Happy New Year! Analytic Geometry Pick up the weekly agenda sheet and the packet for the week. Find your vocabulary match. This is your new team member.
Unit 1: Similarity, Congruence & Proofs
Vocabulary
Postulate A statement that is accepted as true without proof. Also called an axiom.
Theorem A statement that has been proven.
Conjecture An unproven statement, based on observation
Inductive Reasoning Look for patterns and make conjectures
Counterexample An example that shows a conjecture is false
Undefined Terms Point Line Plane
Facts About a Point No Dimension Intersection of two lines Intersection of a line and a plane
Facts About a Line Has length, but no width Intersection of 2 planes
Facts About a Plane Has length and width, but no thickness Extends infinitely in two directions
Collinear Points Points that lie on the same line
Coplanar Points Points that lie in the same plane
AB Line AB Extends in two directions
AB Line segment AB It has definite endpoints on both ends.
Congruent segments are segments that have the same length. In the diagram, PQ = RS, so you can write PQ RS. This is read as segment PQ is congruent to segment RS. Tick marks are used in a figure to show congruent segments.
Lines Parallel Lines: Slope is the same, y- intercepts are different numbers Example y = 3x + 2 and y = 3x 5 Perpendicular Lines: upside down and backwards the slope is the inverse with the opposite sign Example y = 3x + 2 and y = -1/3x + 5 Lines Tangent to a circle: touch the circle at a single point, and are perpendicular to the radius of the circle.
Slope of a Line Slope: Rise over Run (change in y over change in x) Positive: if the line travels upwards (from left to right) Negative: if the line travels downward (from left to right) Zero: the line is flat, parallel to the x-axis Undefined: vertical, parallel to the y-axis
AB Ray AB It has an endpoint at A and extends indefinitely through B
A B C Ray BA and Ray BC are Opposite Rays
Coordinate A number that corresponds to a point on the number line.
AB The distance between points A and B Or The length segment AB
Segment Addition A B C If B is between A and C, then AB + BC = AC, and If AB + BC = AC, then B is between A and C.
A ruler can be used to measure the distance between two points. A point corresponds to one and only one number on a ruler. The number is called a coordinate. The following postulate summarizes this concept.
Ruler Postulate
The distance between any two points is the absolute value of the difference of the coordinates. If the coordinates of points A and B are a and b, then the distance between A and B is a b or b a. The distance between A and B is also called the length of AB, or AB. A a B b AB = a b or b - a
Example 1: Finding the Length of a Segment Find each length. A. BC B. AC BC = 1 3 AC = 2 3 = 1 3 = 2 = 5 = 5
Check It Out! Example 1 Find each length. a. XY b. XZ
In order for you to say that a point B is between two points A and C, all three points must lie on the same line, and AB + BC = AC. Segment Addition Postulate
Example 3A: Using the Segment Addition Postulate G is between F and H, FG = 6, and FH = 11. Find GH. FH = FG + GH 11 = 6 + GH 6 6 5 = GH Seg. Add. Postulate Substitute 6 for FG and 11 for FH. Subtract 6 from both sides. Simplify.
Analytic Geometry Day 2 Classifying Triangles Worksheet Classify each triangle using its angle properties.
Example 3B: Using the Segment Addition Postulate M is between N and O. Find NO. NM + MO = NO 17 + (3x 5) = 5x + 2 3x + 12 = 5x + 2 2 2 3x + 10 = 5x 3x 3x 10 = 2x 2 2 5 = x Seg. Add. Postulate Substitute the given values Simplify. Subtract 2 from both sides. Simplify. Subtract 3x from both sides. Divide both sides by 2.
Example 3B Continued M is between N and O. Find NO. NO = 5x + 2 = 5(5) + 2 = 27 Substitute 5 for x. Simplify.
Check It Out! Example 3a Y is between X and Z, XZ = 3, and XY =. Find YZ. XZ = XY + YZ Seg. Add. Postulate Substitute the given values. Subtract from both sides.
Check It Out! Example 3b E is between D and F. Find DF. DE + EF = DF (3x 1) + 13 = 6x 3x + 12 = 6x 3x 3x 12 = 3x 12 3x = 3 3 4 = x Seg. Add. Postulate Substitute the given values Subtract 3x from both sides. Simplify. Divide both sides by 3.
