Name: ________________________________________________ Period: ____________

Geometry Honors: Midterm Exam Review January 2018

The midterm will cover Chapters 1-6. You WILL NOT receive a formula sheet, but you need to know the following formulas – Make sure you memorize them!!

- Distance - Midpoint - Slope - Slope-Intercept Form

The exam is 1 hour and 55 minutes.

- 40 multiple choice questions (60 points) - 6 open ended questions (40 points)

It is YOUR responsibility to properly prepare for the midterm exam. The answers and work for the review packet will be posted on the class website. Answers will not be provided in class - it is your responsibility to go online and check your work. The schedule for midterm exams is listed in the table below. Each day of exams is a half day. There is no homeroom - report directly to the first exam of the day. You will have a 10 minute break between exams. You may wish to use this time to use the bathroom or have a light snack. You will not be allowed to leave the classroom during exams. If you are absent from an exam, you must provide a doctor’s note to the attendance office. Students who are absent without a note will receive an “F” for the midterm exam.

7:35 – 9:30 9:40 – 11:35

Tuesday January 23rd

Period 1 Period 2

Wednesday January 24th

Period 3 Period 4/5

Thursday January 25th

Period 6/7 or 7/8 Period 8/9 or 9/10

Friday January 26th

Period 11 Period 12

CHAPTER 1: TOOLS OF GEOMETRY

1. Which figure shows 𝐴𝐵 ⃡ and point G contained in plane R?

F G

H J

2. Name the intersection of 𝐴𝐸 ⃡ and 𝐶𝐺 ⃡ .

A line CD C point C

B line AB D point G

3. If point P is between A and M, which is true?

A PA + AM = PM C AM + PM = AP

B AM + AP = PA D AP + PM = AM

4. Find the distance between A(–3, 5) and B(4, 2), to the nearest hundredth.

A 6.75 B 7.62 C 8.06 D 10

√(2 − 5)2 + (4 + 3)2

√(−3)2 + 72

√9 + 49 = √58 ≈ 7.62

5. Find EF if E is the midpoint of 𝐷𝐹̅̅ ̅̅ , DE = 15 – 3x, and EF = x + 3.

F 1 G 3 H 6 J

𝐷𝐸 = 𝐸𝐹

15 − 3𝑥 = 𝑥 + 3

−4𝑥 = −12

𝑥 = 3 EF = 3 + 3 = 6

6. Find the coordinates of B if A has coordinates (3, 5) and Y(–2, 3) is the midpoint of AB.

A B (–7, 1) B B (3, 3) C B (5, –2) D B (–7, –3)

3 + 𝑥

2= −2 → 3 + 𝑥 = −4 → 𝑥 = −7

5 + 𝑦

2= 3 → 5 + 𝑦 = 6 → 𝑦 = 1

7. Find the length of 𝑋𝑍̅̅ ̅̅ if Y(–4, 4) is the midpoint of 𝑋𝑍̅̅ ̅̅ and X has coordinates (2, –4).

𝑋𝑌 = √(2 + 4)2 + (−4 − 4)2 → √62 + (−8)2 → √100 → 10 ∴ 𝑋𝑍 = 2𝑋𝑌 = 20

For Questions 8-10, use the figure.

8. What is another name for ∠2?

A ∠WYX C ∠WXY

B ∠3 D ∠Y

9. Which angles form a linear pair?

F ∠1 and ∠3 H ∠2 and ∠5

G ∠2 and ∠3 J ∠1 and ∠4

10. Name the angle that is vertical to ∠3.

A ∠1 B ∠2 C ∠3 D ∠4

11. If m∠HJK = 7y – 2 and m∠PQR = 133, find the value of y so that ∠HJK is supplementary to ∠PQR.

F –3 G 2 H 4 J 7

7𝑦 − 2 + 133 = 180

7𝑦 = 49

𝑦 = 7

12. The measure of the complement of ∠A is 185 less than two times the measure of the supplement of ∠A.

Find m∠A

𝑚∠𝐴 = 𝑥 𝑠𝑜 𝐶𝑜𝑚𝑝𝑙𝑒𝑚𝑒𝑛𝑡 = 90 − 𝑥 𝑎𝑛𝑑 𝑆𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡 = 180 − 𝑥

