INTRO TO MATH 426
SIMILAR AND CONGRUENT TRIANGLES
BANK OF QUESTIONS FOR EXTRA PRACTICE
1
To draw the plans for a roof, we first draw horizontal segment AC. Then, from the midpoint of
segment AC, we draw perpendicular DB. Finally, we draw segments AB and BC.
B
A
C
D
Using a formal proof, show that triangles ABD and CBD are congruent.
2
In which of the following situations is there sufficient information to conclude that triangles ABC and
DEF are congruent?
A)
B)
C)
D)
m A = 70
m B = 35
m C = 75
m D = 70
m E = 35
m F = 75
m A = 70
m B = 35
m AB = 5 cm
m D = 70
m E = 35
m EF = 5 cm
m A = 70
m C = 75
m AC = 10 cm
m D = 70
m F = 75
m DF = 10 cm
m A = 70
m AB = 5 cm
m AC = 10 cm
m D = 70
m DE = 5 cm
m EF = 10 cm
3
In the figure on the right,
H
G
BF  DH
A
F
B
O
CG  AE
E
AOB  COD
C
D
O is the midpoint of segments BF, DH, CG and AE.
Which property can be used to prove that the 4 triangles are congruent?
4
A)
A.A.
C)
S.A.S.
B)
S.S.S.
D)
A.S.A.
In which of the following situations is there enough information to conclude that triangles ABC and DEF are
congruent?
A)
B)
C)
D)
m B = 45
m AB = 12 cm
m AC = 6 cm
m E = 45
m EF = 12 cm
m DF = 6 cm
m C = 60
m AC = 4 cm
m BC = 9 cm
m F = 60
m DF = 4 cm
m EF = 9 cm
m A = 75
m B = 45
m AB = 8 cm
m D = 75
m E = 45
m DF = 8 cm
m A = 75
m C = 60
m AB = 10 cm
m D = 75
m F = 60
m EF = 10 cm
5
In triangle ABC below, A' C' is parallel to AC .
A
4
A’
15
C
8
C’
B
The following reasoning indicates how the measure of A' C' can be calculated.
STATEMENT
JUSTIFICATION
1.
B  B
1.
By reflexivity
2.
BA'C'  BAC
2.
If a transversal cuts two parallel lines, then the
corresponding angles are congruent.
3.
BA'C' ~ BAC
3.
Two triangles which have two corresponding
angles congruent are similar (AA).
4.
m BA' m C' A'

m BA
m CA
4.
?
m C' A'
8

12
15
Which justification completes the reasoning in step 4?
A)
In similar triangles, corresponding angles are congruent.
B)
In similar triangles, corresponding sides are congruent.
C)
In similar triangles, the measures of corresponding sides are proportional.
D)
In similar triangles, the measures of corresponding angles are proportional.
6
D
In the adjacent figure, AB is parallel to DE .
9
B
C
The following steps can be used to determine the
6
measure of BC .
E
3
A
Statements
Justifications
1
m ACB = m DCE
1
Vertically opposite angles are congruent.
2
m BAC = m CDE
2
If a secant cuts two parallel lines, then the alternate
interior angles are congruent.
3
ABC ~ CDE
3
Two triangles are similar if they have respectively at least
two congruent angles (AA).
4
m AC m BC

