Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Pythagorean theorem wikipedia , lookup
Technical drawing wikipedia , lookup
Rotation formalisms in three dimensions wikipedia , lookup
Perspective (graphical) wikipedia , lookup
Integer triangle wikipedia , lookup
History of trigonometry wikipedia , lookup
Rational trigonometry wikipedia , lookup
Multilateration wikipedia , lookup
Line (geometry) wikipedia , lookup
Trigonometric functions wikipedia , lookup
November 11, 2015 1.1 Angle Pair Relations November 11, 2015 straight angle an angle that is also a straight line, has a measure of 180o linear pair a pair of angles that make a line; 2 angles that are adjacent and supplementary adjacent angles 2 angles that share a common side or ray have a common vertex, and no interior points in common supplementary angles 2 angles whose measures have a sum of 180o 40o 140o *Note: All linear pairs are supplementary angles, but not all supplementary angles are linear pairs!! November 11, 2015 vertical angles 2 nonadjacent (opposite) angles formed by two intersecting lines interior angles angles formed on the inside of two lines cut by a transversal exterior angles angles formed on the outside of two lines cut by a transversal EXTERIOR INTERIOR EXTERIOR November 11, 2015 parallel two or more lines on a flat surface that do not intersect (no matter how far they extend) >> >> Arrowheads at the end of lines indicate that they extend indefinitely. Marks on pairs of lines or segments like > and >> indicate that the lines are parallel. perpendicular two lines or segments that meet (intersect) to form a 90o angle The small box at the point of intersection of two lines or segments indicates that they are perpendicular (form right angles) transversal a line that intersects two or more lines r and w are transversals November 11, 2015 corresponding angles a pair of angles that are in the same "corresponding" location relative to the parallel line and transversal . . . angles on the same side to their respective parallel line and on the same side of the transversal conjecture - (an inference based on incomplete evidence) If two parallel lines are cut by a transversal, then the corresponding angles are . . . November 11, 2015 alternate interior angles a pair of angles that are on the inside of the parallel lines and opposite sides of the transversal . . . conjecture - (an inference based on incomplete evidence) If two parallel lines are cut by a transversal, then the alternate interior angles are . . . November 11, 2015 consecutive interior angles a pair of angles that are on the inside of the parallel lines and same side of the transversal . . . aka. same side interior angles conjecture - (an inference based on incomplete evidence) If two parallel lines are cut by a transversal, then the consecutive interior angles are . . . November 11, 2015 Classify each of the following pairs of angles as corresponding, alternate interior, same side interior, straight, or "none" of these. What conditions are necessary to be able to say that the pairs of corresponding angles or alternate interior angles are congruent? If m<2 = 670, what is m<5? If m<4 = 6x0 and ,<6 = 9x0, find the m<4. Explain your steps. November 11, 2015 Use all that you now know, lets make some more comparisons. Compare the measures of angles e and x. x Compare the measures of angles a and e. Compare m<f and m<y. Compare m<f and m<k. y Compare m<n and m<z. z Compare m<n and m<t. November 11, 2015 alternate exterior angles a pair of angles that are on the outside of the parallel lines and opposite sides of the transversal . . . conjecture If two parallel lines are cut by a transversal, then the alternate exterior angles are . . . consecutive exterior angles a pair of angles that are on the outside of the parallel lines and same side of the transversal . . . aka. same side exterior angles conjecture If two parallel lines are cut by a transversal, then the consecutive exterior angles are . . . November 11, 2015 Use your conjectures about parallel lines and the angles formed by the transversal to find the measures of the labeled angles. Show the step by step procedure you use and name each angles conjecture you use. November 11, 2015 Learning Log a) Label each diagram below with the name of the angle pair that best describes it. b)Describe the angle pair using your geometry vocabulary. c) State the conjecture for the angle pair. linear pair - supplementary a pair of angles that make a line; 2 angles that are adjacent (side by side - share a side) and supplementary (sum of 1800) congruent vertical angles 2 nonadjacent (opposite) angles formed by two intersecting lines corresponding angles - congruent a pair of angles that are in the same "corresponding" location relative to the parallel line and transversal . . . angles on the same side to their respective parallel line and on the same side of the transversal alternate interior angles - congruent a pair of angles that are on the inside of the parallel lines and opposite sides of the transversal . . . alternate exterior angles - congruent a pair of angles that are on the outside of the parallel lines and opposite sides of the transversal . . . consecutive interior angles - supplementary a pair of angles that are on the inside of the parallel lines and same side of the transversal . . . aka. same side interior angles consecutive exterior angles - supplementary a pair of angles that are on the outside of the parallel lines and same side of the transversal . . . aka. same side exterior angles November 11, 2015 1.1 Angle Pair Relations November 11, 2015 straight angle an angle that is also a straight line, has a measure of 180o linear pair a pair of angles that make a line; 2 angles that are adjacent and supplementary adjacent angles 2 angles that share a common side or ray have a common vertex, and no interior points in common supplementary angles 2 angles whose measures have a sum of 180o 40o 140o *Note: All linear pairs are supplementary angles, but not all supplementary angles are linear pairs!! November 11, 2015 vertical angles 2 nonadjacent (opposite) angles formed by two intersecting lines interior angles angles formed on the inside of two lines cut by a transversal exterior angles angles formed on the outside of two lines cut by a transversal EXTERIOR INTERIOR EXTERIOR November 11, 2015 parallel two or more lines on a flat surface that do not intersect (no