose three
angles measure 33, 66, 81 ?
D
|7The trapezoid ABCD shown in the figure is
isosceles, and the rays CP and DP are the bisectors
of its obtuse angles. The acute angles of the
AND
trapezoid are 66° each. Calculate the measure of the B
acute angle a between the bisectors.
65
Repetition IlIl
16 In an isosceles triangle, one angle is twice as large as the other angle. Give the
measures of all the angles in this triangle-Theproblem has two solutions.
Give both.
D
19 The sum of three angles of a rhombus is 300". Prove that the length of one of the diagonals
of this rhombus is the same as the length of its side.
D20 One side of a triangle is 10 cm long, and the height drawn to this side is 5cm.
We extend this height four times, but we leave the side unchanged.
Justify that two of the four
statements beloW are true.
The area of this triangle will increase by 75 cm.
"The area of this triangle will increase by 70 cm'.
"The area of this triangle will increase 4 times. AND
"The area of this triangle willincrease 16 times.
DYT Inside the triangle ABC (see the figure) there is
a point P such that PA =7 cm, PB = 3 cm and PC
Repetition =2 cm. Justifythat the perimeter of this triangle
is less than 24 cm.
line. The length of segment XZ is 12 cm,
22 Points X, Y, Z lie on the same straight
the length of segment YZ?
and the length of segment XY is 9cm. What is 2.
cm, b =2 m, c = 2 km. Can the numbers
23 Given segments of lengths: a = 2
x=
=brc atc22y lengths of the sides of a triangle? a+b
BC + DC = BD.
D24 Points A, B, C, D are loat a8 that AB + CB = AC and
straight line.
Prove that these fou t s S ca one
gtire below, are
28The angles a, B. y, locate: as e s
cartkn i2isgle. Assume
the same as the angles cf a
y.
that a = 20, and calculate B and elblc
times the units digit to
BD 26 Justify that adding four result in a
a two-digit number will
number divisible by 5.
D27 The product of five integers is an odd
that the sum of three numbers,
number. Prove G
five,
chosen arbitrarily from among these
is also an odd number.
an edge length of
D28 In a cube ABCDEFGH withmarked out (see the
16 cm, a triangle ACH is than 1meter?
drawing), Is its perimeter greater AND B
Justify your answer.
66
E
Like in an exam
Task 1. (0-1) March 2021)
measures is 2:3:7.
In atriangle, the ratio of the angle
those given.
Complete the sentence. Choose the correct answer from
triangle with the given properties is
A
A. obtuse. B. rectangular. C. acute-angled. D. isosceles.
Task 2. (0-1) March-April 2020
The shaded figure in the picture is a parallelogram.
Assess the truthfulness of the given sentences. Select
Tif the sentence is true or F - if it is false.
The sum of the measures of angles a and Bis 180. P F (135a
Angle a is 3 times smaller than angle B. P F
Task 3. (0-1) March-April 2020
The figure shows an isosceles triangle KLM with sides KM and LM. The measure
of angle KML is twice the measure of angie KLM.
Complete the sentences below. Choose the answer from thnse rsarked with the letters A
and Band the answer from those marked with the ietters C a Repetition
The measure of angle KLM is equal to A/B.
M
A. 40° B. 45°
Triangle KLM is C/D.
C. obtuse angle D. rectangular
Task 4. (0-1) April 2019
The figure shows a parallelogram ABCD and an D
isosceles triangle AED,where |DE|=|AE].
The measure of angle BCE is 106°. What
is the measure of angle AEC? Choose the
AND
correct answer from those given. B
A. 148° B. 122° C. 74° D. 58°
Task 5. (0-1) December 2018
In triangle ABC, the angle at vertex Chas the greatest measure. The measure of the
angle at vertex Ais equal to 48°, and the measure of the angle at vertex Bis equal to the
difference between the measure of the angle at vertex C and the measure of
the angle at vertex A. Assess the truth value of the given sentences. Select Tif the sentence
is true or F if it is false.
The angle at vertex B is 48°. P F
Triangle ABC is right angled. P
67
