Lesson 2.1 8/5/2014. Perpendicular Lines, Rays and Segments. Let s Draw some examples of perpendicularity. What is the symbol for perpendicular?

Size: px

Start display at page:

Download "Lesson 2.1 8/5/2014. Perpendicular Lines, Rays and Segments. Let s Draw some examples of perpendicularity. What is the symbol for perpendicular?"

Oswald Parsons
6 years ago
Views:

1 8/5/04 Lesson. Perpendicularity From now on, when you write a two-column proof, try to state each reason in a single sentence or less. bjective: Recognize the need for clarity and concision in proofs and understand the concept of perpendicularity This chapter contains more definitions and theorems for you to memorize and use. Perpendicular Lines, Rays and Segments Perpendicularity, right angles and all go together. measurements 90 efinition: Lines, rays, or segments that intersect at right angles are perpendicular. Let s raw some examples of perpendicularity. a b a b E EF E M GH 90 K G M H What is the symbol for perpendicular? F In the figure at the right, the mark inside the angle ( ) indicates that is a right angle. It is also true that and 90 o NT assume perpendicularity from a diagram! In EF it appears that E EF, but we may not assume that they are. In each of the following, name the angles that can be proved to be right angles. Given : M M K L K X PN N MK NX, PX X P ircle L S None W E F

2 8/5/04 Let s practice Find the measure of M M, 5 4'6" M e f Prove: (Hint: This proof takes more than steps. Remember, each reason should be a single sentence or less.) e f 3 is 35 times as large as 4, and XY YZ. Find m 4 to the nearest tenth. X Y 3 S 4 Z Important o NT assume perpendicularity from a diagram The horizontal line is called the x-axis The vertical line is called the y-axis y-axis (3,4) Two perpendicular number lines form a two-dimensional coordinate system, or coordinate plane. x-axis rigin Each point is represented by an ordered pair in the form of (x,y) The values of the x and y are called the points coordinates The intersection of the axes is called the origin. Its coordinates are (0,0). Summary Write three things you learned in this lesson. Homework Lesson. Worksheet

3 8/5/04 Lesson. omplementary and Supplementary ngles bjective: Recognize complementary and supplementary angles efinition: omplementary angles are two angles whose sum is (a right angle) Each of the two angles is called the complement of the other. raw two examples of complementary angles. Which two angles are complementary? is complementary to. If how large is? 6 Find the complement of a angle. 3 8 efinition: Supplementary angles are two angles whose sum is (a straight angle). Each of the two angles is called the supplement of the other. raw two examples of supplementary angles. What is the supplement of a angle? iagram as shown Prove: < is supp. to <. (Hint: This proof takes more than two steps.)

4 8/5/04 <TVK is a right angle. Prove: < is comp. to <. T V X K Steps to Solving Word Problems Read the entire problem. raw a picture or diagram 3. Write down the information given etermine what information is missing 5. Plan how to solve for the missing information 6. Solve for the missing information 7. Make sure that your answer is appropriate and answers any questions presented The measure of one of two complementary angles is three greater than twice the measure of the other. Find the measure of each. Summary Explain how you would find a supplement and a complement. The measure of the supplement of an angle is 60 less than 3 times the measure of the complement of the angle. Find the measure of the complement Homework Lesson. Worksheet

5 8/5/04 Lesson.3 rawing onclusions bjective: fter studying this section, you will be able to follow a five-step procedure to draw logical conclusions Procedures for rawing onclusions Memorize theorems, definitions, and postulates. Look for key words and symbols in the given information. 3. Think of all the theorems, definitions, and postulates that involve those keys. ecide which theorem, definition, or postulate allows you to draw a conclusion. 5. raw a conclusion, and give a reason to justify the conclusion. e certain that you have not used the reverse of the correct reason. Let s Practice bisects onclusion: Thinking process: The key word is bisects. The key symbols are and The definition of bisector (of an angle) contains those keys. n appropriate conclusion is that Let s do a proof : ) bisects onclusion:.. Let s try some more! is a right angle is a right angle onclusion: E is the midpoint of SG onclusion: S E G....

6 8/5/04 is a right angle Summary onclusion: What can you conclude about this lesson? (Hint: you could look back at your notes on how to draw conclusions).. Homework Lesson.3 Worksheet

7 8/5/04 Lesson.4 ongruent Supplements and omplements In the diagram below, is supplementary to, and is also supplementary to. bjective: To prove angles congruent by means of four new theorems 70 How large is? Now calculate. How does compare with? Your results will illustrate (but not prove) the following theorem. Theorem 4 If angles are supplementary to the same angle, then they are congruent Theorem 5 If angles are supplementary to congruent angles, then they are congruent 3 is supplemetary to 4 5 is supplemetary to 4 Prove: F is supplemetary to G H is supplemetary to G Prove: F H F G H Theorem 6 If angles are complementary to the same angle, then they are congruent. Let s try so more is supplemetary to 3 is supplemetary to 4 Prove: Theorem 7 If angles are complementary to congruent angles, then they are congruent

8 8/5/04 is complemetary to is complemetary to iagram as shown H G onclusion:? Prove: HFE GF E F KM M K R P P M KMR PR Prove: RM RM M Summary You need to memorize the theorems. How are you going to remember them? Homework Lesson.4 Worksheet