90 − 𝑥 = 2(180 − 𝑥) − 185

90 − 𝑥 = 360 − 2𝑥 − 185

𝑥 = 85 ∴ 𝑚∠𝐴 = 85

13. In the figure, 𝑄𝑃 and 𝑄𝑇 are opposite rays. Find m∠PQR, m∠RQS, and m∠SQT. Then classify each

angle as right, acute, or obtuse.

9𝑥 = 180 → 𝑥 = 20 𝑚∠𝑃𝑂𝑅 = 120 𝑜𝑏𝑡𝑢𝑠𝑒

𝑚∠𝑅𝑄𝑆 = 40 𝑎𝑐𝑢𝑡𝑒

𝑚∠𝑆𝑄𝑇 = 20 𝑎𝑐𝑢𝑡𝑒

For questions 14 and 15, use the figure below. 𝑯𝑳 bisects ∠KHI and 𝑯𝑮 and 𝑯𝑰 are opposite rays.

14. If ∠1 ≅ ∠2, m∠KHG = 70, and m∠1 = 3d + 2, find the value of d.

3𝑑 + 2 = 35 𝑑 = 11

15. If m∠2 = a + 15 and m∠3 = a + 35, find the value of a so that 𝐻𝐿 ⊥ 𝐻𝐽 .

𝑎 + 15 + 𝑎 + 35 = 90

2𝑎 + 50 = 90 2𝑎 = 40 𝑎 = 20

16. Which is not a polygon?

F G H J

17. Name this polygon by its number of sides and then classify it as convex or concave and regular or irregular.

Quadrilateral – Concave Irregular

18. Find the length of one side of a regular hexagon whose perimeter is 75 feet.

F 25 ft G 18.75 ft H 15 ft J 12.5 ft

𝟔𝒔 = 𝟕𝟓 → 𝒔 = 𝟏𝟐. 𝟓

19. Find the perimeter of a regular octagon if one of its sides is x + 6 and another side is 14 – x.

A 4 B 40 C 8 D 80

𝑥 + 6 = 14 − 𝑥

2𝑥 = 8 𝑥 = 4

Each side is 10 units

Perimeter is 80

CHAPTER 2: REASONING AND PROOF

1. Make a conjecture about the next letter in the sequence.

L M N P Q R T . . .

U

2. Find a counterexample for the statement.

Five is the only whole number between 4.5 and 6.1.

6 is another whole number between 4..5 and 6.1

3. Which statement has the same truth value as 3 = 5?

A 3 = x C AB = BC

B AB = 3 D BC = 3 + x

4. Determine whether the following statement is always, sometimes, or never true.

Points X, Y, and Z determine two lines.

sometimes 5. If ∠1 ≅ ∠2 and ∠2 ≅ ∠3, then which is a valid conclusion?

I m∠1 = m∠2

II ∠1 ≅ ∠3 NOTE: I is a true statement, but valid conclusion is II

III m∠1 + m∠2 = m∠3

F I, II, and III G II only H I and II J I and III

For Questions 6 and 7, name the property that justifies the given statement.

