husd high school geometry semester 1 study guide · husd high school geometry semester 1 study...

HUSD High School Geometry Semester 1 Study Guide

of 18 MCC@WCCUSD (HUSD) 12/06/12

1 What are the three undefined terms of Geometry? Point, Line and Plane are undefined terms. If two lines intersect, they intersect in _______? If two lines intersect, they intersect in a point. Any three noncolinear points are ________? Any three noncolinear points are coplanar. What is the definition of a midpoint?

A midpoint of a line segment is the point that bisects the line segment. That is, M is the midpoint of AB if .AM MB≅

1.0

1´ You try:

1a) What is the definition of an angle bisector?

1b) Name the intersection of plane BPQ and plane CPQ.

1c) Supplementary angles are two angles whose measures have the sum _______.

Complementary angles are two angles whose measures have the sum _______.

1d) Are O, N, and P collinear? If so, name the line on which they lie. If not, name all the lines that pass through them.

M

N

O

P



2 m AOB∠ =_______.

1.0/4.MD.6

3 BD

is the bisector of .ABC∠ If

52m DBC∠ = ° , then m ABD m ABC∠ + ∠ = __

1.0/7.G.5,G.CO.1

2´ You try: 2)

3´ You try:

3) BD

is the bisector of .ABC∠ If 46 ,m DBC∠ = ° then m ABD m ABC∠ + ∠ = __

140 5090 .

m AOB∠ = °− °= °

m AOB∠ = ________.

A

B

O

A

O

B

C

D

B

A

BD

is the bisector of ABC∠

52 .m ABD m DBC⇒ ∠ = ∠ = °

52 52104

m ABD m DBC m ABCm ABCm ABC

∠ + ∠ = ∠° + ° = ∠

° = ∠

52 104156

m ABD m ABC∴ ∠ + ∠= °+ °= °

C

D

B

A



4 Points A, B, and C are collinear, but they do not necessarily lie on the same line in the order named. If 3AB = and 8BC = , what is AC ? There are two possibilities:

OR

1.0/G.CO.1

5 Give an example of the Symmetric Property

of Congruence.

€

If AB ≅ CD, then CD ≅ AB

1.0

6 What kind of angles are AOB∠ & BOC∠ ?

AOB∠ and BOC∠ are adjacent angles.

They also form a linear pair and are, therefore, supplementary.

180m AOB m BOC∠ + ∠ = °

13.0/7.G.5

4´ You try: 4) Points A, B, and C are collinear, but they do not necessarily lie on the same line in the order named. If 9AB = and 5,BC = what is AC ?

5´ You try:

5) Give an example of each of the following properties of angle congruence.

a) Reflexive

b) Transitive c) Symmetric

6´ You try:

6) If 2 5m AOB x∠ = − and

3 10m BOC x∠ = + , what is the value

of x?

3

C A B

8 3 811

AC = +=

3

C B A

8

8 35

AC = −=

C O

B

A

C O

B

A



7 In the figure shown,

€

m∠AEB = 55°. What is

€

m∠DEC? Find the value of x and

€

m∠AED and m∠BEC.

The

€

m∠DEC = 55° because vertical angles are congruent.

€

m∠AED = m∠BEC7x − 8 = 6x +11

x =19

€

∴

€

m∠AED = 7 19( ) − 8=125

m∠BEC = 6 19( ) +11=125

13.0/7.G.5

7´ You try: 7a)

€

m∠3 = 37. Find m∠1.

7b) In the figure below, name two pairs of vertical angles. Find the values of x and y. Find

€

m∠AED and

€

m∠CED.

1 4 2

3



8 Find the value of k. The diagram is not to scale.

The sum of the measure of the angles of a triangle is 180º 45º + 62º + kº = 180º 107º + kº = 180º kº = 73º

Find the value of x. The diagram is not to scale.

The measure of each of the exterior angles of a triangle is equal to the sum of the measures of its two remote interior angles.

€

x° + 59° =106°

x° = 47°

12.0/G.CO.10

8´ You try: 8a) Find the values of x, y, and z. The diagram is not to scale.

8b) Find the smallest angle in the triangle shown below. The diagram is not to scale.

8c) Find the measure of

€

∠A in the triangle shown below. The diagram is not to scale.



9 Find the distance from ( )5, 2− to ( )1, 4− − .

OR

17.0/N.CN.6

10

The coordinates of the midpoint are the averages of the coordinates of the endpoints:

17.0/N.CN.6

9´ You try:

9) Find the distance from ( )6, 1− to ( )4, 6− − .

