Honors Geometry
Semester 1 Midterm Review
1.
The supplement of the complement of an angle is 132˚. What is the supplement of the angle?
90 + x = 132
x= 42
supplement = 138
2.
ΔABC  ΔDEF, DF = 10, AB = 18, and the perimeter of ΔABC is 40. Find DE + EF.
Hint: Draw a picture and identify corresponding congruent parts.
DE + EF = 30
3.
The sides of the rectangle are in a ratio 5:7 and the perimeter is 72. Find the area of the
rectangle.
5x + 5x + 7x + 7x = 72
x=3
Arectangle = (15)(21) = 315 u2
4.
An acute angle of a triangle measures (40 – 2x)˚. Find all restrictions on the value of x.
0 < acute angle < 90
0 < 40 – 2x < 90
-40 < -2x < 50
20 > x > -25
5.
6.
-25 < x < 20
What is the measure of the supplement of a 39˚46’ angle?
180˚ - 39̊ 46’ = 140˚14’
The sum of an angle plus its complement plus its supplement is 200 more than the angle. Find
the measure of the supplement of the angle.
x + 90 – x + 180 – x = x + 200
x = 35
supplement = 145̊
C
7.
CAT is obtuse
NAP is acute
AT bisects PAN
N
(72 – 3x)˚
(3x + 54)˚
T
A
(x2)˚
Is CAN a right angle?
P
Justify your reasoning
2
x = 3x + 54
x2 – 3x – 54 = 0
(x – 9)(x + 6) = 0
x = 9 or x = -6 (x = 9 does not work because mNAP = 162̊ and therefore NAP is not acute)
mCAN = 72 – 3(-6) = 72 + 18 = 90̊ → yes, CAN is a right angle.
8.
Given: Circle P
ONE  TEN
R
H
O 5
6 T
Prove: 5  6
P
1. given
3
4
1
2
2. All radii of a circle are
N
congruent.
E
3. 1  2
3. If sides then angles.
4. 3  4
4. Congruent angles subtracted from congruent angles → differences
congruent.
5. OPN  TPE 5. Vertical angles are congruent.
6. ∆OPN  ∆TPE 6. ASA (5,2,4)
7. PON  PTE 7. CPCTC
8. 5  6
8. Angles supplementary to congruent angles are congruent.
1.--------------------2. PN  PE
9.
Given:
Circle O
OM  XY
Prove:
OM bisects XY
O
●
X
1.--------------------2. OX and OY
1. given
2. Two points determine a unique line.
3. OX  OY
4. OM  OM
5. OMX and
OMY rt s
6. ∆OMX and
∆OMY are right ∆
7. ∆OPX  ∆OMY
8. MX  MY
9. OM bisects XY
3. All radii of a circle are congruent.
4. Reflexive
5. Perpendicular → right angles
6. ∆ with one right angle  right ∆
7. HL
8. CPCTC
9. Two congruent segments → bisects
M
Y
Answer “Always,” “Sometimes,” or “Never” to each of the following statements:
10 ______
Supplements of supplementary angles are supplementary.
ALWAYS
11______
The intersection of two rays is a segment.
SOMETIMES (Point, Line or a Ray)
12_______
The union of two rays is a line.
SOMETIMES (Angle, Line or a Ray)
13______
A triangle can have exactly one acute angle.
NEVER
14______
ΔABC  ΔCBA the triangle is equilateral.
SOMETIMES (definitely isosceles based on the congruence statement)
15______
If two sides of one right triangle are congruent to two corresponding sides of
another, then the triangles are congruent.
ALWAYS (either by SAS or HL)
16______
All three altitudes of a triangle fall outside the triangle.
NEVER
17 ______
The median of a triangle is perpendicular to the opposite side.
SOMETIMES (only the median drawn to the base in an isosceles ∆ or in any equilateral ∆)
Supply the reasons for each statement.
18.
AB  BC and P and Q are midpoints of AB and BC respectively.
Why is AP  QC ?
If segments are congruent their like divisions are congruent.
B
12
19.
20.
If 1  2, why is 5  6?
5  6 because they are vertical angles – the congruence
does not depend on 1  2
If AB  BC and BP  BQ , why is AP  CQ ?
If congruent segments are subtracted from congruent
segments, then the difference are congruent.
P 3
4
5
9 10
6
7 Q
8
R
21.
A
If AR  BR and CR  BR , why is AR  CR ?
Segments congruent to the same segment are congruent.
22.
If 3  7, why is 4  8?
Angles supplementary to congruent angles are congruent.
23.
If 9  10, and 5  6, why is ARB  CRB?
If congruent angles are added to congruent angles then the sums are congruent.
24.
If BR bisects PBQ, why is PR  QR ?
This is not necessarily true. If “sides” then “angles” cannot be used here.
25.
If PC and QA bisect ARB and BRC respectively, why is ARB  BRC?
The angles are not necessarily congruent.
26.
If P and Q are midpoints of AB and BC respectively, why is BP  BQ ?
If segments are congruent then their like divisions are congruent.
27.
If ΔBAQ  ΔBCP, why is PR  QR ?
The segments are not necessarily congruent.
C
28.
29.
ΔABC is isosceles.
What is the perimeter?
One of the following must be true if ∆ABC is isosceles
AB  AC
AB  BC
AC  BC
4x+8 = 3x+4
3x+4 = 5x-8
4x+8=5x-8
x = -4
x=6
x = 16
AB = 4(-4) + 8
P∆ABC = 76
P∆ABC = 196
AB = -8
Is ΔABC isosceles?
Why?
A = 180 – 38 – 58 = 84̊
Since no two angles are congruent, the triangle
is not isosceles.
A
3x + 4
4x + 8
B
C
5x - 8
A
3x + 18
4x - 2
B
38°
58°
48
C
Solve:
30.
31.
3x + 10 = 2x + y
2x + y + 2y + 30 = 180
x – y = -10
2x + 3y = 150
y = 2x - 3
3y – x = 7
x – 34 = -10
x = 24
-2x + 2y = 20
2x + 3y = 150
5y = 170
y = 34
32.
Find the perimeter
ΔABC  ΔDEF
AB = 3x2 + 2x
BC = 4x + 8
AC = 18
ED = 16
-2x + y = -3
-x + 3y = 7
6x – 3y = 9
-x + 3y = 7
5x = 16
x = 16/5
of ΔDEF
AB = DE
3x2 + 2x = 16
3x2 + 2x – 16 = 0
(3x + 8)(1x – 2 ) = 0
x = -8/3 or x = 2
x = -8/3 results in BC having a negative length
For x = 2, P∆DEF = 50 u
y = 2(16/5) – 3
y = 17/5
33.
An angle has a measure of (90 – a)°,
What is the measure of its complement?
90 – (90 – a) = 90 – 90 + a = a
What is the measure of its supplement?
180 – (90 – a) = 180 – 90 + a = 90 + a
34.
The ratio of an angle to its complement is 7:3, what is the ratio of the angle to its
supplement?
x
7

90  x 3
3x = 7(90 – x)
3x = 630 – 7x
10x = 630
x = 63
angle:supplement = 63:117
35.
The measure of the supplement of an angle exceeds three times the measure of the
complement of the angle by 40°. Find the measure of half the complement.
180 – x = 3(90 – x) + 40
180 – x = 270 – 3x + 40
180 – x = 310 – 3x
2x = 120
x = 60
complement = 30̊
½ complement = 15̊
38.
mACB = 42°
mADC = 23°
AD is perpendicular to BC
mABD = 109̊
B
.
A
All radii of a circle are congruent.
Therefore, ∆ABC  ∆ABD by SSS.
C
.
D