6. If AB = CD and CD = 11, then AB = 11.

A Transitive B Symmetric C Congruence D Reflexive

7. If ∠XYZ ≅ ∠PQR, then ∠PQR ≅ ∠XYZ.

F Transitive G Symmetric H Congruence J Reflexive

8. Given: 𝑥 + 3 = 15𝑥 − 53

Prove: 𝑥 = 4

Statements Reasons

1. x + 3 = 15x – 53 1. Given

2. -14x = -56 2. Subtraction Property of Equality

3. X =4 3 Division Property of Equality

9. If B is in the interior of ∠DEF, m∠DEB = 27.2, and m∠DEF = 92.5, find m∠BEF.

𝒎∠𝑩𝑬𝑭 = 𝒎∠𝑫𝑬𝑭 − 𝒎∠𝑫𝑬𝑩

𝟗𝟐. 𝟓 − 𝟐𝟕. 𝟑 = 𝟔𝟓. 𝟑 → 𝒎∠𝑩𝑬𝑭 = 𝟔𝟓. 𝟑

10. If ∠1 is supplementary to ∠2 and ∠3 is complementary to ∠2, find m∠3 if m∠1 is 145.

A 35 C 55

B 45 D 90

𝑚∠1 + 𝑚∠2 = 180 𝑎𝑛𝑑 𝑚∠3 + 𝑚∠2 = 90

145 + 𝑚∠2 = 180 𝑠𝑜 𝑚∠2 = 35

35 + 𝑚∠3 = 90 𝑠𝑜 𝑚∠3 = 55

11. Refer to the following figure to answer the questions below.

a. Name a pair of supplementary angles. ∠𝑊𝑈𝑆 𝑎𝑛𝑑 ∠𝑆𝑈𝑅

b. Name a pair of complementary angles. ∠𝑆𝑈𝑅 𝑎𝑛𝑑 ∠𝑅𝑈𝑉

c. Find m∠RUV.

𝑚∠𝑆𝑈𝑉 = 𝑚∠𝑆𝑈𝑅 + 𝑚∠𝑅𝑈𝑉

90 = 42 + 𝑚∠𝑅𝑈𝑉

48 = 𝑚∠𝑅𝑈𝑉

B E

B

D

F

12. Given: ∠1 ≅ ∠2

Prove: 𝑚∠𝐴𝐵𝐶 = 2(𝑚∠1)

Statements Reasons

1. ∠1 ≅ ∠2 Given

2. 𝑚∠1 = 𝑚∠2 Definition of Congruent Angles

3. 𝑚∠𝐴𝐵𝐶 = 𝑚∠1 + 𝑚∠2 Angle Addition Postulate

4. 𝑚∠𝐴𝐵𝐶 = 𝑚∠1 + 𝑚∠1 Substitution 5. 𝑚∠𝐴𝐵𝐶 = 2(𝑚∠1) Substitution

CHAPTER 3: PARALLEL AND PERPENDICULAR LINES

For Questions 1-3, use the figure below.

1. What type of angles are ∠3 and ∠∠10?

F alternate interior angles

G alternate exterior angles

H corresponding angles

J consecutive interior angles

2. State the transversal that forms ∠11 and ∠13.

A ℓ B m C p D q

3. If m∠1 = 120, find m∠8.

F 60 G 110 H 120 J 140

4. Find m∠HJK.

A 33 C 78

B 45 D 147

5. Find the value of x so that 𝒦 ║ ℓ.

3𝑥 + 11 = 4𝑥 − 1 𝑥 = 12

6. Find the value of x so that ℓ ║ m.

7𝑥 + 12 + 25 = 180

𝑥 = 20

7. Two lines ℓ and k are cut by a transversal forming two pairs of alternate interior angles: ∠4 and ∠5

and ∠3 and ∠6. Which condition below is necessary to make lines ℓ and k parallel?

