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SECONDARY 4 MATHEMATICS
PROOFS IN GEOMETRY - REVIEW
1
You are asked to prove the following proposition :
The segment joining the midpoints of two sides of a triangle is parallel to the third side and is half as long as
the third side.
To prove : in ABC
A(0, 0)
B(10, 0)
y
C(8, 8)
C(8, 8)
D
E
D is the midpoint of AC
E is the midpoint of BC
Prove :
B(10, 0) x
A(0, 0)
DE // AB
m DE =
1
m AB
2
Statements
1. The coordinates of point D are (4, 4) and those
of point E are (9, 4)
Justifications
1.
?
2. The slope of DE is :
44 0
= =0
94 5
2. Formula for slope
3. The slope of AB is :
00
0
= =0
10  0 10
3. Formula for slope
4. Therefore DE // AB
4.
5. d(D, E) = |9  4| = 5
5. Distance formula
6. d(A, B) = |10  0| = 10
6. Distance formula
7. Therefore m DE =
1
m AB
2
?
7. By substitution of the measures
What are the appropriate justifications to support statements 1 and 4?
A)
B)
C)
D)
1.
Formula for the distance between a point and a line
4.
Two lines are parallel if their slopes are equal.
1.
Formula for midpoint
4.
Two lines are parallel if their slopes are equal.
1.
Formula for the distance between a point and a line
4.
Two lines, not parallel to the vertical axis, are perpendicular if and only if their slopes
are opposite of the reciprocals of each other.
1.
Formula for midpoint
4.
Two lines, not parallel to the vertical axis, are perpendicular if and only if their slopes
are opposite of the reciprocals of each other.
2
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.
3
To determine the measure of angle D in the figure at right, the
following facts are used:
B
C
- the measure of angle BAC is 30,
- triangle ADE is the image of
D
triangle ABC by a rotation of
30
-70 about point A.
E
A
Below is the reasoning used to determine m D.
STATEMENT
1. ADE  ABC
JUSTIFICATION
By hypothesis, ΔADE is the image of ΔABC by a
rotation and every rotation is an isometry.
2. m D = m B
3. m B = 180  (90 + 30)
m B = 60
4. m D = 60
?
The sum of the measures of the interior angles of a
triangle is 180.
The equality relation is transitive.
Which of the following is the justification for statement 2?
A)
The measure of the side opposite the 30 angle is equal to half the measure of the
hypotenuse.
B)
The acute angles in a right triangle are complementary.
C)
Congruent figures have the same measures of angles.
D)
The relation of congruence is reflexive.
4
In triangle ABC, parallelograms AJMK and JMCK are drawn such that KJ is congruent to KM and m A = 42.
B
J
A
M
C
K
The following reasoning can be used to determine the measure of C.
STATEMENT
JUSTIFICATION
1.
m KJ = m KM
Given
2.
m MJK = m JMK
In an isosceles triangle, the angles opposite the congruent sides
have equal measures.
3.
m A = m JMK
_____________________________________________
m MJK = m C
_____________________________________________
4.
m A = m C
By transitivity
5.
m C = 42
m A = 42 (ABC has two congruent angles.)
Which statement completes step 3 of the reasoning?
A)
When a transversal cuts parallel lines, alternate interior angles are congruent.
B)
When a transversal cuts parallel lines, alternate exterior angles are congruent.
C)
When a transversal cuts parallel lines, corresponding angles are congruent.
D)
In a parallelogram, opposite angles are congruent.
5
In the figure below, line MN is parallel to line OP, lines EC and BD intersect at point A and AE  AC .
O
M
E
D
A
B
C
P
N
The following proof can be used to show that triangles ADE and ABC are congruent.
Statements
Justifications
1.
AED  ACB
1.
If a secant cuts two parallel lines, then the alternate interior
angles are congruent.
2.
AE  AC
2.
Given
3.
EAD  CAB
3.
Vertical angles formed by two intersecting lines are
congruent.
4.
ADE  ABC
4.
?
Which of the following can be used to justify the fourth statement of this proof?
A)
If two angles and the included side of one triangle are congruent to two angles and the
included side of another triangle, then the triangles are congruent. (ASA)
B)
If three sides of one triangle are congruent to three sides of another triangle, then the
triangles are congruent. (SSS)
C)
If two sides and the included angle of one triangle are congruent to two sides and the
included angle of another triangle, then the triangles are congruent. (SAS)
D)
If two angles of one triangle are congruent to two angles of another triangle, then the
triangles are similar. (AA)
6
In the figure below, BC  DC and AC is the bisector of angle BCD.
B
A
C
D
To prove triangles ABC and ADC congruent, the following reasoning is used :
STATEMENTS
JUSTIFICATIONS
1.
ACB  ACD
1.
AC is the bisector of angle BCD.
2.
BC  DC
2.
Given
3.
AC  AC
3.
Reflexive property
4.
ABC  ADC
4.
?
Which of the following reasons can be used to justify the fourth statement?
A)
If two angles and the contained side of one triangle are congruent to two angles and the
contained side of another triangle, then the triangles are congruent. (ASA)
B)
If three sides of one triangle are congruent to three sides of another triangle, then the
triangles are congruent. (SSS)
C)
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. (SAS)
D)
If two angles of one triangle are congruent to two angles of another triangle, then the
triangles are similar. (AA)
7
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'
4.
m BA