Check It Out! Example 3b Continued E is between D and F. Find DF. DF = 6x = 6(4) = 24 Substitute 4 for x. Simplify.
Distance Formula The distance from (x 1,y 1 ) to (x 2,y 2 ) is 2 2 2 1 2 1 ( x x ) ( y y )
Angle The union of two rays with a common endpoint
Vertex The common endpoint of the two rays forming an angle
Angles Angles with equal measure
A transit is a tool for measuring angles. It consists of a telescope that swivels horizontally and vertically. Using a transit, a survey or can measure the angle formed by his or her location and two distant points. An angle is a figure formed by two rays, or sides, with a common endpoint called the vertex (plural: vertices). You can name an angle several ways: by its vertex, by a point on each ray and the vertex, or by a number.
The set of all points between the sides of the angle is the interior of an angle. The exterior of an angle is the set of all points outside the angle. Angle Name R, SRT, TRS, or 1 You cannot name an angle just by its vertex if the point is the vertex of more than one angle. In this case, you must use all three points to name the angle, and the middle point is always the vertex.
The measure of an angle is usually given in degrees. Since there are 360 in a circle, one degree is of a circle. When you use a protractor to measure angles, you are applying the following postulate. Protractor Postulate
You can use the Protractor Postulate to help you classify angles by their measure. The measure of an angle is the absolute value of the difference of the real numbers that the rays correspond with on a protractor. If OC corresponds with c and OD corresponds with d, m DOC = d c or c d.
Angle Addition Postulate If P is a point on the interior of RST, then m RSP + m PST = m RST
Example 1: Naming Angles A surveyor recorded the angles formed by a transit (point A) and three distant points, B, C, and D. Name three of the angles. Possible answer: BAC CAD BAD
Check It Out! Example 1 Write the different ways you can name the angles in the diagram. RTQ, T, STR, 1, 2
Acute Angle An angle that measures between 0 o and 90 o
Right Angle An angle that measures 90 o
Obtuse Angle An angle that measures between 90 o and 180 o
Straight Angle An angle that measures 180 o
Types of Angles
Example 2: Measuring and Classifying Angles Find the measure of each angle. Then classify each as acute, right, or obtuse. A. WXV m WXV = 30 WXV is acute. B. ZXW m ZXW = 130-30 = 100 ZXW = is obtuse.
Check It Out! Example 2 Use the diagram to find the measure of each angle. Then classify each as acute, right, or obtuse. a. BOA m BOA = 40 BOA is acute. b. DOB m DOB = 125 DOB is obtuse. c. EOC m EOC = 105 EOC is obtuse.
Congruent angles are angles that have the same measure. In the diagram, m ABC = m DEF, so you can write ABC DEF. This is read as angle ABC is congruent to angle DEF. Arc marks are used to show that the two angles are congruent. The Angle Addition Postulate is very similar to the Segment Addition Postulate that you learned in the previous lesson.
Angle Addition Postulate
Example 3: Using the Angle Addition Postulate m DEG = 115, and m DEF = 48. Find m FEG m DEG = m DEF + m FEG Add. Post. 115 = 48 + m FEG Substitute the given values. 48 48 Subtract 48 from both sides. 67 = m FEG Simplify.
Check It Out! Example 3 m XWZ = 121 and m XWY = 59. Find m YWZ. m YWZ = m XWZ m XWY Add. Post. m YWZ = 121 59 m YWZ = 62 Substitute the given values. Subtract.
Angle Bisector An angle bisector is a ray that divides an angle into two congruent angles. JK bisects LJM; thus LJK KJM.
Example 4: Finding the Measure of an Angle KM bisects JKL, m JKM = (4x + 6), and m MKL = (7x 12). Find m JKM.
Example 4 Continued Step 1 Find x. m JKM = m MKL (4x + 6) = (7x 12) +12 +12 4x + 18 = 7x 4x 4x 18 = 3x 6 = x Def. of bisector Substitute the given values. Add 12 to both sides. Simplify. Subtract 4x from both sides. Divide both sides by 3. Simplify.
Example 4 Continued Step 2 Find m JKM. m JKM = 4x + 6 = 4(6) + 6 = 30 Substitute 6 for x. Simplify.
