Postulate 1-1-2: Through any three noncollinear points there is exactly one plane containing them.

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1 Unit Definitions Term Labeling Picture Undefined terms Point Dot, place in space Line Plane Series of points that extends in two directions forever Infinite surface with no thickness Defined Terms Collinear Points that lie on the same line points Coplanar Points that lie in the same plane Segment Ray Opposite Rays Part of a line consisting of two points and all points between them Part of a line that starts at an endpoint and extends forever in one direction Two rays that have a common endpoint and form a line. Skew non-coplanar lines. Postulate Theorem Accepted statement of fact. Statement proven to be true. Postulate --: Through any two points there is exactly one line. Postulate --: Through any three noncollinear points there is exactly one plane containing them. Postulate --3: If two points lie in a plane, then the line containing those points lies in the plane. Postulate --4: If two lines intersect, then they intersect in exactly one point. Postulate --5: If two planes intersect, then they intersect in exactly one line.

2 Example a) Name line m in three other ways. b) Name 3 pairs of collinear points. c) Name the intersection of lines l and m. d) Name the intersection of lines l and n. e) Name 3 noncollinear points Example a) Name 3 noncollinear points. b) Name the intersection of lines HG and EH. c) Name the intersection of planes HAD and CBD. d) Name the intersection of planes ACB and HGC. e) Name the plane that represents the right side of the box. f) Name the plane that represents the front of the box. g) Name a fourth point on plane EHB. Draw and Label a) A segment with endpoints M and N. b) Opposite rays with a common endpoint A c) A ray with endpoint M that contains N. Always, Sometimes, Never ) Two points lie in exactly one line. ) Three points lie in exactly one line. 3) Three collinear points lie in exactly one plane. 4) Two intersecting planes intersect in a segment. 5) Three points determine a plane. 6) Two intersecting lines determine a plane. 7) Two non-intersecting lines determine a plane.

3 .6 Distance and Midpoint Distance Formula: Midpoint Formula: ) Y is the midpoint of XZ. Find the coordinate of the 3 rd point for each example. a) coord. Of X is 8 b) coord. Of X is c) coord. Of Y is -8 coord. Of Z is 6 coord. Of Y is 5 coord. Of Z is -9 coord. Of Y = coord. Of Z = coord. Of X = ) the numbers given are the coordinates of two points. Fin the distance between the points. a) -8 and 4 b) -3 and -4 c) 0 and 7.5 d) and 37 e) -5.6 and 7.4 f) a and b 3) Find the midpoint of AG if A(4, -7) and G(-9, 4). 4) Find B if M (, ) is the midpoint of BT and T(7,-4) 5) Use the Distance Formula to find the distance between K(-7, -4) and L(-, 0). 6) 6) Find the length of F(9, 5) and G(, ).. Segments Midpoint a point that bisects, or divides, the segment into two congruent segments. Segment Bisector: A segment, line, plane or ray that intersects a segment at its midpoint. Example- Line a bisects MS at point D. What are conclusions one may make? 3

4 Example -. KM = MT HL and KT intersect at the midpoint of HL. True or False.. KT is a bisector of LH. 3. MT bisects LH. 4. HL is a bisector of KT. 5. M is the midpoint of KT. Postulate -- Segment Addition Postulate: If B is between A and C, then AB + BC = AC. Examples: Example If R is between S and I, find the following. a) If SR = 5 and RI =, find SI. b) If SR = 5 and SI = 8, find RI. c) If SI = 60, SR = x 8, RI = 3x, find x and then find SR and RI. d) If SR = x + 7, RI = 4x 5, and SI = 34, find x. Example B is the midpoint of AC. AB = x + and BC = 5x + 0. Draw the example, come up with an equation, and find x and AC. Example Given RS = x 4, ST = 3x + 7, and RT = 43. Draw the example, come up with an equation, and find x, RS, and ST. Example Given C is the midpoint of AB. AC= 5x-6 and CB = x. Draw the example, come up with an equation, and find x and AC. Example In the figure, RS = 3x, and ST = x 8, and RT = 60. Draw the example, come up with an equation, and find x, RS, and ST. Example G is the midpoint of EF. EG = 3y, and GF = 36 y. Draw the example, come up with an equation, and find x and EG. 4

