Quarterly 3 Review

2. Find the value of x: 2𝑥 + 5 5 = 𝑥 + 10 4 . The ratio of two supplementary angles is 4:5. Find the measure of the smaller angle. 2. Find the value of x: 2𝑥 + 5 5 = 𝑥 + 10 4 .   smaller angle = 4x 4x + 5x = 180 smaller angle = 4 (20) 9x = 180 smaller angle = 80° x = 20 4(2x + 5) = 5(x + 10) 8x + 20 = 5x +50 3x = 30 x = 10

For #3-5, use the diagram below.   trapezoid ABCD is similar to trapezoid EHGF. 3. Find x. 4. Find y. 𝑥 21 = 2 3 3𝑥=42 x = 14 10 𝑦 = 2 3 y = 15 2𝑦=30 𝐴𝐵 𝐸𝐻 = 𝐵𝐶 𝐻𝐺 = 𝐶𝐷 𝐺𝐹 = 𝐴𝐷 𝐸𝐹 𝑥 21 = 𝐵𝐶 𝐻𝐺 = 8 12 = 10 𝑦 Scale Factor = 8 12 = 2 3

For #3-5, use the diagram below.   trapezoid ABCD is similar to trapezoid EHGF. 3. Find x. 4. Find y. 5. Find m∠A. x = 14 m∠𝐸=143° y = 15 𝑚∠𝐴=𝑚∠𝐸 𝑚∠𝐵=𝑚∠𝐻 𝑚∠𝐶=𝑚∠𝐺 𝑚∠𝐷=𝑚∠𝐹 m∠𝐴=143° ≫ ≫

For #6 and #7, use the diagram below. 𝐴𝐼 ∥ 𝐷𝑆 6. Find x. 7. Find y. 𝐴𝐼 ∥ 𝐷𝑆 6. Find x. 7. Find y. 𝑦 12 = 5 13 = 3 𝑥+3 3 𝑥 + 3 = 5 13 𝑥= 24 5 = 4 4 5 39=5(𝑥+3) 𝑦 12 = 5 13 𝑦= 60 13 = 4 8 13 13𝑦=60

8. In isosceles ∆XYZ, 𝑋𝑌 ≅ 𝑌𝑍 . m∠Y = 40°, find m∠X and m∠Z. x + x + 40 = 180 Y 2x + 40 = 180 40° 2x = 140 x = 70 m∠X = 70° x° x° m∠Z = 70° X Z

9. ∆MON is similar to ∆QOP. Find the scale factor. 𝑀𝑂 𝑄𝑂 = 𝑂𝑁 𝑂𝑃 = 𝑀𝑁 𝑄𝑃 9 12 = 15 20 = 𝑀𝑁 𝑄𝑃 9 12 = 3 4 15 20 = 3 4 Scale Factor is 3 4

10. Given: a ∥ b ∥ c Find x. 12 16 = 21 𝑥 12x = 336 x = 28

AA similarity postulate 11. Are the triangles shown similar? If so, which postulate or theorem justifies the similarity. AA similarity postulate

For #12 and #13, WXYZ is a parallogram. 2y + 10 = 4y – 12 22 = 2y 11 = y 32° 148° y = 11 10x – 2 = 148 2y + 10 15 10x = 150 2(11) + 10 x = 15 22 + 10 11 32

14. In ∆XYZ, P and Q are midpoints of 𝑋𝑌 and 𝑋𝑍 14. In ∆XYZ, P and Q are midpoints of 𝑋𝑌 and 𝑋𝑍 . PQ = (5x + 2) and YZ = (3x + 18). Find PQ. 3x + 18 = 2(5x + 2) 3x + 18 = 10x + 4 18 = 7x + 4 7x = 14 x = 2 5x + 2 PQ = 5x + 2 PQ = 5(2) + 2 3x + 18 PQ = 10 + 2 PQ = 12

15. ABCD is a trapezoid. 𝐸𝐹 is the median 15. ABCD is a trapezoid. 𝐸𝐹 is the median. AB = (x – 3), EF = 10, and DC = (2x – 4). Find x. (x – 3) + (2x – 4) = 2(10) 3x – 7 = 20 3x = 27 x = 9 x – 3 10 2x – 4