Check It Out! Example 4a Find the measure of each angle. QS bisects PQR, m PQS = (5y 1), and m PQR = (8y + 12). Find m PQS. Step 1 Find y. Def. of bisector Substitute the given values. 5y 1 = 4y + 6 y 1 = 6 y = 7 Simplify. Subtract 4y from both sides. Add 1 to both sides.
Check It Out! Example 4a Continued Step 2 Find m PQS. m PQS = 5y 1 = 5(7) 1 = 34 Substitute 7 for y. Simplify.
Check It Out! Example 4b Find the measure of each angle. JK bisects LJM, m LJK = (-10x + 3), and m KJM = ( x + 21). Find m LJM. Step 1 Find x. LJK = KJM ( 10x + 3) = ( x + 21) +x +x 9x + 3 = 21 3 3 9x = 18 x = 2 Def. of bisector Substitute the given values. Add x to both sides. Simplify. Subtract 3 from both sides. Divide both sides by 9. Simplify.
Check It Out! Example 4b Continued Step 2 Find m LJM. m LJM = m LJK + m KJM = ( 10x + 3) + ( x + 21) = 10( 2) + 3 ( 2) + 21 Substitute 2 for x. = 20 + 3 + 2 + 21 Simplify. = 46
Lesson Quiz: Part I 1. M is between N and O. MO = 15, and MN = 7.6. Find NO. 22.6 2. S is the midpoint of TV, TS = 4x 7, and SV = 5x 15. Find TS, SV, and TV. 25, 25, 50 3. Sketch, draw, and construct a segment congruent to CD. Check students' constructions
Lesson Quiz: Part II 4. LH bisects GK at M. GM = 2x + 6, and GK = 24. Find x. 3 5. 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. Always K M L
The midpoint M of AB is the point that bisects, or divides, the segment into two congruent segments. If M is the midpoint of AB, then AM = MB. So if AB = 6, then AM = 3 and MB = 3.
Example 4: Recreation Application 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? XY = 3(5280 ft) Convert race distance to feet. = 15,840 ft
Example 4 Continued XC + CY = XY Seg. Add. Post. Substitute 6000 for XC and 15,840 6000 + CY = 15,840 for XY. 6000 6000 Subtract 6000 from both sides. CY = 9840 = 4920 ft Simplify. D is the mdpt. of CY, so CD = CY. The checkpoints are 4920 ft apart.
Check It Out! Example 4 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? The distance XY is 1182.5 m. The midpoint would be.
Example 5: Using Midpoints to Find Lengths D is the midpoint of EF, ED = 4x + 6, and DF = 7x 9. Find ED, DF, and EF. E 4x + 6 D 7x 9 F Step 1 Solve for x. ED = DF D is the mdpt. of EF. 4x + 6 = 7x 9 Substitute 4x + 6 for ED and 7x 9 for DF. 4x 4x 6 = 3x 9 Subtract 4x from both sides. Simplify. +9 + 9 Add 9 to both sides. 15 = 3x Simplify.
Example 5 Continued D is the midpoint of EF, ED = 4x + 6, and DF = 7x 9. Find ED, DF, and EF. E 4x + 6 D 7x 9 F 15 = 3x 3 3 Divide both sides by 3. x = 5 Simplify.
Example 5 Continued D is the midpoint of EF, ED = 4x + 6, and DF = 7x 9. Find ED, DF, and EF. E 4x + 6 D 7x 9 F Step 2 Find ED, DF, and EF. ED = 4x + 6 = 4(5) + 6 = 26 DF = 7x 9 = 7(5) 9 = 26 EF = ED + DF = 26 + 26 = 52
Check It Out! Example 5 S is the midpoint of RT, RS = 2x, and ST = 3x 2. Find RS, ST, and RT. R 2x S 3x 2 T Step 1 Solve for x. RS = ST S is the mdpt. of RT. 2x = 3x 2 Substitute 2x for RS and 3x 2 for ST. +3x +3x Add 3x to both sides. x = 2 Simplify.
Check It Out! Example 5 Continued S is the midpoint of RT, RS = 2x, and ST = 3x 2. Find RS, ST, and RT. R 2x S 3x 2 T Step 2 Find RS, ST, and RT. RS = 2x = 2( 2) = 4 ST = 3x 2 = 3( 2) 2 = 4 RT = RS + ST = 4 + 4 = 8
Construction You can make a sketch or measure and draw a segment. These may not be exact. A construction is a way of creating a figure that is more precise. One way to make a geometric construction is to use a compass and straightedge.