10´ You try:

10) Find the midpoint of the segment whose

endpoints are ( )3, 9− and ( )5, 2− − .

2 2 2

2 2 2

2

2

2 64 3640

40

2 10

a b cccc

c

c

+ =+ =+ =

=

=

=

d = x2 − x1( )2+ y2 − y1( )

2

d = (−1−5)2 + (−4−−2)2

= (−6)2 + (−4+ 2)2

= −6 ⋅−6+ (−2)2

= 36+ 4

= 40

= 2 10

( )5, 2−

( )1, 4− −

c a

b

Find the midpoint of the segment whose endpoints are ( )2, 7− and ( )3, 1− − .

The coordinates of the midpoint are the averages of the coordinates of the endpoints:

x1 + x22

,y1 + y22

!

"#

$

%&

2+−32,−7+−12

!

"#

$

%&

=−12,−82

"

#$

%

&'

= −12,−4

"

#$

%

&'



11

7.0/G-CO.9

12 What does the Isosceles Triangle, or Base

Angles, Theorem say aboutABC ?

1.0/G.CO.10

11´ You try: 11)

12´ You try:

12) In the triangle below, what is the value of x?

p

q

p 5 18x +

10 12x +

p q . Solve for x. p q . Solve for x.

2 5x+ p

q

p

5 4x −

If AB BC≅ , then A C∠ ≅ ∠ .

OR

If BC CA≅ , then B A∠ ≅ ∠ .

OR

If CA AB≅ , then B C∠ ≅ ∠

(3 1)x + ° 61° A

B

C

The angles whose measurements are given are same-side, or consecutive, interior angles. Because the lines are parallel, these angles are supplementary. Therefore,

(5x +18)+ (10x +12) =18015x +30 =180

15x =150x =10



13 In ABC , if A B C∠ ≅ ∠ ≅ ∠ , then what do you know about AB , BC , and AC ?

They are all congruent.

1.0,12.0/7.G.5

14 ABC ≅DEF by which postulate or

theorem?

5.0/G.SRT.5

13´ You try: 13) If ABC below is equiangular, what is the value of x?

14´ You try:

Name the postulate or theorem by which the congruence statement is true. 14a) ABC ≅DEF 14b) ACB ≅DCB 14c) ABC ≅DEC

A

B

3 6x +

C

5x

C B

A

F E

D

The ASA Congruence Postulate.

C A

B

F D

E

A

B

C D

Bx

C

D

E A



15

Which statement is correct for the diagram above?

5.0/G.SRT.5

16 Given:

Prove:ABC ≅ADC

5.0/G.SRT.5

15´ You try:

C

B

A D

15) Which statement is correct for the diagram above?

A) ABD ≅BDC

B) ADB ≅CDB

C) ABC ≅BCD

D) DBA≅DCB

E) None of the above.

16´ You try:

16) Given:

A B

C

D E

A) ABC ≅CDE

B) ABC ≅DEC

C) ACB ≅DCE

D) ABC ≅EDC

E) both B) and C)

By the Alternate Interior Angle Theorem, A D∠ ≅ ∠ and B E∠ ≅ ∠ . Therefore, A must

correspond to D and B to E. That happens in both B) and C). ∴ E) is the correct answer.

D

C

B

A

Statements Reasons

1. AB AD≅

BC DC≅

1. Given as true.

2. AC AC≅

2. Reflexive Prop. of ≅ .

3. ABC ≅ADC 3. SSS ≅ Post.

E

B

D

A

C

Prove: ABE ≅DCE



17

16.0/G.CO.12

17´ You try: Do each of the following constructions: 17a) Copy a line segment. 17b) Bisect a line segment. 17c) Copy an angle. 17d) Bisect an angle.

Construct a line through P parallel to line l.

l

P

Draw any line through P and l. We will copy PAB∠ using only a straightedge and compass.

Make an arc from A (using A as center). Make the same arc from P:

X

Y

B A l

P

“Measure” how much PAB∠ opens up by putting the compass point and pencil point at X and Y. Make a mark to show that you measured correctly:

X

Y

B A

P

Z

Make that same mark from Z, crossing the arc you made from P:

X

Y

B A

P

Z

W

Draw PW

. PW

will be parallel to line l because ZPW YAB∠ ≅ ∠ and are corresponding angles.

We will construct

PW

parallel to l using corresponding angles:



18 In a 30 60 90°− °− ° triangle what is the relationship between the leg opposite the 30° angle and a) the hypotenuse? b) the long leg? Answers: a) The hypotenuse is twice as long as the leg opposite the 30° angle.

b) The longer leg is 3 times as long as the shorter leg. Side length ratios of

Short leg: Long leg: Hypotenuse are 1: 3 : 2

20.0/G.SRT.6,7

19

?x = By the SAS Similarity Theorem 64=32and vertical angles are congruent

!