m CD m CE
4
?
3 m BC

9
6
Which justification completes the reasoning in step 4?
A)
If two triangles are similar, their corresponding angles are proportional.
B)
If two triangles are similar, their corresponding sides are proportional.
C)
Triangles are similar if the measures of their corresponding sides are proportional.
D)
Triangles are similar if the measures of their corresponding angles are proportional.
7
In the figure below, diagonal AC bisects angles A and C.
B
C
A
D
Which property can be used to prove that triangles ABC and ADC are isometric?
8
A)
If the corresponding sides of two triangles are isometric, then the triangles are isometric.
B)
If two angles and the contained side of one triangle and the corresponding two angles and
contained side of another triangle are isometric, then the triangles are isometric.
C)
If two sides and the contained angle of one triangle and the corresponding two sides and
contained angle of another triangle are isometric, then the triangles are isometric.
D)
If two angles of one triangle and the two corresponding angles of another triangle are
isometric, the triangles are isometric.
Which of the four pairs of triangles below consists of two triangles that are definitely congruent?
A)
C)
4 cm
98°
3 cm
3 cm
40°
5 cm
4 cm
45°
5 cm
45°
98°
40°
B)
D)
5 cm
45°
10 cm
105°
30°
10 cm
105°
5 cm
9
In the following figure:
A
Lines RC and DQ are parallel.
Lines BD and CP intersect at point A.
?
R
B
m  CBD = 120
m  QEP = 40
C
120
D
Q
E
40
P
The following is part of a procedure used to determine the measure of angle BAC.
Step 1
m  ABC + m  CBD = 180
because ...
m  ABC + 120 = 180
m  ABC = 60
Step 2
m  BCA = m  QEP = 40
because ...
Step 3
m  ABC + m  BCA + m  BAC = 180
because the sum of the measures of the
interior angles of a triangle is 180.
60 + 40 + m  BAC = 180
m  BAC = 80
Complete steps 1 and 2 of this procedure.
10
In the diagram below, line segments AE and BD intersect at C.
In addition:
D
A
m AC  50 cm
C
m BC  40 cm
m DC  100 cm
m EC  125 cm
B
E
The following is part of a procedure used to show that  ABC   EDC.
Step 1
 BCA   DCE
because vertically opposite angles are congruent.
Step 2
m AC m BC
50 cm
40 cm

because

125 cm 100 cm
m EC m DC
Step 3
 ABC ~  EDC
because ...
Step 4
 ABC   EDC
because ...
Complete steps 3 and 4 of this procedure.
11
In the figure to the right, segments DE and CB are
parallel and they measure 6 units and 10 units
respectively. Segment EB measures 3 units.
A
?
E
6
3
D
B
10
C
What is the measure of segment AE?
12
A)
2 units
C)
5 units
B)
4.5 units
D)
7.5 units
State the property by which you can conclude that triangle Z is congruent to triangle Y.
C
m A = 56
m AC = 2.5 cm
m BC = 3 cm
m B = 36
Y
m AB = 4 cm
B
A
m AC = 2.5 cm
C'
m AB = 4 cm
Z
m BC = 3 cm
A'
B'
m C = 88
13
The figure below shows triangle ABC and triangle ADE. The data given on the figure can be used to prove that
these triangles are congruent.
C
B
D
4.5 cm
65
25
E
4.5 cm
A
Below is the reasoning which shows that triangle ABC is congruent to triangle ADE.
STATEMENT
1.
m C = m E = 90
JUSTIFICATION
1.
These data are given in the problem.
m AC  m AE  4.5 cm
m BAC = 25
m ADE = 65
2.
m DAE = 25
2.
The acute angles in a right triangle are
complementary.
3.
m BAC = m DAE
3.
By transitivity
4.
ABC  ADE
4.
?
Which of the following is the justification for statement 4?
A)
Two triangles are congruent if they have one congruent angle bounded by two
corresponding congruent sides.
B)
Two triangles are congruent if they have three corresponding congruent sides.
C)
Two triangles are congruent if they have one congruent side between two corresponding
congruent angles.
D)
Two triangles are congruent if they have two corresponding congruent angles.
Correction key
1
Work : (example)
B
A
C
D
Statements
Justifications
1.
AD  DC
1.
The hypothesis states that D is the midpoint of segment AC.
2.
BD  BD
2.
Reflexive property.
3.
ADB  CDB
3.
The hypothesis states that DB  AC .
4.
ABD  CBD
4.
If two sides and the contained angle of one triangle are
congruent to two sides and the contained angle of another
triangle, then the triangles are congruent.
or
S-A-S.
2
C
3
C
4
B
6
B
7
B
8
D
5
C
9
Step 1
m  ABC + m  CBD = 180
because adjacent angles whose external sides are in a
straight line are supplementary.
m  ABC + 120 = 180
m  ABC = 60
Step 2
m  BCA = m  QEP = 40
Step 3
m  ABC + m  BCA + m  BAC = 180
because alternate exterior angles are congruent when
formed by a transversal intersecting two parallel lines.
because the sum of the measures of the
interior angles of a triangle is 180.
60 + 40 + m  BAC = 180
m  BAC = 80
Step 3
10
 ABC ~  EDC
because
Step 4
if the lengths of two sides of one triangle are proportional to the lengths
of the two corresponding sides of another triangle and the contained
angles are congruent, then the triangles are similar.
 ABC   EDC
because
the corresponding angles of similar figures are congruent.
11
B
12
If three sides of one triangle are congruent to three sides of another triangle, the two triangles are congruent.
13
C