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Honors Geometry Review
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Hour:
Chapter 5
Answers
Show all work. Complete this review without a calculator.
Temporary Assumption: ABC with BC > AC and A  B.
1. Write an indirect proof.
Given: ΔABC with BC > AC
Prove: A  B
Argument:
Since A  B, by the Converse of the Base Angles Theorem,
AC  BC and AC = BC (definition of congruent segments). But it is
given that BC > AC. 
A
B
Therefore, in ABC, if BC > AC, A  B
C
2. Write an indirect proof.
Given: ΔABC is scalene, mABX = 36°,
mCBX = 36°
Prove: XB


AC
B
A
X
Temporary Assumption: ABC is scalene, mABX = 36°,
mCBX = 36°, XB  AC
Argument:
Since XB  AC, AXB and CXB are right angles by the definition
of perpendicular segments. So, AXB  CXB by the Right
Angles Congruence Theorem. By the Reflexive Property of
Congruence, XB  XB, so ΔABX  ΔCBX by ASA . By CPCTC,
AB  CB. Therefore, by the definition of isosceles triangle, ΔABC
is isosceles. But, it is given that ΔABC is scalene. 
C
Therefore, if ABC is scalene, mABX = 36°, mCBX = 36°,
then XB  AC
3. Perform the given construction.
a. Construct the circle containing the three vertices
Construct the circumcenter
(point of concurrency of the perpendicular
bisectors of the sides of the triangle) then
construct the circle through the 3 points.
c. Construct the incenter of the triangle shown.
Construct the point of concurrency of
the angle bisectors of the triangle.
b. Construct an Inscribed circle in the triangle
First construct the
incenter (point of
concurrency of the
angle bisectors)
Next, from that center
point, construct the
perpendicular through
that center to one of the
sides in order to get the
radius of the circle.
Then construct the
inscribed circle.
d. Construct the orthocenter of the triangle shown.
Construct the point of concurrency of
the altitudes of the triangle.
4. Construct the centroid of the triangle shown.
Construct the point of concurrency of
the medians of the triangle.
Construct the perpendicular bisectors
of the sides in order to find the
midpoint of each side.
With a straightedge, draw the
segment connecting each vertex to
the midpoint of the opposite side.
5. List all of the properties of a centroid of a triangle.
• Centroid is the point of concurrency of the medians of a triangle
• Centroid must be inside of the triangle
• Centroid is 2/3 length of the median (measured from the vertex)
6. List all of the properties of the incenter of a triangle.
• Incenter is the point of concurrency of the angle bisectors of a triangle
• Incenter must be inside of the triangle and is equidistant from the sides of the triangle
• Incenter is the center of the inscribed circle in a triangle
7. Two sides of a triangle are 7.2 and 3.5.
What are the possible lengths of the third side?
3.7 < x < 10.7
8. Can 4, 4, and 8 be lengths of sides of a triangle?
If so, tell why. If not, tell why not. Be very specific.
No. By the Triangle Inequality Theorem, the sum of any two sides of a triangle must be greater than the
length of the third side. In this triangle, 4 + 4 = 8, which is not greater than 8.
9. Which segment is the longest? Why?
OA is the longest segment.
A
B
C
8°
9°
10
°
O
In ΔCOD, CD < OD < OC
In ΔBOC, BC < OC < OB
In ΔAOC, AB < OB < OA
D
10. List the angles in order from largest to smallest. Explain your answer.
3, 1, 2
98 cm
1
2
97 cm
3
m1 = 45°. In the right triangle on the right,
m3 > m2 (3 is opposite the longer leg).
If m2 is greater than (or equal to) 45°, then
m3 would have to be smaller than (or equal to)
m2 (their sum has to be 90°), which would
contradict the fact that m3 > m2.
So it must be true that m2 < 45°.
You should be able to complete the following problem without a calculator.
11. Consider ΔABC shown. Solve each part algebraically. Show all steps.
Leave fractions in simplest form (no decimals).
a. Find the coordinates of the circumcenter.
C
(−5, 6)
B (5, 2)
1 27
−
,
4 8
A
(−1, −2)
b. Find the coordinates of the centroid.
−
1
, 2
3
c. Find the coordinates of the orthocenter.
−
1
3
, −
2
4
continue 
B
12. Prove the Concurrency of Medians of a Triangle Theorem
GIVEN: ΔABC with centroid P, and midsegment M N .
2
PROVE: A P   A M
3
3
M
P
2
4
1
A
C
N
Statements
Reasons
1. ΔABC with centroid P, midsegment MN
1. Given
2. MN // AB
2. Midsegment Theorem (triangles)
3. 1  2
3. Alternate Interior Angles Theorem
4. 3  4
4. Vertical Angles Congruence Theorem
5. ΔMNP ~ ΔABP
5. AA ~
6. MP = MN
6. Definition of similar triangles
AP
AB
1
7. MN = • AB
2
8. MN = 1
AB
2
MP = 1
9.
AP
2
7. Midsegment Theorem
8. Division Property of Equality and simplify
9. Transitive Property of Equality (steps 6, 8)
10.
MP + 1 = 1 + 1
AP
2
10. Addition Property of Equality
11.
MP + AP= 3
AP
2
11. Simplify
12. AP + MP = AM
12. Segment Addition Postulate
AM 3
13. AP = 2
13. Substitution (steps 11, 12)
14. 2AM = 3 AP
14. Cross Products Property
15. AP =
15. Division Property of Equality and simplify
2 • AM
3
not drawn to scale
C
13. Prove the Concurrency of Angle Bisectors of a Triangle Theorem
5 6
GIVEN: ΔABC, A D bisects CAB, BD bisects CBA, C D bisects ACB
G
DE  A B, DF  BC, DG  C A
3
D
PROVE: DG = DE = DF
A
Statements
Reasons
1. ΔABC, AD bisects CAB, BD bisects CBA,
1. Given
2
1
F
7
4
8
•
11 12
9
10
E
CD bisects ACB, DE  AB, DF  BC, DG  CA
2. 1  2
2. Definition of angle bisector
3. 3 and 12 are right angles
3. Definition of perpendicular segments
4. 3  12
4. Right Angles Congruence Theorem
5. AD  AD
5. Reflexive Property of Congruence
6. ΔADE  ΔADG
6. AAS 
7. DG  DE
7. CPCTC
8. DG = DE
8. Definition of congruent segments
9. 9  10
9. Definition of angle bisector
10. 8 and 11 are right angles
10. Definition of perpendicular segments
11. 8  11
11. Right Angles Congruence Theorem
12. BD  BD
12. Reflexive Property of Congruence
13. ΔDEB  ΔDFB
13. AAS 
14. DE  DF
14. CPCTC
15. DE = DF
15. Definition of congruent segments
16. DG = DE = DF
16. Transitive Property of Equality (steps 8, 15)
B