Bell Work

Outcomes I will be able to: 1) Define and Use new vocabulary: midpoint, bisector, segment bisector, construction, Midpoint Formula and angle bisector through CONSTRUCTION! 2) Bisect a segment/angle by measuring, by folding, and by algebraic reasoning. 3) Review skills 1-12 for the chapter 1 test.

Agenda Bell Work Outcomes Agenda Construction Activity Whiteboards! Begin Study Guide

Assessment Review Let’s spend a few minutes looking over the tests. All tests must be collected again and are not to go home. SO… Open your notebooks and take notes of anything that you might want to study further to get a better grade on next time. Collect all tests.

Constructions The compass, like the straight edge, has been a useful geometry tool for thousands of years. The ancient Egyptians used a compass to mark off distances. During the Golden Age of Greece, Greek mathematicians made a game of geometric constructions. In his 13 volume work Elements, Euclid (325-265 BC) established the basic rules for constructions using only a compass and straight edge. He proposed definitions and constructions about points, lines, angles, surfaces and solids. He also showed why the constructions were correct with deductive reasoning. You will learn many of these constructions using the same tools.

School of Athens by Raphael

School of Athens Euclid is represented here teaching by showing a geometric construction to his fellow mathematicians. Notice the globes being held. One is of the earth and one of the heavens. They thought the earth was the center and the heavens a sphere around them.

Constructions Compass and a straight edge only In partners- please use the compasses and straightedge as math learning tools and not “weapons” This game of trying to draw figures with only these two tools dates back to the classical Greeks. Constructions develop deductive reasoning while giving insight into geometry relationships.

Bisecting a Segment Draw a segment Set compass to be more than the midpoint Strike an arc above and below the segment from each endpoint. DON’T change the compass setting. Connect the points where the arcs intersect. You have created a perpendicular bisector of the segment.

Bisecting an Angle Draw an Angle Set compass on the vertex and strike an arc that touches both sides of the angle. Move the compass to the point made by the first arc touching the side. Strike an arc between the sides but beyond the first arc. DON’T change the compass setting. Repeat with compass on other side of angle. Connect the point where the arcs intersect to the vertex of your angle. You have created a ray that is an angle bisector.

Questions to Think About! How can we verify that our segment is REALLY bisected? Describe two things you could do. How can we verify that our angle is REALLY bisected? Describe two things you could do. Brainstorm: what else do you think you could draw with the rules of construction to challenge a fellow mathematician with? Example: Construct a perfect square.

Whiteboard Angle Problems HINT: Consider writing the name of the angle pair to help you motivate the setup. Think about what type of angles you set equal to each other, add to equal to 180 and add to 90

Whiteboards 1 On your whiteboards, construct and label the following (or write out the definition): Linear Pair Complementary Angles Vertical Angles

Whiteboards 2 Complete the chart below: Key words that make you add angles to 180 Key words that make you add angles to 90 degrees Key words that make you set the angles equal to each other.

Whiteboards 3 (Find Q)

Whiteboards 4:

Whiteboard 5

Whiteboard 6: Find X

Whiteboard 7

Whiteboard

Simplifying Radicals When simplifying radicals, sometimes it helps to use a Prime Factoring Tree. Look for factors that are written twice and circle them. When we take the square root, we write only one of the circled numbers. The numbers uncircled, without a pair, remain under the radical, multiplied back together. Square root of 72 2 x 2 x 2 x 3 x 3 = 6√2

Simplifying Radicals To add or subtract them, they must be the same. To Multiply or Divide, work the whole numbers separately from the radical. Multiply under the radical, stays under the radical. No final answer may have a radical in the denominator.

Whiteboard 8

Whiteboard 9

Whiteboard 10

Whiteboard 11

Study Guide Work Time Please work on your study guides so that we can target some questions that you may have before the end of class. You may work quietly with a neighbor.

Exit Quiz Solve for x if Ray QS is an angle bisector: 1) 2) Name the angle relationship and solve for x a) b)