16. The given figure is a parallelogram with its diagonals drawn, find the values of x and y.   2x + 6 = 26 4y – 10 = 6 2x = 20 4y = 16 x = 10 y = 4

17. Find x. 9x = 3x + 54 6x = 54 x = 9

18. A regular polygon has 18 sides 18. A regular polygon has 18 sides. Find the measure of each interior angle. 𝑛−2 180 𝑛 18−2 180 18 16 180 18 2880 18 160°

TRUE ; 8 + 8 > 15 TRUE TRUE For #19-21, answer TRUE or FALSE. 19. A triangle may have the sides measuring 8, 8, 15.   20. The diagonals of a rectangle are congruent. 21. All equilateral triangles are similar polygons. TRUE ; 8 + 8 > 15 TRUE TRUE

22. If AM > AN, then m∠M ____ m∠N. < 22. If AM > AN, then m∠M ____ m∠N. 5 4

23. L is the midpoint of 𝑀𝑁. ML = (2x + 3) and MN = (7x – 12). Find MN 23. L is the midpoint of 𝑀𝑁 . ML = (2x + 3) and MN = (7x – 12). Find MN. (Draw your own picture.) 2x + 3 ● M L N 7x – 12 2(2x + 3) = 7x – 12 MN = (7x – 12) MN = 7(6) – 12 4x + 6 = 7x – 12 MN = 42 – 12 18 = 3x MN = 30 6 = x x = 6

rotation translation reflection dilation 24. Draw an example of all the transformations for the figure below. rotation translation reflection dilation

For #25-27, use the diagram below. 25 For #25-27, use the diagram below. 25. Name a pair of alternate interior angles, a pair of same-side interior angles, and a pair of corresponding angles. alternate interior ∠’s: ∠3 and ∠5 same-side interior ∠ ′ s: ∠3 and ∠4 corresponding ∠’s: ∠2 and ∠5

26. Solve for x: m∠3 = (7x – 10)° and m∠4 = (15x – 8)°. 27 26. Solve for x: m∠3 = (7x – 10)° and m∠4 = (15x – 8)°. 27. Solve for y: m∠2 = ( 3(y – 4) )° and m∠5 = (y + 14)°. (7x – 10) + (15x – 8) = 180 22x – 18 = 180 22x = 198 x = 9 3(y – 4) = y + 14 3y – 12 = y + 14 2y = 26 y = 13

28. What type of transformation is shown below? rotation

29. Write the definitions and draw a diagram of: Midpoint of a segment, Segment bisector, Median of a triangle, and Angle bisector. Midpoint of a segment – The point that divides the segment into two congruent segments. Segment bisector – A line, segment, ray, or plane that intersects the segment at its midpoint.

Median of a triangle – A segment from a vertex to the midpoint of the opposite side. Angle bisector – The ray that divides the angle into two congruent adjacent angles.

30. Given: 𝐴𝐶 ⊥ 𝐵𝐹 ; 𝐵𝐸 ≅ 𝐸𝐹 ; BAD  FAD Which of the following is the altitude, median, angle bisector of ∆ABF? altitude – 𝐴𝐶 median – 𝐴𝐸 angle bisector – 𝐴𝐷

31. Given: c ⊥ d. If m∠1 = (3x – 20)° and m∠2 = (5x + 6)°, find x. (3x – 20) + (5x + 6) = 90 8x – 14 = 90 8x = 104 x = 13

32. Write a two-column proof 32. Write a two-column proof. Given: 𝐸𝐷 ≅ 𝐷𝐺 ; F is the midpoint 𝐸𝐺 Prove: ∠EDF ≅ ∠GDF Statements Reasons 1. 𝐸𝐷 ≅ 𝐷𝐺 1. Given 2. F is the midpoint of 𝐸𝐺 2. Given 3. 𝐸𝐹 ≅ 𝐹𝐺 3. Definition of midpoint 4. Reflexive property 4. 𝐷𝐹 ≅ 𝐷𝐹 5. SSS Post. 5. ∆EDF ≅ ∆GDF 6. CPCTC 6. ∠EDF ≅ ∠GDF