F ∠4 ≅ ∠3 H ∠4 ≅ ∠5 and ∠3 ≅ ∠6

G m∠3 + m∠6 = 180 J m∠3 + m∠6 = 90

8. If 𝐽𝐾̅̅ ̅ || 𝐿𝑀̅̅ ̅̅ , then ∠4 must be supplementary to 𝒎∠𝟔__ ?

9. Given: ∠1 and ∠3 are supplementary

Prove: 𝑗 ∥ 𝑘

Statements Reasons

1. ∠1 𝑎𝑛𝑑 ∠3 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 Given

2. 𝑚∠1 + 𝑚∠3 = 180 Definition of supplementary angles

3. ∠1 ≅ ∠2 Vertical Angle Theorem

4. 𝑚∠1 = 𝑚∠2 Definition of congruent angles

5. 𝑚∠2 + 𝑚∠3 = 180 Substitution

6. ∠2 𝑎𝑛𝑑 ∠3 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 Definition of supplementary angles

7. 𝑗 ∥ 𝑘 Consecutive Interior Angles Converse Theorem

10. Find the slope of the line that passes through points A(–7, 14) and B(5, –2).

F – 4

3 G –

3

4 H

3

4 J

4

3

𝑚 = −2 − 14

5 + 7=

−16

12=

−4

3

11. Find the slope of a line parallel to 3y – 6x = 9.

𝟑𝒚 = 𝟔𝒙 + 𝟗

𝒚 = 𝟐𝒙 + 𝟗

Slope of parallel line is 2

12. Determine whether 𝑄𝑅 ⃡ and 𝑆𝑇 ⃡ are parallel, perpendicular, or neither for Q(–4, –4), R(5, 2), S(4, –5),

and T(0, 1).

𝑄𝑅 = 2 + 4

5 + 4=

6

9=

2

3

𝑆𝑇 =1 + 5

0 − 4=

6

−4=

3

−2

𝑄𝑅 ⊥ 𝑆𝑇

13. Find the distance between two lines that have equations y = 3x + 1 and y = 3x – 19.

𝐴 → 𝑦 = 3𝑥 + 1

𝐵 → 𝑦 = 3𝑥 − 19

𝑃 → 𝑦 = −1

3𝑥 + 1

3𝑥 − 19 = −1

3𝑥 + 1

10

3𝑥 = 20

𝑥 = 6

3(6) − 19 = −1 𝑠𝑜 𝑦 = −1

Distance between ( 0, 1) and (6, -1) √(6 − 0)2 + (−1 − 1)2 = √(6)2 + (−2)2 = √40 = 2√10 ≈ 6.3

14. Find the distance from 𝐴(−1,5) to the line whose equation is 4𝑥 − 5𝑦 = 12.

4𝑥 − 5𝑦 = 12

−5𝑦 = −4𝑥 + 12

𝑦 = 4

5𝑥 −

12

5

Use perpendicular slope and point A to write equation,

𝑦 − 5 = −5

4(𝑥 + 1)

𝑦 = −5

4𝑥 +

15

4

Find point where two equation meet: 4

5𝑥 −

12

5=

−5

4𝑥 +

15

4

20 (4

5𝑥 −

12

5 =

−5

4𝑥 +

15

4 )

16𝑥 − 48 = −25𝑥 + 75

41 𝑥 = 123

𝑥 = 3 𝑡ℎ𝑒𝑛 𝑦 = −5

4(3) +

15

4 = 0

Find distance between: (-1,5) and (3,0)