5 .3 &.4 Angles and Angle Bisectors Angle: the union of rays with a common endpoint Vertex: the common endpoint of the sides of the angle. Interior of an angle: set of all points between the sides of the angle. Exterior of an angle: set of all points outside of the angle Measure: given in degrees B A C Name three of the angles. Protractor Postulate: Given line AB and a point O on AB, all rays that can be drawn from O can be put into a one-to-one correspondence with the real numbers from 0 to 80. Acute Angle- Right Angle- Obtuse Angle- Straight Angle- Perpendicular intersecting to form 90ᵒ angles Postulate -3-: Angle Addition Postulate If S is in the interior of <PQR, then m<pqs + m<sqr = m<pqr Congruent Angles: angles that have the same measure. Angle Bisector: a ray that divides an angle into two congruent angles. Example Draw a picture. If CD bisects ACB, then. Example Draw a picture and solve. M<DEG = 5 and m<def = 48. Find m<feg. Example 3- Draw a picture for the following and write an equation to solve for x. RQ bisects PRS. If m PRQ = x + 40 and m QRS = 3x, solve for x. Then find the measure of each angle Example 4 - Write an equation and solve for x. x = 5 m BAD =, x =

6 Example 5 - Draw a picture for the following problem. E is on the interior of <SRP. M SRP = 57 m ERP = 98 Find m PRS = and m SRE = Example 6- Draw a picture for the following and write an equation to solve for x. If SU bisects <RST, and m<rsu = x, and m<rst = 3x+3, find x = m<tsu = m<rst = Example 7 Draw a picture for the following and write an equation to solve for x. <DEF is a right angle. M<DEG = (3x + 8) and m<gef = (6x 7). Find the m<deg Write the equation for #s 9-3. Then solve. Pairs of Angles *Two angles are if their sides form two pairs of opposite rays. Vertical angles are always 4 3 *Two angles are if they have a common side, a common vertex, and no common interior points. 6 3

7 *Two angles are if the sum of their measures is 90. Each angle is a complement of the other. A 35 *Two angles are if the sum of their measures is 80. Each angle is a supplement of the other. C 55 B E D *Two angles are a if they are adjacent and their non-common sides are opposite rays. A linear pair is always and Example State whether the numbered angles shown in -8 are adjacent, vertical, or neither. If NOT, explain ABC, ABD D C A B Verifying Angle Relationships-Label the diagram below so that m BGC=33 and m DGE=57. FGA. M CGD = 3. BGF and are supplementary 4. M AGF = 5. EGC and are supplementary 6. M AGB = 7. m AGC = 8. M BGD= 9a. Are AGF and CGD supplementary? why/why not? b. Are they a linear pair? why/why not? 7 B A C G F E D

8 Solve the following for the indicated variable. Set up an equation using the angle relationships.. 6x 3 3x 5 x + 6 x x y 7 3x 8 x 50 3x y x 36 Fill in the blank.. The supplement of a right angle is a angle.. The supplement of an obtuse angle is a angle. 3. The supplement of an acute angle is a angle. 4. Vertical angles are. 5. lines form right angles. 6. Congruent supplementary angles each have a measure of. 7. Congruent complementary angles each have a measure of. 8. The angles in a linear pair are and. 9. m< = 40. What is the measure of its complement? Its supplement? 0. m< = 0. What is the measure of its complement? Its supplement?. m<a = x. What is the measure of its complement? Its supplement? Complete with Always, Sometimes, or Never. Two < s that are supplementary form a linear pair.. Two < s that form a linear pair are supplementary. 3. Two congruent angles are right. 4. Two right angles are congruent. 5. Two angles that are vertical are adjacent. 6. Two angles that are non-adjacent are vertical. 7. Two angles that form a linear pair are congruent. 8. Two angles that form a right angle are complementary. 9. Two angles that form a right angle are supplementary. 0. Two angles that are supplementary are congruent.. Vertical angles have a common vertex. 8

9 . Two right angles are complementary. 3. Right angles are vertical angles. 4. Angles A, B, and C are supplementary. 5. Vertical angles have a common supplement. Given: is a right angle, m 5=30, and 3. Find the following m = m = m 3= m 4= m 6= m 7= m 3 + m +m = a. Acute b. Right c. Obtuse d. Linear Pair e. Congruent f. Vertical g. Complementary h. Supplementary i. Adjacent EOA is EOA and DOB are COB is EOA and AOB are EOD and AOB are DOC is DOC and <AOC DOC and COB are COB and BOA are 9