Review Test 2
Math 236 Spring 2005
1. a. Given ∆RST ≅ ∆JKL , complete the following statement ∆SRT ≅ __________
b. You are given ∆RST and ∆XYZ with ∠S ≅ ∠Y . To show that ∆RST ≅ ∆XYZ by the
ASA (Angle Side Angle) congruence property, what more would you need to know?
2. i. List all combinations of corresponding sides and angle measurements
where congruence of two triangles is assured.
ii. . List all combinations of corresponding sides and angle measurements
where congruence of two triangles is not assured.
D
E
3. The figure to the right can be used to show Side Side Angle is not a
a condition of congruence.
a. Name the two triangles that satisfy SSA (Side Side Angle) but are not congruent.
b. Mark on the figure the sides that are congruent and angles that are congruent.
A
B
4. Matt constructs a triangle with one side of length 6 cm and one side of length 8 cm and
included angle of measure 40° and Esmeralda also constructs a triangle with one side of length
6 cm and one side of length 8 cm and included angle of measure 40°. Must the two triangles
must be congruent. WHY OR WHY NOT?
5. Refer to the figure. Using only the indicated information, can it be shown
that the two triangles are congruent? WHY OR WHY NOT?
6. For each of the situations below, draw two triangles, ABC and MLN, and mark
them and state why they are congruent (just use abbreviations for the congruence
properties.)
i. ∠A ≅ ∠M , ∠B ≅ ∠L, AB ≅ ML
ii. AB ≅ ML; BC ≅ LN ; AC ≅ MN
A
P
7. Which two triangles to the right are congruent?
Be careful not to rely on your perception.
a. State the congruence correctly
+ _______ ≅ + _________
b. State the property you used to determine that
the two are congruent.
N
28°
28°
3.8
4
4
2
3.8
28 °
L
4
M
Q
U
2
C
B
c. Now explain why you can not state conclusively that the third triangle is congruent to the other two.
8. Use the two triangles below to explain whether this statement is true or false “Any two isosceles
triangles are similar”.
2
2
2
2
1
9. Provide justifications in the proof below
Given: DA ≅ AB ≅ BC ≅ CD
A
D
2
1
4
3
B
Prove. Quad ABCD parallelogram
C
STATEMENTS
1. DA ≅ AB ≅ BC ≅ CD .
2. DB ≅ DB
∆DAB ≅ ∆BCD
3.
4. ∠1 ≅ ∠ 4 and ∠ 2 ≅ ∠ 3
REASONS
1. Given
2.
3.
4.
5.
6.
5.
6.
↔ ↔
↔ ↔
AB ll CD and AD ll CB
Quad ABCD is a
parallelogram
10. Draw a circle. Inscribe an equilateral triangle in the circle using only a straightedge and
compass. Show all your construction marks.
11. Name all these constructions, dot in the rhombus being used in the construction, and
describe the property of a rhombus that justifies the construction.
a.
b.
c.
A
d.
e.
B
j
12. Name this construction on the right.
How are lines j and k constructed?
What does point D represent?
What is the final step in this
construction?
k
D
2
13. Name this construction on the right.
How are lines l and m constructed?
What does point E represent?
What are the final steps in this construction?
l
m
E
14. Perform the following constructions using only a straightedge and compass. Be sure to
show all construction marks.
• Dot in the RHOMBUS that makes the construction work.
• State the property (or properties) of the rhombus that makes the construction work.
(a) Bisect an angle.
b) Construct a line perpendicular to the given line through the point P on the line.
(c) Given a point Q not on a line m construct a line through point Q that is parallel to line m.
d) Given a line segment, construct its perpendicular bisector.
e) Construct a line perpendicular to a given line through the point P, not on the line.
f. Circumscribe a circle about the given scalene triangle. Use a straightedge and compass.
Show all of your construction marks.
g. Inscribe a circle in this triangle using only a straightedge and compass. Show all your
construction marks.
3
P
15. . In the given diagram LM || PO , and LN ≅ ON
Note:
∆LMN is
5
L
congruent to ∆ OPN
2
a. State the two congruence properties you could use to justify the congruence,
1
N 4
6
b. Justify your reasoning for each property. (why certain sides would be congruent
or angle measures congruent)
3
O
M
16. Look back at the construction in problem 12. Since each vertex of the
triangle lies on the circle, all three vertices are equidistant from the center and we use that idea
to find the center of the circle. We use perpendicular bisectors (actually the intersection of two
of them) to locate the center. Why does this work? We need to prove that any point on the
perpendicular bisector of a line segment is equidistant from the endpoints of the line
segment.
C
What make up our line segments are the sides of the triangle!
Provide the following reasons for the proof we need:
A
1. AD = BD and ∠ADC ≅ ∠BDC by construction
D
2. CD = CD equal to itself
3. + ADC ≅+ BDC
(Give reason here!)
4. AC = BC
(give reason here!)
B
17. Look back at the construction in problem 13. Here the center of the circle must be
equidistant from each side of the triangle. We use angle bisectors to find the center – why?
Two sides of the triangle make up an angle
B
Construct the angle bisector
Then
∠1 ≅ ∠2 by construction
1
C
Pick any point C on the bisector
A
2
Look at the distance from C to each side of the angle (the distance is
measured as the length of the line from C perpendicular to the side of the triangle.
D
Provide a justification that + ABC ≅+ ADC and that BC = DC.
18. Using measurements given, find the distance PQ across the
river.
Explain any major mathematical ideas you are using for
this indirect measurement.
P
10m
20m
4m
river
Q
19. Some hunters want to know the distance across a canyon
and found the following measurements. Help them find the distances
BE and AC.
60 ft
A
B
25 ft
E
20. In the following situations, explain why the pairs of triangles are/or are not
similar. Find any missing side lengths.
.
H
4.8
85°
10
a.
b.
3
75°
20°
G
40 ft
47.2 ft
C
20°
J
15
6
16
K
10
G
4