𝑑 = √(3 + 1)2 + (0 − 5)2 = √41 ≈ 6.4

CHAPTER 4: CONGRUENT TRIANGLES

1. Classify △DEF with vertices D(2, 3), E(5, 7) and F(9, 4).

F acute G equiangular H obtuse J right

𝐷𝐸 = 7 − 3

5 − 2=

4

3

𝐷𝐸̅̅ ̅̅ ⊥ 𝐸𝐹̅̅ ̅̅

𝐸𝐹 =4 − 7

9 − 5=

−3

4

2. Find PR if △PQR is isosceles, ∠Q is the vertex angle, PQ = 4x – 8, QR = x + 7, and PR = 6x – 12.

Since ∠𝑄 is the vertex angle then PQ = QR

4𝑥 − 8 = 𝑥 + 7

3𝑥 = 15

𝑥 = 5

𝑃𝑅 = 6(5) − 12 = 18

3. In the figure, ∠1 ≅ ∠2. Find the measures of the numbered angles.

𝑚∠1 = 25, 𝑚 ∠2 = 25, 𝑚∠3 = 130

4. Find m∠PQR.

𝑚∠𝑃𝑄𝑅 = 125 − 63 = 62°

5. If m∠D = 42, what is m∠E?

A 18 C 43

B 40 D 81

𝑚∠𝐸 = 123 − 42 = 81°

6. If PQ = QS, QS = SR, and m∠R = 20, find m∠PSQ.

𝑚∠𝑃𝑆𝑄 = 40

180 – (70+65) = 45

20

140

7. Let △ABC be an isosceles triangle with △ABC ≅ △PQR. If m∠B = 154, find m∠R.

F 154 G 126 H 26 J 13

8. If △ABC ≅ △WXY, AB = 72, BC = 65, CA = 13, XY = 7x – 12, and WX = 19y + 34, find the values of x

and y.

19𝑦 + 34 = 72 7𝑥 − 12 = 65

19𝑦 = 38 7𝑥 = 77

𝑦 = 2 𝑥 = 11

9. In the figure, 𝐿𝐾̅̅ ̅̅ bisects ∠JKM and ∠KLJ ≅ ∠KLM. Determine which theorem or postulate can be used

to prove that △JKL ≅ △MKL.

ASA

10. Which postulate or theorem can be used to prove △ABD ≅ △CBD?

A SAS C SSS

B ASA D AAS

11. Which of the following theorems can be used to prove △ABC ≅ △DEC?

A SSS C SAS

B AAS D ASA

12. Given: 𝐴𝐵̅̅ ̅̅ ∥ 𝐷𝐸̅̅ ̅̅ , 𝐴𝐷̅̅ ̅̅ bisects 𝐵𝐸̅̅ ̅̅

Prove: ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐶

Statements Reasons

1. 𝐴𝐵̅̅ ̅̅ ∥ 𝐷𝐸̅̅ ̅̅ , 𝐴𝐷̅̅ ̅̅ bisects 𝐵𝐸̅̅ ̅̅ Given

2. C is midpoint of 𝐵𝐸̅̅ ̅̅ Definition of segment bisector

3. 𝐵𝐶̅̅ ̅̅ ≅ 𝐸𝐶̅̅ ̅̅ Midpoint Theorem

4. ∠𝐵 ≅ ∠𝐸 Alternate Interior Angle Theorem

5. ∠𝐵𝐶𝐴 ≅ ∠𝐸𝐶𝐷 Vertical Angle Theorem

6. : ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐶 ASA

13. Given: ∠1 ≅ ∠2, ∠3 ≅ ∠4, ∠5 ≅ ∠6, ∠7 ≅ ∠8

Prove: ∠𝐴 ≅ ∠𝐶

Statements Reasons

7. ∠1 ≅ ∠2, ∠3 ≅ ∠4,

∠5 ≅ ∠6, ∠7 ≅ ∠8

Given

2. ∠𝐴 ≅ ∠C Third Angle Theorem

14. △PQR is an isosceles triangle with base 𝑄𝑅̅̅ ̅̅ . If m∠P = 6x + 40 and m∠Q = x – 10, find x.