matter how far they extend) >> >> Arrowheads at the end of lines indicate that they extend indefinitely. Marks on pairs of lines or segments like > and >> indicate that the lines are parallel. perpendicular two lines or segments that meet (intersect) to form a 90o angle The small box at the point of intersection of two lines or segments indicates that they are perpendicular (form right angles) transversal a line that intersects two or more lines r and w are transversals November 11, 2015 a)Take a sheet of patty paper and place it over angle a. Using a ruler as a guide, make a precise copy of the angle and label it. Slide the copy of <a to <c and compare the sizes of the two angles. What do you observe? b)Trace a copy of <b. Slide it to <d and compare. What do you observe? c)Trace a copy of <e. Slide it to <g and compare. What do you observe? d)Trace a copy of <f. Slide it to <h and compare. What do you observe? e) Both figures show two lines that are intersected by a transversal. Summarize your findings by describing the relationship between pairs of angles created by lines cut by a transversal. When the angles are congruent, what must be true about the two intersected lines? If the lines are parallel, then what must be true about the angles? November 11, 2015 Use what you know about straight angles and vertical angles to find the measures of missing angles b and d, f and k, and r and s. Label each angle with its measurement. Be prepared to explain your thinking. Compare the measures of angles a and d, and then compare the measures of angles b and e. Compare the measures of angles f and j, and then compare the measures of angles g and k. Compare the measures of angles t and r, and then compare the measures of angles n and s. November 11, 2015 corresponding angles a pair of angles that are in the same "corresponding" location relative to the parallel line and transversal . . . angles on the same side to their respective parallel line and on the same side of the transversal conjecture - (an inference based on incomplete evidence) If two parallel lines are cut by a transversal, then the corresponding angles are . . . November 11, 2015 Use what you know about straight angles, vertical angles, and what you learned from the previous example to find the measures of angles c, h and w. Compare the measures of angles c and d. Compare m<h and m<j. Compare m<w and m<s. November 11, 2015 alternate interior angles a pair of angles that are on the inside of the parallel lines and opposite sides of the transversal . . . conjecture - (an inference based on incomplete evidence) If two parallel lines are cut by a transversal, then the alternate interior angles are . . . November 11, 2015 Use all that you now know, lets make some more comparisons. Compare the measures of angles b and d. Compare m<g and m<j. Compare m<r and m<s. November 11, 2015 consecutive interior angles a pair of angles that are on the inside of the parallel lines and same side of the transversal . . . aka. same side interior angles conjecture - (an inference based on incomplete evidence) If two parallel lines are cut by a transversal, then the consecutive interior angles are . . . November 11, 2015 Classify each of the following pairs of angles as corresponding, alternate interior, same side interior, straight, or "none" of these. What conditions are necessary to be able to say that the pairs of corresponding angles or alternate interior angles are congruent? If m<2 = 670, what is m<5? If m<4 = 6x0 and ,<6 = 9x0, find the m<4. Explain your steps. November 11, 2015 Use all that you now know, lets make some more comparisons. Compare the measures of angles e and x. x Compare the measures of angles a and e. Compare m<f and m<y. Compare m<f and m<k. y Compare m<n and m<z. z Compare m<n and m<t. November 11, 2015 alternate exterior angles a pair of angles that are on the outside of the parallel lines and opposite sides of the transversal . . . conjecture If two parallel lines are cut by a transversal, then the alternate exterior angles are . . . consecutive exterior angles a pair of angles that are on the outside of the parallel lines and same side of the transversal . . . aka. same side exterior angles conjecture If two parallel lines are cut by a transversal, then the consecutive exterior angles are . . . November 11, 2015 Use your conjectures about parallel lines and the angles formed by the transversal to find the measures of the labeled angles. Show the step by step procedure you use and name each angles conjecture you use. November 11, 2015 Learning Log a) Label each diagram below with the name of the angle pair that best describes it. b)Describe the angle pair using your geometry vocabulary. c) State the conjecture for the angle pair. November 11, 2015 Learning Log a) Label each diagram below with the name of the angle pair that best describes it. b)Describe the angle pair using your geometry vocabulary. c) State the conjecture for the angle pair. linear pair - supplementary a pair of angles that make a line; 2 angles that are adjacent (side by side - share a side) and supplementary (sum of 1800) congruent vertical angles 2 nonadjacent (opposite) angles formed by two intersecting lines corresponding angles - congruent a pair of angles that are in the same "corresponding" location relative to the parallel line and transversal . . . angles on the same side to their respective parallel line and on the same side of the transversal alternate interior angles - congruent a pair of angles that are on the inside of the parallel lines and opposite sides of the transversal . . . alternate exterior angles - congruent a pair of angles that are on the outside of the parallel lines and opposite sides of the transversal . . . consecutive interior angles - supplementary a pair of angles that are on the inside of the parallel lines and same side of the transversal . . . aka. same side interior angles consecutive exterior angles - supplementary a pair of angles that are on the outside of the parallel lines and same side of the transversal . . . aka. same side exterior angles November 11, 2015 Find the measure of each angle below. Justify your answer by naming the angle pair conjecture used. 86o V V a c 128o i j f d b e g h k l November 11, 2015 b d c 59o f g e i k l p q h j 43o m r n s V a V Find the measure of each angle below. Justify your answer by naming the angle pair conjecture used. November 11, 2015