1  

Unit 1: Foundations of Geometry Section 1: Points, Lines & Planes

The most basic figures in geometry are ____________________________.

  2  

Intersections: Lines Planes

Ex #1

  3  

1a. Name four coplanar points.

1b. Name three lines. ________________________________________________________ 2.Use the diagram to name two planes. Draw and label the following. 3a. a segment with endpoints M and N 3b. opposite rays with a common endpoint T.

SEGMENT RAY OPPOSITE RAYS

  4  

4. Draw and label a ray with endpoint M that contains N. 5a. Sketch two lines intersecting in exactly one point.

5b. Sketch a figure that shows a line

that lies in a plane.

6. Sketch a figure that shows two lines intersect in one point in a plane, but only one of the lines lies in the plane. Lesson Quiz (use the diagram) 7. Name two opposite rays. 8. Name a point on BC

u ruu.

9. Name the intersection of plane N and plane T. 10. Name a plane containing E, D, and B. Draw each of the following. 11. a line intersecting a plane at one point

12. a ray with endpoint P that passes through Q

  5  

Unit 1: Foundations of Geometry Section 2: Segments

The distance between any two points… Find each length. 13a. XY 13b. XZ

______________________ are segments that have the same length.

14. G is between F and H, FG = 6, and FH = 11. Find GH. 15. M is between N and O. Find NO.

  6  

The ____________ M of AB is the point that __________, or divides, the segment into two congruent segments. 16. The map shows the route for a race. You are at X, 6000 ft from the first checkpoint C. The second checkpoint D is located at the midpoint between C and the end of the race Y. The total race is 3 miles. How far apart are the 2 checkpoints? 17. You are 1182.5 m from the first-aid station. What is the distance to a drink station located at the midpoint between your current location and the first-aid station? 18. D is the midpoint of EF , ED = 4x + 6, and DF = 7x – 9. Find ED, DF, and EF. Lesson Quiz: 19. S is the midpoint of TV , TS = 4x – 7, and SV = 5x – 15. Find TS, SV, and TV. 20. LH bisects GK at M. GM = 2x + 6, and GK = 24. Find x.

21. Tell whether the statement below is sometimes, always, or never true. Support your answer with a sketch.

If M is the midpoint of KL, then M, K, and L are collinear.

  7  

Unit 1: Foundations of Geometry Section 3: Angles

22. A surveyor recorded the angles formed by a transit (point A) and three distant points, B, C, and D. Name three of the angles.

23. Write the different ways you can name the angles in the diagram.

Types of Angles:

  8  

Find the measure of each angle. Then classify each as acute, right, or obtuse. 24a. ∠BOA 24b. ∠DOB 24c. ∠EOC Congruent angles are… 25. m∠DEG = 115°, and m∠DEF = 48°. Find m∠FEG. An angle bisector is… JKu ruu

bisects ∠LJM; thus ∠LJK ≅ ∠KJM.

   

9  

26. KMu ruu

bisects ∠JKL, m∠JKM = (4x + 6)°, and m∠MKL = (7x – 12)°. Find m∠JKM. 27. Find the measure of each angle. QSu ruu

bisects ∠PQR, m∠PQS = (5y – 1)°, and m∠PQR = (8y + 12)°. Find m∠PQS. 28. m∠WYZ = (2x – 5)° and m∠XYW = (3x + 10)°. Find the value of x.

   

10  

   

11  

Unit 1: Foundations of Geometry Section 4: Pairs of Angles

Tell whether the angles are only adjacent, adjacent and form a linear pair, or non-adjacent. 29. ∠AEB and ∠BED

30. ∠AEB and ∠BEC

31. ∠DEC and ∠AEB

Tell whether the angles are only adjacent, adjacent and form a linear pair, or non-adjacent. 32: ∠5 and ∠6

33. ∠7 and ∠SPU

34. ∠7 and ∠8

35. Find the complement of ∠F

36. Find the supplement of ∠G

37. An angle is 10° more than 3 times the measure of its complement. Find the measure of the complement.

   

12  

38. An angle’s measure is 12° more than 12

the measure of its supplement. Find the

measure of the angle. 39. Name the pairs of vertical angles. 40. Light passing through a fiber optic cable reflects off the walls of the cable in such a way that ∠1 ≅ ∠2, ∠1 and ∠3 are complementary, and ∠2 and ∠4 are complementary. If m∠1 = 47°, find m∠2, m∠3, and m∠4. 41. Suppose m∠3 = 27.6°. Find m∠1, m∠2, and m∠4. 42. The supplement of an angle is 3 more than 6 times its complement. Find the measure of the angle.

   

13  

Unit 1: Foundations of Geometry Section 5: Formulas

Base – Height – Find the perimeter and area of each figure. 43.

44.

45. The Queens Quilt block includes 12 blue triangles. The base and height of each triangle are about 4 in. Find the approximate amount of fabric used to make the 12 triangles. 46. Find the amount of fabric used to make four rectangles. Each rectangle has a

length of 162

in. and a width of 122in.

   

14  

47. Find the circumference and area of a circle with radius 8 cm. Use the π key on your calculator. Then round the answer to the nearest tenth.

48. Find the circumference and area of a circle with diameter 28m.

Lesson Quiz: Find the area and perimeter of each figure. 49.

50.

51. The area of a rectangle is 74.82 in2, and the length is 12.9 in. Find the width.

   

15  

Unit 1: Foundations of Geometry Section 6: Midpoint & Distance Formula

Finding the Coordinates of an Endpoint 52. M is the midpoint of XY . X has coordinates (2, 7) and M has coordinates (6, 1). Find the coordinates of Y. 53. S is the midpoint of RT . R has coordinates (–6, –1) and S has coordinates (–1, 1). Find the coordinates of T.

   

16  

54. Use the Distance Formula and the Pythagorean Theorem to find the distance, to the nearest tenth, from D(3, 4) to E(–2, –5). 55. Use the Distance Formula and the Pythagorean Theorem to find the distance, to the nearest tenth, from R to S. R(3, 2) and S(–3, –1)