F 20 G 25 H 30 J 100

𝑚∠𝑃 + 𝑚∠𝑄 + 𝑚∠𝑅 = 180

6𝑥 + 40 + 𝑥 − 10 + 𝑥 − 10 = 180

8𝑥 + 20 = 180

8𝑥 = 160

𝑥 = 20

C

15. Given: ∆𝐴𝐶𝐷 is isosceles with vertex A

𝐵𝐶̅̅ ̅̅ ≅ 𝐷𝐸̅̅ ̅̅

Prove: ∆𝐴𝐵𝐸 is isosceles

Statements Reasons

1. 𝐴𝐶𝐷 is isosceles with vertex A

𝐵𝐶̅̅ ̅̅ ≅ 𝐷𝐸̅̅ ̅̅

Given

2. 𝐴𝐶̅̅ ̅̅ ≅ 𝐴𝐷̅̅ ̅̅ Definition of Isosceles Triangle

3. ∠𝐴𝐶𝐷 ≅ ∠𝐴𝐷𝐶 Isosceles Triangle Theorem

4. 𝐶𝐷̅̅ ̅̅ ≅ 𝐶𝐷̅̅ ̅̅ Reflexive Property

5. BC = DE, CD=CD Defiiniton of Congruent Segments

6. BC + CD = DE + CD Addition Property of Equality

7. BD = BC + CD ; EC = DE + CD Segment Addition Postulate

8. BD = EC Substitution

9. 𝐵𝐷̅̅ ̅̅ ≅ 𝐸𝐶̅̅ ̅̅ Def of Congruent Segments

10. ∆𝐴𝐵𝐷 ≅ ∆𝐴𝐸𝐶 SAS

11. 𝐴𝐵̅̅ ̅̅ ≅ 𝐴𝐸̅̅ ̅̅ CPCTC

12. ∆𝐴𝐵𝐸 is isosceles Definition of Isosceles Triangle

16. Find the coordinates of B, the midpoint of 𝐴𝐶̅̅ ̅̅ , if A(2a, b) and C(0, 2b).

F (2a, 2b) G (a, b) H (𝑎,3

2𝑏) J (

3

2 𝑎, 𝑏)

B = (2𝑎+0

2,𝑏+2𝑏

2) = (𝑎,

3

2𝑏)

17. Write a coordinate proof showing that the segments joining the midpoints of the sides of a right triangle

form a right triangle.

A = (0, 2b) Midpoint of AB = X = (0,b)

B = (0,0) Midpoint of BC = Y = (a, 0)

C = (2a, 0) Midpoint of AC – Z = (a, b)

Slope of XZ = 𝑏−𝑏

𝑎−0=

0

𝑎= 0

Slope of ZY = 𝑏−0

𝑎−𝑎=

𝑏

𝑜= 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑

Lines with 0 and undefined are perpendicular and by definition perpendicular lines form a right angle

∴ ∆𝑋𝑌𝑍 𝑖𝑠 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒

A

B

CHAPTER 5: RELATIONSHIPS IN TRIANGLES

1. In △XYZ, which type of line is ℓ ?

F perpendicular bisector H altitude

G angle bisector J median

2. If 𝑅𝑉̅̅ ̅̅ is an angle bisector, find m∠UVT.

A 10 C 68

B 34 D 136

3. If 𝐵𝐷 is an altitude of △ABC, find the value of x.

𝟑𝟓 + 𝟐𝒙 + 𝟏𝟕 + 𝟑𝒙 − 𝟐 = 𝟗𝟎

𝟓𝒙 + 𝟓𝟎 = 𝟗𝟎 𝟓𝒙 = 𝟒𝟎

𝒙 = 𝟖

For Questions 4 and 5, refer to the figure.

4. Find the value of a and m∠ZWT if 𝑍𝑊̅̅ ̅̅ ̅ is an altitude of △XYZ, mvZWT = 3a + 5, and

m∠TWY = 5a + 13.

