Points, Lines, Planes, and Angles 7-1 Points, Lines, Planes, and Angles Warm Up Problem of the Day Lesson Presentation Course 3

Points, Lines, Planes, and Angles Course 3 7-1 Points, Lines, Planes, and Angles Warm Up Solve. 1. x + 30 = 90 2. 103 + x = 180 3. 32 + x = 180 4. 90 = 61 + x 5. x + 20 = 90 x = 60 x = 77 x = 148 x = 29 x = 70

Points, Lines, Planes, and Angles Course 3 7-1 Points, Lines, Planes, and Angles Problem of the Day Mrs. Meyer’s class is having a pizza party. Half the class wants pepperoni on the pizza, of the class wants sausage on the pizza, and the rest want only cheese on the pizza. What fraction of Mrs. Meyer’s class wants just cheese on the pizza? 1 3 1 6

Points, Lines, Planes, and Angles Course 3 7-1 Points, Lines, Planes, and Angles Learn to classify and name figures.

Insert Lesson Title Here Course 3 7-1 Points, Lines, Planes, and Angles Insert Lesson Title Here Vocabulary point line plane segment ray angle right angle acute angle obtuse angle complementary angles supplementary angles vertical angles congruent

Points, Lines, Planes, and Angles Course 3 7-1 Points, Lines, Planes, and Angles Points, lines, and planes are the building blocks of geometry. Segments, rays, and angles are defined in terms of these basic figures.

Points, Lines, Planes, and Angles Course 3 7-1 Points, Lines, Planes, and Angles A point names a location. • A Point A

Points, Lines, Planes, and Angles Course 3 7-1 Points, Lines, Planes, and Angles A line is perfectly straight and extends forever in both directions. B C l line l, or BC

Points, Lines, Planes, and Angles Course 3 7-1 Points, Lines, Planes, and Angles A plane is a perfectly flat surface that extends forever in all directions. P E plane P, or plane DEF D F

Points, Lines, Planes, and Angles Course 3 7-1 Points, Lines, Planes, and Angles A segment, or line segment, is the part of a line between two points. H GH G

Points, Lines, Planes, and Angles Course 3 7-1 Points, Lines, Planes, and Angles A ray is a part of a line that starts at one point and extends forever in one direction. J KJ K

Additional Example 1: Naming Points, Lines, Planes, Segments, and Rays Course 3 7-1 Points, Lines, Planes, and Angles Additional Example 1: Naming Points, Lines, Planes, Segments, and Rays A. Name 4 points in the figure. Point J, point K, point L, and point M B. Name a line in the figure. KL or JK Any 2 points on a line can be used.

Additional Example 1: Naming Points, Lines, Planes, Segments, and Rays Course 3 7-1 Points, Lines, Planes, and Angles Additional Example 1: Naming Points, Lines, Planes, Segments, and Rays C. Name a plane in the figure. Plane , plane JKL Any 3 points in the plane that form a triangle can be used.

Additional Example 1: Naming Points, Lines, Planes, Segments, and Rays Course 3 7-1 Points, Lines, Planes, and Angles Additional Example 1: Naming Points, Lines, Planes, Segments, and Rays D. Name four segments in the figure. JK, KL, LM, JM E. Name four rays in the figure. KJ, KL, JK, LK

Points, Lines, Planes, and Angles Course 3 7-1 Points, Lines, Planes, and Angles Check It Out: Example 1 A. Name 4 points in the figure. Point A, point B, point C, and point D B. Name a line in the figure. DA or BC Any 2 points on a line can be used. A B C D

Points, Lines, Planes, and Angles Course 3 7-1 Points, Lines, Planes, and Angles Check It Out: Example 1 C. Name a plane in the figure. Plane , plane ABC, plane BCD, plane CDA, or plane DAB Any 3 points in the plane that form a triangle can be used. A B C D

Points, Lines, Planes, and Angles Course 3 7-1 Points, Lines, Planes, and Angles Check It Out: Example 1 D. Name four segments in the figure AB, BC, CD, DA E. Name four rays in the figure DA, AD, BC, CB A B C D

Points, Lines, Planes, and Angles Course 3 7-1 Points, Lines, Planes, and Angles An angle () is formed by two rays with a common endpoint called the vertex (plural, vertices). Angles can be measured in degrees. One degree, or 1°, is of a circle. m1 means the measure of 1. The angle can be named XYZ, ZYX, 1, or Y. The vertex must be the middle letter. 1 360 X Y Z 1 m1 = 50°

Points, Lines, Planes, and Angles Course 3 7-1 Points, Lines, Planes, and Angles The measures of angles that fit together to form a straight line, such as FKG, GKH, and HKJ, add to 180°. F K J G H

Points, Lines, Planes, and Angles Course 3 7-1 Points, Lines, Planes, and Angles The measures of angles that fit together to form a complete circle, such as MRN, NRP, PRQ, and QRM, add to 360°. P R Q M N

Points, Lines, Planes, and Angles Course 3 7-1 Points, Lines, Planes, and Angles A right angle measures 90°. An acute angle measures less than 90°. An obtuse angle measures greater than 90° and less than 180°. Complementary angles have measures that add to 90°. Supplementary angles have measures that add to 180°.