9 8/5/04 Lesson.5 ddition and Subtraction Properties 7cm 3cm 7cm bjective: fter studying this lesson you will be able to apply the addition and subtraction properties of segments and angles. In the diagram above =. o you think that =? Suppose that were 5 cm. Would =? oes the length of have any effect on whether =? Theorem 8 If a segment is added to two congruent segments, the sums are congruent. (addition property of equality) Theorem 9 If an angle is added to two congruent angles, the sums are congruent. (ddition Property of Equality) P Q R S PQ RS Prove: PR QS PQ RS Given. PQ RS. ef. of congruent segments 3. PQ QR RS QR 3. ddition Property of Equality PR QS If a seg. Is added to congruent segs. The sums are congruent K R M P S o you think that KM is necessarily congruent to P? Theorem 0 If congruent segments are added to congruent segments, the sums are congruent. (ddition Property of Equality) Y T o you think that TWX is necessarily congruent to TXW? Theorem If a segment (or angle) is subtracted from congruent segments (or angles), the differences are congruent. (Subtraction property) W X If K = KP and N = RP Is KN = KR? K Theorem If congruent angles are added to congruent angles, the sums are congruent. (ddition Property of Equality) N R P

10 8/5/04 Theorem 3 If congruent segments (or angles) are subtracted from congruent segments (or angles), the differences are congruent. (Subtraction property) NP NP RP RP onclusion: NR NPR N R P onclusion:? E F HEF is supplementary to EHG H GFE is supplementary to FGH EHF FGE GHF HGE onclusion: HEF GFE E G F Summary How are the addition and subtraction theorems that you just learned similar/different? Homework Lesson.5 Worksheet

11 8/5/04 Lesson.6 Multiplication and ivision Properties In the figure below, E,,, and F are trisection points. E T U F If E = U = 3, what can we say about T and U? If E U, is T congruent to U? bjective: fter studying this lesson you will be able to apply the multiplication and division properties of segments and angles. K and PS are angle bisectors If m K m NPS 5, what can we say about KM and NPR? If K NPS, is KM NPR? K M S N R P Theorem 4 If segments (angles) are congruent, their like multiples are congruent (multiplication property of equality) U E S is the midpoint of U N is the midpoint of E Prove: S N S N U E Theorem 5 If segments (angles) are congruent, their like divisions are congruent (division property of equality) TRY E RW and RX trisect TRY and trisect E onclusion: TRW If angles are congruent, their like divisions (thirds) are congruent. (ivision property) R T W X Y E M K MK FG KG bisects M and FH F G H Prove: M FH If segments are, their like multiples (doubles) are. (multiplication property)

12 8/5/04 YZ XZY Prove: ZS bisects XZY YU bisects YZ UY is complementary to XSZ is complementary to 3 UY XSZ YZ XZY. ZS bisects XZY Given. Given 3. YU bisects YZ 3. Given 3 Halves of ' s are. (n alternative form of the division property). 5. UY is complementary to 5. Given 6. XSZ is complementary to 3 6. Given 7. UY XSZ 7. omplements of ' s are Y S U 3 X Z Summary How are the addition and subtraction theorems that you just learned similar/different? Homework Lesson.6 Worksheet

13 8/5/04 Lesson.7 Transitive and Substitution Properties S U If S, and U is S U? bjective: fter studying this lesson you will be able to apply the transitive properties of segments and angles. You will also be able to apply the substitution property. Theorem 6 If angles (or segments) are congruent to the same angle (or segment), they are congruent to each other. (transitive property) The Substitution Property If, find m Theorem 7 If angles (or segments) are congruent to congruent angles (or segments), they are congruent to each other. (transitive property) ( x 0) (x 4) Prove: FG K GH K KG bisects FH F K G H Prove: If segments are congruent to the same segment, they are congruent If a line divides a segment into two congruent segments, it bisects the segment

14 8/5/04 If P R and Q R, express m Q in terms of x and a ( x y a) ( y a) P Q R y + a = x + y + a y = x + y y = x m P x y a m P x x a m P x a m Q x a Summary Explain how the transitive property and the substitution property will work in proofs? Homework Lesson.7 Worksheet

15 8/5/04 Lesson.8 Vertical ngles and are opposite rays bjective: fter studying this lesson you will be able to recognize opposite rays and vertical angles. efinition Two collinear rays that have a common endpoint and extend in different directions are called opposite rays. re all pairs of rays called opposite rays? Vertical ngles M K Key questions: re they on the same line? Whenever two lines intersect, two pairs of vertical angles are formed. 4 3 and 3 and 4 are vertical angles o the rays share the same endpoint? efinition Two angles are vertical angles if the rays forming the sides of one and the rays forming the sides of the other are opposite rays re angle 3 and angle vertical angles? 4 3 How do vertical angles compare in size? Prove: iagram as shown Theorem 8 Vertical angles are congruent 9 Find the missing angles

16 8/5/04 Prove: is complementary to is complementary to Prove: H K M m 4 x 5 m 5 x 30 Find: m Summary Explain what opposite rays are and how they relate to vertical angles? Homework Lesson.8 Worksheet

Geometry-Chapter 2 Notes

2.1 Perpendicularity Geometry-Chapter 2 Notes Vocabulary: perpendicular symbol parallel symbol Definition #16 Lines, rays or segments that intersect at right angles are perpendicular. Examples and label