3𝑎 + 5 + 5𝑎 + 13 = 90

8𝑎 + 18 = 90

8𝑎 = 72

𝑎 = 9 𝑠𝑜 𝑚∠𝑍𝑊𝑇 = 3(9) + 5 = 32°

5. Determine which angle has the greatest measure: ∠YWZ, ∠WZY, or ∠ZYW.

∠YWZ

6. In △XYZ, point M is the centroid. If XM = 8, find the length of 𝑀𝐴.

MA = 4

7. The vertices of △ABC are A(–2, 3), B(4, 3), and C(–2, –3). Find the coordinates of each of the following

points of concurrency of △ABC.

a. circumcenter

𝑀𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐴𝐵 = (−2 + 4

2,3 + 3

2) = (1,3)

𝑆𝑙𝑜𝑝𝑒 𝑜𝑓 𝐴𝐵 = 3 − 3

4 + 2= 0

𝑃𝑒𝑟𝑝𝑑𝑒𝑛𝑖𝑐𝑢𝑙𝑎𝑟 𝑠𝑙𝑜𝑝𝑒 − 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 ==> 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝒙 = 𝟏

𝑀𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐵𝐶 = (4 − 2

2,3 − 3

2) = (1,0)

𝑆𝑙𝑜𝑝𝑒 𝑜𝑓 𝐵𝐶 =−3 − 3

−2 − 4=

−6

−6= 1

𝑃𝑒𝑟𝑝𝑑𝑒𝑛𝑖𝑐𝑢𝑙𝑎𝑟 𝑠𝑙𝑜𝑝: −1 ==> 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝒚 = −𝒙 + 𝟏

𝑦 = −(1) + 1 = 0

𝐶𝑖𝑟𝑐𝑢𝑚𝑐𝑒𝑛𝑡𝑒𝑟 𝑖𝑠 (1,0)

b. orthocenter

Write equation using perpendicular slope of the side and the opposite vertex to write the equations,

Vertex C with perpendicular slope of AB:

(-2,-3) and Undefined slope x=-2

Vertex A with perpendicular slope of BC

(-2,3) and -1 slope 𝑦 − 3 = −1(𝑥 + 2) 𝑦 = −𝑥 + 1

y = -(-2) +1 y = 3

Orthocenter is ( -2,3)

For Questions 8 and 9 refer to the figure.

8. Which line segment is the shortest?

F 𝑃𝑄 H 𝑄𝑅

G 𝑅𝑆 J 𝑃𝑆

9. Which line segment is the longest?

A 𝑃𝑄 B 𝑄𝑅 C 𝑅𝑆 D 𝑃𝑆

10. Find x and the measure of each angle. Then list the sides of the triangle in order from shortest to

longest.

𝟓𝒙 + 𝟐𝟓 + 𝟏𝟖𝟎

𝒙 = 𝟑𝟏

𝒎∠𝑨 = 𝟒𝟏, 𝒎∠𝑩 = 𝟓𝟕, 𝒎∠𝑪 = 𝟖𝟐

𝑩𝑪 < 𝑨𝑪 < 𝑨𝑩

11. Which of the following sets of numbers cannot be lengths of the sides of a triangle?

A 1, 2, 3 B 2, 3, 4 C 3, 4, 5 D 4, 5, 6

12. The measures of two sides of △ABC are 19 and 15. The range for measure of the third side n would be

4 < n < ? .

4 < 𝑛 < 34

13. Determine whether 128 feet, 136 feet, and 245 feet can be the lengths of the sides of a triangle.

Yes, because 126 + 136 > 245

14. Write an inequality to describe the possible values of x.

2𝑥 − 5 > 11

2𝑥 > 16

𝒙 > 8

15. Which inequality describes the possible values of x?

F x > 6 H x ≮ 12

G x < 6 J 6 < x < 12 𝑥 + 5 > 3𝑥 − 7

−2𝑥 > −12

𝑥 < 6

CHAPTER 6: QUDRILATERALS

1. Find the sum of the measures of the interior angles for a convex heptagon.

(𝒏 − 𝟐)𝟏𝟖𝟎 ==> (𝟕 − 𝟐)𝟏𝟖𝟎 = 𝟗𝟎𝟎°

2. The measure of an interior angle of a regular polygon is 140. Find the number of sides in the polygon.

If interior angle is 140° then exterior angle is 40°. 360° is total of the exterior angles then