"#

$

%& ,

The two triangles are similar. Therefore,

32 32 9

92

x

x

x

=

=

=

5.0/G.SRT.5

18´ You try: 18a) Find AB.

18b) Find AB.

19´ You Try:

19) Solve for x:

3

30°

C B

A

xx

6 2

3

4 3

3

2

xx

3

A

B

C 30°

9



20 Complete the statement:

If DE BC , then 10 ?10 x

=+

Because DE BC , C AED∠ ≅ ∠ (corresponding ∠ ’s are ≅ if the lines are parallel) and for the same reason, B ADE∠ ≅ ∠ .

By the AA Similarity postulate, ADE ABC :

10 510 7x

∴ =+

5.0/G.SRT.5

20´ You try: 20a) Name the triangles that are similar. 20b) How do you know the triangles are similar?

7

x

10

A

C

E

D

5

E

10 5

A

E

D A B

C

7 10 + x

x

B

A C

B

E

D



21 The Law of Syllogism: If

€

p→ q and

€

q→ r are true statements then

€

p→ r is a true statement. From the given true statements, draw a valid conclusion. If Kegan misses practice, she will not play in the game. If Kegan does not play in the game, she will not score. Kegan missed practice. The valid conclusion is that Kegan did not score.

2.0/G.MP3.2

21´ You try: 21) Use the Law of Syllogism to draw a conclusion from the two given statements. If a number is a multiple of 64, then it is a multiple of 8. If a number is a multiple of 8, then it is a multiple of 2. A. If a number is a multiple of 64, then it is a multiple of 2. B. The number is a multiple of 2. C. The number is a multiple of 8. D. If a number is not a multiple of 2, then the number is not a multiple of 64.

End of Study Guide



You Try Solutions:

1´ You try:

1a) An angle bisector is a ray that is equidistant from the sides of an angle, OR An angle bisector is a ray that bisects an angle. OR An angle bisector is a ray that cuts an angle into two congruent angles. 1b)

€

PQ 1c) Supplementary angles are two angles whose measures have the sum

€

180°. Complementary angles are two angles whose measures have the sum

€

90°. 1d) No.

€

ON, OM, NM, NP

2´ You try:

2)

130 10120

m AOB∠ = °− °= °

3´ You try: 3)

4´ You try:

4) There are two possibilities: or

5´ You try:

5) (Answers may vary.)

a. A A∠ ≅ ∠ or AB AB≅ b. If A B∠ ≅ ∠ and B C∠ ≅ ∠ , then A C∠ ≅ ∠ . c. If A B∠ ≅ ∠ , then B A∠ ≅ ∠ .

BD

is the bisector of ABC∠ ABD DBC⇒∠ ≅∠ . 46m ABD∴ ∠ = ° . By the Angle Addition Postulate,

46 4692

m ABD m DBC m ABCm ABCm ABC

∠ + ∠ = ∠° + ° = ∠

° = ∠

46 92138

m ABD m ABC∴ ∠ + ∠= °+ °= °

5 9

C A B

5

B A C

9

9 5 14AC = + =

9 5 4AC = − =



6´ You try: 6)

7´ You try:

7a)

€

m∠1= 37 Vertical Angle Theorem 7b)

€

∠AEB and ∠DEC, ∠AED and ∠BEC are pairs of vertical angles.

€

m∠AED =112°

7x + 7 =1127x =105x =15

€

m∠CED =180°−112°

m∠CED = 68°, ∴4y = 68y =17

8´ You try:

8a)

€

38° + 56° + x° =180°

94° + x° =180°

x° = 86°

€

x° + y° =180°

86° + y° =180°

y° = 94°

€

y° + z° +19° =180°

94° + z° +19° =180°

z° +113° =180°

z° = 67°

8´ You try continued: 8b)

€

2x + x + 25 + x − 5 =1804x + 20 =180

4x =160x = 40

€

2 40( ) = 8040 + 25 = 65

40 − 5 = 35, ∴35° is the smallest angle.