m C' A'
m CA
?
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.
8
In the adjacent figure, AB is parallel to DE .
D
9
B
C
6
The following steps can be used to determine the
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
4
m CD

m BC
?
m CE
3 m BC

9
6
Which justification completes the reasoning in step 4?
A)
Triangles are similar if their corresponding angles are congruent.
B)
Triangles are similar if their corresponding sides are congruent.
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.
9
Polygons ABCDE and A'B'C'D'E' are similar. Segment
AC and A'C' are diagonals.
B
B’
C
A
C’
A’
The following proof can be used to show that
triangles ABC and A'B'C'D' are similar.
STATEMENT
D’
D
E’
JUSTIFICATION
1.
B  B'
In similar polygons, corresponding angles are congruent.
2.
m AB
m BC

m A' B' m B' C'
In similar polygons, corresponding sides are proportional.
3.
ABC ~ A'B'C'
?
Which of the following can be used to justify the third statement of this proof?
A)
Two triangles are similar if an angle of one is congruent to an angle of the other and the
sides about these angles are proportional.
B)
Two triangles are similar if their three corresponding sides are proportional.
C)
Two triangles are similar if a side of one is proportional to a side of the other and the angles
about these sides are congruent.
D)
Two triangles are similar if two angles of one triangle are congruent to two angles of the
other triangle.
10
In the figure on the right,
m DA = 8 cm
A
m EB = 12 cm

3 m EB
m EC =
4
m DC =

C

3 m DA
4
E

Show that triangle ABC is similar to triangle DEC.
Justify each step followed.
B
D
11
In the parallelogram ABCD on the right,
ACB  FED
A
E
D
m AB = 22 cm
F
m BC = 47 cm
B
m AC = 45 cm
m EF = 18 cm
Show that segment DF measures 8.8 cm.
Justify each step of your proof.
C
12
Diagonal BD is drawn in parallelogram ABCD shown below.
A
D
B
C
The following is part of a procedure used to prove that triangle ABD and CDB are congruent.
Step 1
AB  DC
because the opposite sides of a parallelogram are congruent.
Step 2
DAB  BCD
because…
Step 3
ABD  CDB
because…
Step 4
In conclusion, triangles ABD and CDB are congruent because two angles and the
contained side of one triangle are congruent to the corresponding angles and contained
side of the other triangle.
Complete steps 2 and 3 of this procedure.
13
In the following figure:
Lines RC and DQ are parallel.
Lines BD and CP intersect at point A.
m  CBD = 120
m  QEP = 40
A
?
R
B
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
In your answer booklet, complete steps 1 and 2 of this procedure.
14
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
B
m EC  125 cm
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
Step 3
m AC
m EC

m BC
m DC
because
50 cm
40 cm

125 cm 100 cm
 ABC ~  EDC
because ...
Step 4
 ABC   EDC
because ...
In your Answer Booklet, complete steps 3 and 4 of this procedure.
SECONDARY 4 MATHEMATICS
PROOFS IN GEOMETRY – REVIEW
CORRECTION KEY
1
B
2
C
3
C
4
D
5
A
6
C
7
C
8
C
9
A
10
Work : (example)
m DA = 8 cm
A
m EB = 12 cm

3 m EB
4
m EC =


3 m DA
m DC =
4
C
E

Statements
1.
2.
3.
3
 8 cm = 6 cm
4
m AC = 2 cm
8 cm  6 cm = 2 cm
m EC = 9 cm
3
 12 cm = 9 cm
4
m BC = 3 cm
12 cm  9 cm = 3 cm
m AC
2 3

6 9
m DC
m BC
m EC
D
Justifications
m DC = 6 cm

B
4.
ACB  DCE
Vertically opposite angles are congruent.
5.
ABC ~ DEC
Two triangles are similar if the angles included between
corresponding proportional sides are congruent (S.A.S.
Similarity Theorem).
11
Work : (example)
A
E
D
F
B
C
Statements
1.
ABC ~ FDE
Justifications
Two pairs of corresponding angles are congruent :
ACB  FED given in the hypothesis,
B  D because the opposite angles of a parallelogram are
congruent.
2.
m DF = 8.8 cm
m DF
m BA

m EF
In similar triangles, the corresponding sides are proportional.
m CA
m DF 18

22
45
hence, m DF = 8.8 cm
By substitution
12
Step 1
AB  DC
because the opposite sides of a parallelogram are congruent.
Step 2
DAB  BCD
because the opposite angles of a parallelogram are congruent.
Step 3
ABD  CDB
because they are alternate interior angles formed by a transversal intersecting two parallel


segments AB // CD .
Step 4
In conclusion, triangles ABD and CDB are congruent because two angles and the contained side of
one triangle are congruent to the corresponding angles and contained side of the other triangle.
Note:
Give two marks if only one of the two steps is correctly completed.
To earn marks for Step 3, students must specify that the angles are formed by a transversal
intersecting parallel segments.
13
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
14
Step 3
 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.