Points, Lines, Planes, and Angles Course 3 7-1 Points, Lines, Planes, and Angles A right angle can be labeled with a small box at the vertex. Reading Math

Additional Example 2: Classifying Angles Course 3 7-1 Points, Lines, Planes, and Angles Additional Example 2: Classifying Angles A. Name a right angle in the figure. TQS B. Name two acute angles in the figure. TQP, RQS

Additional Example 2: Classifying Angles Course 3 7-1 Points, Lines, Planes, and Angles Additional Example 2: Classifying Angles C. Name two obtuse angles in the figure. SQP, RQT

Additional Example 2: Classifying Angles Course 3 7-1 Points, Lines, Planes, and Angles Additional Example 2: Classifying Angles D. Name a pair of complementary angles. TQP, RQS mTQP + m RQS = 47° + 43° = 90°

Additional Example 2: Classifying Angles Course 3 7-1 Points, Lines, Planes, and Angles Additional Example 2: Classifying Angles E. Name two pairs of supplementary angles. TQP, RQT mTQP + m RQT = 47° + 133° = 180° SQP, SQR mSQP + m SQR = 137° + 43° = 180°

Points, Lines, Planes, and Angles Course 3 7-1 Points, Lines, Planes, and Angles Check It Out: Example 2 A. Name a right angle in the figure. BEC E D C B A 90° 75° 15°

Points, Lines, Planes, and Angles Course 3 7-1 Points, Lines, Planes, and Angles Check It Out: Example 2 B. Name two acute angles in the figure. AEB, CED C. Name two obtuse angles in the figure. BED, AEC E D C B A 90° 75° 15°

Points, Lines, Planes, and Angles Course 3 7-1 Points, Lines, Planes, and Angles Check It Out: Example 2 D. Name a pair of complementary angles. AEB, CED mAEB + m CED = 15° + 75° = 90° E D C B A 90° 75° 15°

Points, Lines, Planes, and Angles Course 3 7-1 Points, Lines, Planes, and Angles Check It Out: Example 2 E. Name two pairs of supplementary angles. AEB, BED mAEB + mBED = 15° + 165° = 180° CED, AEC mCED + mAEC = 75° + 105° = 180° E D C B A 90° 75° 15°

Points, Lines, Planes, and Angles Course 3 7-1 Points, Lines, Planes, and Angles Congruent figures have the same size and shape. Segments that have the same length are congruent. Angles that have the same measure are congruent. The symbol for congruence is , which is read “is congruent to.” Intersecting lines form two pairs of vertical angles. Vertical angles are always congruent, as shown in the next example.

Additional Example 3A: Finding the Measure of Vertical Angles Course 3 7-1 Points, Lines, Planes, and Angles Additional Example 3A: Finding the Measure of Vertical Angles In the figure, 1 and 3 are vertical angles, and 2 and 4 are vertical angles. If m1 = 37°, find m 3. The measures of 1 and 2 are supplementary. m2 = 180° – 37° = 143° The measures of 2 and 3 are supplementary. m3 = 180° – 143° = 37° So m1 = m3 or m1 = m3. ~

Additional Example 3B: Finding the Measure of Vertical Angles Course 3 7-1 Points, Lines, Planes, and Angles Additional Example 3B: Finding the Measure of Vertical Angles In the figure, 1 and 3 are vertical angles, and 2 and 4 are vertical angles. If m4 = y°, find m2. m3 = 180° – y° m2 = 180° – (180° – y°) = 180° – 180° + y° Distributive Property m2 = m4 = y° So m4 = m2 or m4  m2.

Points, Lines, Planes, and Angles Course 3 7-1 Points, Lines, Planes, and Angles Check It Out: Example 3A In the figure, 1 and 3 are vertical angles, and 2 and 4 are vertical angles. 2 3 If m1 = 42°, find m3. 1 4 The measures of 1 and 2 are supplementary. m2 = 180° – 42° = 138° The measures of 2 and 3 are supplementary. m3 = 180° – 138° = 42° So m1 = m3 or m1  m3.

Points, Lines, Planes, and Angles Course 3 7-1 Points, Lines, Planes, and Angles Check It Out: Example 3B In the figure, 1 and 3 are vertical angles, and 2 and 4 are vertical angles. 2 3 If m4 = x°, find m2. 1 4 m3 = 180° – x° m2 = 180° – (180° – x°) = 180° –180° + x° Distributive Property m2 = m4 = x° So m4 = m2 or m4  m2.

Points, Lines, Planes, and Angles Course 3 7-1 Points, Lines, Planes, and Angles Lesson Quiz In the figure, 1 and 3 are vertical angles, and 2 and 4 are vertical angles. 1. Name three points in the figure. Possible answer: A, B, and C 2. Name two lines in the figure. Possible answer: AD and BE 3. Name a right angle in the figure. Possible answer: AGF 4. Name a pair of complementary angles. Possible answer: 1 and 2 5. If m1 = 47°, then find m 3. 47°