Example 2: Copying a Segment Sketch, draw, and construct a segment congruent to MN. Step 1 Estimate and sketch. Estimate the length of MN and sketch PQ approximately the same length. P Q
Example 2 Continued Sketch, draw, and construct a segment congruent to MN. Step 2 Measure and draw. Use a ruler to measure MN. MN appears to be 3.5 in. Use a ruler to draw XY to have length 3.5 in. X Y
Example 2 Continued Sketch, draw, and construct a segment congruent to MN. Step 3 Construct and compare. Use a compass and straightedge to construct ST congruent to MN. A ruler shows that PQ and XY are approximately the same length as MN, but ST is precisely the same length.
Check It Out! Example 2 Sketch, draw, and construct a segment congruent to JK. Step 1 Estimate and sketch. Estimate the length of JK and sketch PQ approximately the same length.
Check It Out! Example 2 Continued Sketch, draw, and construct a segment congruent to JK. Step 2 Measure and draw. Use a ruler to measure JK. JK appears to be 1.7 in. Use a ruler to draw XY to have length 1.7 in.
Check It Out! Example 2 Continued Sketch, draw, and construct a segment congruent to JK. Step 3 Construct and compare. Use a compass and straightedge to construct ST congruent to JK. A ruler shows that PQ and XY are approximately the same length as JK, but ST is precisely the same length.
Triangles Interior angles: add up to 180 Equilateral: all 3 sides are equal, all angles equal 60 Isosceles: 2 sides are equal, 2 angles are equal Triangle Inequality: The sum of the lengths of any 2 sides of a triangle is greater than the length of the third side.
Classifying Triangles Triangle A figure formed when three noncollinear points are connected by segments. E Side Angle D The sides are DE, EF, and DF. The vertices are D, E, and F. The angles are D, E, F. Vertex F
Triangles Classified by Angles Acute Obtuse Right 60º 50º 120º 17º 30 70º All acute angles 43º One obtuse angle 60º One right angle
Triangles Classified by Sides Scalene Isosceles Equilateral no sides congruent at least two sides congruent all sides congruent
Classify each triangle by its angles and by its sides. E C 45 60 F 45 G A 60 60 B EFG is a right isosceles triangle. ABC is an acute equilateral triangle
Parts of Isosceles Triangles The two angles formed by the base and one of the congruent sides are called base angles. base angle The angle formed by the congruent sides is called the vertex angle. The side opposite the vertex is the base. leg leg The congruent sides are called legs. base angle
HYPOTENUSE LEG LEG
Interior Angles Exterior Angles
Triangle Sum Theorem The measures of the three interior angles in a triangle add up to be 180º. x + y + z = 180 x y z
Find m T R in RST. 54 m R + m S + m T = 180º 54º + 67º + m T = 180º S 67 T 121º + m T = 180º m T = 59º
Find the value of each variable in DCE A B 85 C y x 55 E m D + m DCE + m E = 180º 55º + 85º + y = 180º 140º + y = 180º D y = 40º
Corollary to the Triangle Sum Theorem The acute angles of a right triangle are complementary. x + y = 90º x y
Find m A and m B in right triangle ABC. A m A + m B = 90 C 2x 3x B 2x + 3x = 90 5x = 90 x = 18 m A = 2x = 2(18) = 36 m B = 3x = 3(18) = 54
Lesson Quiz: Part I Classify each angle as acute, right, or obtuse. 1. XTS 2. WTU acute right 3. K is in the interior of LMN, m LMK =52, and m KMN = 12. Find m LMN. 64
Lesson Quiz: Part II 4. BD bisects ABC, m ABD =, and m DBC = (y + 4). Find m ABC. 32 5. Use a protractor to draw an angle with a measure of 165.
Lesson Quiz: Part III 6. m WYZ = (2x 5) and m XYW = (3x + 10). Find the value of x. 35
Formulas to Memorize 2, 2 2 1 2 1 y y x x 2 1 2 2 1 2 ) ( ) ( y y x x d Midpoint Formula Distance Formula (Based on the Pythagorean Theorem)