Ch. 1 Essentials of Geometry Review

Lesson 1-1 Point, Line, Plane Coplanar Objects **Remember: Any 3 non-collinear points determine a plane! Coplanar objects (points, lines, etc.) are objects that lie on the same plane. The plane does not have to be visible. Are the following points coplanar? A, B, C ? Yes A, B, C, F ? No H, G, F, E ? Yes E, H, C, B ? Yes A, G, F ? Yes C, B, F, H ? No Lesson 1-1 Point, Line, Plane

Front Side – True or False Naming points, lines, rays, and planes Finding intersections Lesson 1-1 Point, Line, Plane

Lesson 1-1 Point, Line, Plane Example 3 Point S is between point R and point T. Use the given information to write an equation in terms of x. Solve the equation. Then find both RS and ST. RS = 3x – 16 ST = 4x – 8 RT = 60 I---------------60 -------------------I 3x-16 I-----------4x-8-----------I Lesson 1-1 Point, Line, Plane

Lesson 1-1 Point, Line, Plane EXAMPLE 2 Use algebra with segment lengths Point M is the midpoint of VW . Find the length of VM . ALGEBRA Lesson 1-1 Point, Line, Plane 5

Lesson 1-1 Point, Line, Plane GUIDED PRACTICE line l Identify the segment bisector of . Then find PQ. Lesson 1-1 Point, Line, Plane

Lesson 1-1 Point, Line, Plane MIDPOINT FORMULA The midpoint of two points P(x1, y1) and Q(x2, y2) is M(X,Y) = M(x1 + x2, x2 +y2) 2 2 Think of it as taking the average of the x’s and the average of the y’s to make a new point. Lesson 1-1 Point, Line, Plane 7

Lesson 1-1 Point, Line, Plane EXAMPLE 3 Use the Midpoint Formula a. FIND MIDPOINT The endpoints of RS are R(1,–3) and S(4, 2). Find the coordinates of the midpoint M. Lesson 1-1 Point, Line, Plane 8

Lesson 1-1 Point, Line, Plane EXAMPLE 3 Use the Midpoint Formula SOLUTION a. FIND MIDPOINT Use the Midpoint Formula. 2 5 1 + 4 – 3 + 2 = , M – 1 The coordinates of the midpoint M are 1 , – 5 2 ANSWER Lesson 1-1 Point, Line, Plane 9

Lesson 1-1 Point, Line, Plane EXAMPLE 3 Use the Midpoint Formula b. FIND ENDPOINT The midpoint of JK is M(2, 1). One endpoint is J(1, 4). Find the coordinates of endpoint K. FIND ENDPOINT Let (x, y) be the coordinates of endpoint K. Use the Midpoint Formula. STEP 1 Find x. STEP 2 Find y. 1+ x 2 = 4+ y 1 2 = 1 + x = 4 4 + y = 2 x = 3 y = – 2 The coordinates of endpoint K are (3, – 2). ANSWER Lesson 1-1 Point, Line, Plane 10

Lesson 1-1 Point, Line, Plane Distance Formula The distance between two points A and B is Lesson 1-1 Point, Line, Plane 11

Lesson 1-1 Point, Line, Plane EXAMPLE 4 Standardized Test Practice SOLUTION Use the Distance Formula. You may find it helpful to draw a diagram. Lesson 1-1 Point, Line, Plane 12

Lesson 1-1 Point, Line, Plane Naming Angles Name the three angles in diagram. Name this one angle in 3 different ways. WXY, WXZ, and YXZ The vertex of the angle What always goes in the middle? Lesson 1-1 Point, Line, Plane

Lesson 1-1 Point, Line, Plane EXAMPLE 2 Find angle measures o ALGEBRA Given that m LKN =145 , find m LKM and m MKN. SOLUTION STEP 1 Write and solve an equation to find the value of x. m LKN = m LKM + m MKN Angle Addition Postulate 145 = (2x + 10) + (4x – 3) o Substitute angle measures. 145 = 6x + 7 Combine like terms. 138 = 6x Subtract 7 from each side. 23 = x Divide each side by 6. Lesson 1-1 Point, Line, Plane

Lesson 1-1 Point, Line, Plane EXAMPLE 2 Find angle measures STEP 2 Evaluate the given expressions when x = 23. m LKM = (2x + 10)° = (2 23 + 10)° = 56° m MKN = (4x – 3)° = (4 23 – 3)° = 89° So, m LKM = 56° and m MKN = 89°. ANSWER Lesson 1-1 Point, Line, Plane

Lesson 1-1 Point, Line, Plane GUIDED PRACTICE Find the indicated angle measures. 3. Given that KLM is straight angle, find m KLN and m NLM. SOLUTION STEP 1 Write and solve an equation to find the value of x. m KLM + m NLM = 180° Straight angle (10x – 5)° + (4x +3)° = 180° Substitute angle measures. 14x – 2 = 180 Combine like terms. 14x = 182 Subtract 2 from each side. x = 13 Divide each side by 14. Lesson 1-1 Point, Line, Plane

Lesson 1-1 Point, Line, Plane GUIDED PRACTICE STEP 2 Evaluate the given expressions when x = 13. m KLM = (10x – 5)° = (10 13 – 5)° = 125° m NLM = (4x + 3)° = (4 13 + 3)° = 55° ANSWER m KLM = 125° m NLM = 55° Lesson 1-1 Point, Line, Plane

Lesson 1-1 Point, Line, Plane EXAMPLE 3 Double an angle measure In the diagram at the right, YW bisects XYZ, and m XYW = 18. Find m XYZ. o SOLUTION By the Angle Addition Postulate, m XYZ = m XYW + m WYZ. Because YW bisects XYZ you know that XYW WYZ. ~ So, m XYW = m WYZ, and you can write M XYZ = m XYW + m WYZ = 18° + 18° = 36°. Lesson 1-1 Point, Line, Plane