360 /40 = 9 sides

3. Which statement ensures that quadrilateral QRST is a parallelogram?

A ∠Q ≅ ∠S C 𝑄𝑇̅̅ ̅̅ ∥ 𝑅𝑆̅̅̅̅

B 𝑄𝑅̅̅ ̅̅ ≅ 𝑇𝑆̅̅̅̅ and 𝑄𝑅̅̅ ̅̅ ∥ 𝑇𝑆̅̅̅̅ D m∠Q + m∠S = 180

4. For parallelogram JKMH, find m∠JHK, m∠HMK, and the value of x.

7𝑥 − 24 = 3𝑥 + 8

4𝑥 = 32

𝑥 = 8

𝑚∠𝐽𝐻𝐾 = 52 𝑎𝑛𝑑 𝑚∠𝐻𝑀𝐾 = 108

5. Determine whether the vertices of quadrilateral DEFG form a parallelogram given D(–3, 5), E(3, 6),

F(–1, 0), and G(6, 1).

Using slope:

𝐷𝐸 = 6−5

3+3=

1

6

𝐸𝐹 = 0−6

−1−3=

−6

−4=

3

2

𝐹𝐺 = 1−0

6+1=

1

7

𝐷𝐺 = 1−5

6+3=

−4

9

Not a parallelogram

6. Prove that quadrilateral PQRS is NOT a parallelogram.

Opposite sides do not have the same slope:

𝑺𝒍𝒐𝒑𝒆 𝒐𝒇 𝑷𝑸 = 𝟏

𝟐, 𝑺𝑹 = 𝟏, 𝑷𝑺 =

−𝟏

𝟐, 𝑸𝑹 = −𝟏

7. For rectangle WXYZ with diagonals 𝑊𝑌̅̅ ̅̅ ̅ and 𝑋𝑍̅̅ ̅̅ , WY = 3d + 4 and XZ = 4d – 1, find the value of d.

In rectangles diagonal are congruent.

3𝑑 + 4 = 4𝑑 − 1 ==> −𝑑 = = 5 ==> 𝑑 = 5

8. Rectangle ABCD has vertices A(–3, 0), B(–2, 3), C(4, 1), and D(3, –2). Determine where the diagonals

of the rectangle intersect.

𝑀𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐴𝐶 = (−3 + 4

2,0 + 1

2) = (

1

2,1

2)

𝑀𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐵𝐷 = (−2 + 3

2,3 − 2

2) = (

1

2,1

2)

9. If m∠BEC = 9z + 45 in rhombus ABCD, find the value of z.

In a rhombus diagonal are perpendicular.

𝟗𝒛 + 𝟒𝟓 = 𝟗𝟎 ==> 𝟗𝒛 = 𝟒𝟓 ==> 𝒛 = 𝟓

10. In trapezoid HJLK, M and N are midpoints of the legs. Find KL.

45+𝐾𝐿

2= 28

𝐾𝐿 = 11

11. What is the value of x?

F 2 H 5.5

G 4 J 7

16+4𝑥−4

2 = 20 12 + 4x = 40 4x = 28 x=7

12. What is m∠T in kite STVW?

F 100 H 95

G 130 J 260

Sum of interior angles = 360 and 𝒎∠𝑾 = 𝒎 ∠𝑻. 𝟖𝟓 + 𝟏𝟓 + 𝟐𝒙 = 𝟑𝟔𝟎

𝟐𝒙 = 𝟐𝟔𝟎

𝒙 = 𝟏𝟑𝟎

13. JKLM is a kite. Complete each statement.

a. 𝑀𝐽̅̅ ̅̅ ≅ __𝑀𝐿̅̅ ̅̅ ___

b. 𝑀𝐾̅̅ ̅̅ ̅ ⊥ ___𝐽𝐾𝐿̅̅ ̅̅ ̅___

c. m∠L = m∠𝐽_______