8c)

€

2x +15 + 45 = 4x − 202x + 60 = 4x − 20

80 = 2xx = 40, ∴m∠A = 2 40( ) +15

€

m∠A = 95°

9´ You try:

9) OR

By the Angle Addition Postulate, m AOB m BOC m AOC∠ + ∠ = ∠ Therefore,

2 5 3 10 1805 5 1805 175

35

x xxxx

− + + =+ =

==

5 5 180x + =

2 2 2

2 2 2

2

2

10 5100 25

125

125

25 5

5 5

a b cccc

c

c

c

+ =+ =+ =

=

=

⋅ =

=

( ) ( )2 22 1 2 1

2 2

2 2

2 2

(6 ( 4)) ( 1 ( 6))

(6 4) ( 1 6)

10 5

100 25

125

25 5

5 5

d x x y y

d

= − + −

= − − + − − −

= + + − +

= +

= +

=

= ⋅

=

( )6, 1−

( )4, 6− −



10´ You try: 10) The coordinates of the midpoint are the averages of the coordinates of the endpoints:

x1 + x22

,y1 + y22

!

"#

$

%&

3+−52,−9+−2

2!

"#

$

%&

=−22,−112

"

#$

%

&'

= −1,−112

"

#$

%

&'

11´ You try:

11)

12´ You try:

12) By the Base Angles, or Isosceles Triangle, Theorem, A C∠ ≅ ∠ .

13´ You try:

13) ABC is equiangular A C∴∠ ≅∠

5 3 62 6

3

x xxx

= +==

14´ You try:

14a) HL Congruence Theorem 14b) SSS Congruence Postulate 14c) ASA Congruence Postulate

15´ You try:

15) BD BD≅ , so ABD ≅CBD by the HLrt. Theorem. ADB ≅CDB is the same as ABD ≅CBD because corresponding angles are written in corresponding places. ∴ B) is the correct answer.

16´ You try:

16)

The angles whose measurements are given are alternate interior angles. Because the lines are parallel, these angles are congruent. Therefore,

2 5 5 49 33

x xxx

+ = −==

3 1 613 60

20

xxx

+ ===

Statements Reasons 1. A D∠ ≅ ∠ AE DE≅

1. Given as true.

2. AEB DEC∠ ≅ ∠

2. Vertical angles are congruent.

3. ABE ≅DCE

3. ASA Congruence Postulate.



17´ You try:

17a) Given

17b) Given

17c) Given 17d) Given

A B To copy AB , first draw a ray or line and pick a point to call A´:

A´ “Measure” AB with a compass by putting the point

on one end and drawing an arc on the other end. Make that same arc from A´:

' 'A B will be congruent to AB

A B

A´ B´

To bisect AB , make an arc above and below AB with the compass at A (using A as center) and opened

more than halfway (with radius 12mAB> ):

A B

A B

Do the same thing using B as center, using the same radius as the previous step. The intersections of the arcs will be on the perpendicular bisector. Draw a line segment, ray, or line on the line that connects the two points of intersection, and you have bisected AB :

A B

C

D

CD bisects AB .

A

B C

First draw a ray or line and pick a point to call B´:

Then draw an arc using B as the center and draw that same arc from B´:

B´

A

B C

D

E

B´

“Measure” DE with your compass and use that same setting to make a mark from E´: A

B C

D

E

B´

Draw a ray from B´ through the intersection of the two arcs, and D B E′ ′ ′∠ will be congruent to ABC∠ :

B´

E´

D´

A

B C

Make an arc from B:

A

B

D

E

Using the same radius (compass opening), make arcs from the points of intersection, D and E:

A

B C

D

E

BF

will bisect ABC∠ : A

B C

D

E

F



18´ You try: This is a (30-60-90)º triangle. The hypotenuse is twice as long as the side opposite the 30º angle. 18a)

2 2 2

2

2

2

3 ( ) 69 ( ) 36( ) 36 9( ) 27

27

3 3 3

3 3

ABABABAB

AB

AB

AB

+ =+ =

= −=

=

= ⋅ ⋅

=

OR By the 30-60-90 Theorem: AB is 3 ishort leg

AB = 3 i3

= 3 3

18b) Using a proportion

30 :60 :90

1: 3 : 2

° ° °

9AB

=32

3 AB( ) = 2 9( )AB = 18

3

=183i 33

=3i6 33

= 6 3

19´ You try: 19) The top triangle and the whole triangle are similar because they share an angle and corresponding angles are congruent because the lines are parallel: Therefore,

( )

35 3

3 3 53 9 3 2

9 29 22 292

xx

x xx x x

xx

x

=+

+ =+ = +

=

=

=

20´ You try:

20a) AED ABC 20b) A A∠ ≅ ∠ and AED ABC∠ ≅ ∠ (corresponding angles are

congruent when the line are parallel). Therefore, the triangles are similar by the AA Similarity Postulate.

21´ You try:

21) The answer is choice A.

long legHypotenuse

3

2

xx

3

3 x

x

3

2

xx

3

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