Lesson 1-1 Point, Line, Plane Example 4 Lesson 1-1 Point, Line, Plane

Lesson 1-1 Point, Line, Plane EXAMPLE 2 Find measures of a complement and a supplement a. Given that 1 is a complement of 2 and m 1 = 68°, find m 2. SOLUTION a. You can draw a diagram with complementary adjacent angles to illustrate the relationship. m 2 = 90° – m 1 = 90° – 68° = 22 Lesson 1-1 Point, Line, Plane

Lesson 1-1 Point, Line, Plane EXAMPLE 3 Find angle measures Sports When viewed from the side, the frame of a ball-return net forms a pair of supplementary angles with the ground. Find m BCE and m ECD. Remember in geometry we take are knowledge of the diagram and properties of the angle pairs to set up an equation. Lesson 1-1 Point, Line, Plane

Lesson 1-1 Point, Line, Plane EXAMPLE 3 Find angle measures SOLUTION STEP 1 Use the fact that the sum of the measures of supplementary angles is 180°. m BCE + m ∠ ECD = 180° Write equation. (4x+ 8)° + (x + 2)° = 180° Substitute. 5x + 10 = 180 Combine like terms. 5x = 170 Subtract 10 from each side. x = 34 Divide each side by 5. Lesson 1-1 Point, Line, Plane

Lesson 1-1 Point, Line, Plane EXAMPLE 3 Find angle measures STEP 2 Evaluate: the original expressions when x = 34. m BCE = (4x + 8)° = (4 34 + 8)° = 144° m ECD = (x + 2)° = ( 34 + 2)° = 36° The angle measures are 144° and 36°. ANSWER Lesson 1-1 Point, Line, Plane

Angles Formed by the Intersection of 2 Lines Click on picture!! What angles always are equal, which ones add to equal 180 degree? What are properties of linear pairs (adjacent angles) and vertical angles? How might this be useful? Time saving? When would the vertical and adjacent angles all be equal?  Click Me! Lesson 1-1 Point, Line, Plane

Lesson 1-1 Point, Line, Plane EXAMPLE 4 Identify angle pairs Identify all of the linear pairs and all of the vertical angles in the figure at the right. SOLUTION To find linear pairs, look for adjacent angles whose noncommon sides are opposite rays. 1 and 4 are a linear pair. 4 and 5 are also a linear pair. ANSWER To find vertical angles, look or angles formed by intersecting lines. 1 and 5 are vertical angles. ANSWER Lesson 1-1 Point, Line, Plane

Lesson 1-1 Point, Line, Plane Example 5 Two angles form a linear pair. The measure of one angle is 5 times the measure of the other. Find the measure of each angle. Lesson 1-1 Point, Line, Plane

Lesson 1-1 Point, Line, Plane Example 6 Given that m5 = 60 and m3 = 62, use your knowledge of linear pairs and vertical angles to find the missing angles. Lesson 1-1 Point, Line, Plane

Lesson 1-1 Point, Line, Plane EXAMPLE 1 Identify polygons Tell whether the figure is a polygon and whether it is convex or concave. a. b. c. d. SOLUTION Some segments intersect more than two segments, so it is not a polygon. a. b. The figure is a convex polygon. Part of the figure is not a segment, so it is not a polygon. c. d. The figure is a concave polygon. Lesson 1-1 Point, Line, Plane

Lesson 1-1 Point, Line, Plane # of sides Type of Polygon triangle quadrilateral pentagon hexagon heptagon octagon nonagon decagon dodecagon n-gon 3 4 5 6 7 8 9 10 12 n What is a polygon with 199 sides called? 199-gon Lesson 1-1 Point, Line, Plane

Lesson 1-1 Point, Line, Plane EXAMPLE 2 Classify polygons Classify the polygon by the number of sides. Tell whether the polygon is equilateral, equiangular, or regular. Explain your reasoning. a. b. SOLUTION The polygon has 6 sides. It is equilateral and equiangular, so it is a regular hexagon. a. The polygon has 4 sides, so it is a quadrilateral. It is not equilateral or equiangular, so it is not regular. b. Lesson 1-1 Point, Line, Plane

Lesson 1-1 Point, Line, Plane EXAMPLE 3 Find side lengths A table is shaped like a regular hexagon.The expressions shown represent side lengths of the hexagonal table. Find the length of a side. ALGEBRA SOLUTION First, write and solve an equation to find the value of x. Use the fact that the sides of a regular hexagon are congruent. 3x + 6 4x – 2 = Write equation. 6 = x – 2 Subtract 3x from each side. 8 = x Add 2 to each side. Lesson 1-1 Point, Line, Plane

Lesson 1-1 Point, Line, Plane EXAMPLE 3 Find side lengths Then find a side length. Evaluate one of the expressions when x = 8. 30 3(8) + 6 = 3x + 6 The length of a side of the table is 30 inches. ANSWER Lesson 1-1 Point, Line, Plane

Lesson 1-1 Point, Line, Plane Perimeter/Area Rectangle Square Triangle Circle Lesson 1-1 Point, Line, Plane

Lesson 1-1 Point, Line, Plane Area The area of the triangle is 14 square inches and its height is 7 inches. Find the base of the triangle. Lesson 1-1 Point, Line, Plane

Lesson 1-1 Point, Line, Plane Perimeter The perimeter of a rectangle 84.6 centimeters. The length of the rectangle is twice as long as its width. Find the length and width of the rectangle. Lesson 1-1 Point, Line, Plane