91Ó°ÊÓ

Planes \(M\) and \(N\) are known to intersect. a. What kind of figure is the intersection of \(M\) and \(N ?\) b. State the postulate that supports your answer to part (a).

State whether it is possible for the figure described to exist. Write yes or no. Two points both lie in each of two lines.

State whether it is possible for the figure described to exist. Write yes or no. Three points all lie in each of two planes.

Points \(A, B, C,\) and \(D\) are four noncoplanar points. a. State the postulate that guarantees the existence of planes \(A B C, A B D, A C D,\) and \(B C D\) b. Explain how the Ruler Postulate guarantees the existence of a point \(P\) between \(A\) and \(D\) c. State the postulate that guarantees the existence of plane \(B C P\) d. Explain why there -are an infinite number of planes through \(\overline{B C}\).

Without measuring, sketch each angle. Then use a protractor to check your accuracy. \(45^{\circ}\) angle

Parts (a) through (d) justify Theorem 1-2: Through a line and a point not in the line there is exactly one plane. a. If \(P\) is a point not in line \(k,\) what postulate permits us to state that there are two points \(R\) and \(S\) in line \(k ?\) b. Then there is at least one plane \(X\) that contains points \(P, R,\) and \(S .\) Why? c. What postulate guarantees that plane \(X\) contains line \(k ?\) Now we know that there is a plane \(X\) that contains both point \(P\) and line \(k\). d. There can't be another plane that contains point \(P\) and line \(k,\) because then two planes would contain noncollinear points \(P, R,\) and \(S .\) What postulate does this contradict?

Using a ruler, draw a large triangle. Then use a protractor to find the approximate measure of each angle and compute the sum of the three measures. Repeat this exercise for a triangle with a different shape. Did you get the same result?

a. Name two planes that do not intersect. b. Name two other planes that do not intersect.

You can think of the ceiling and floor of a room as parts of horizontal planes. The walls are parts of vertical planes. Vertical planes are represented by figures like those shown in which two sides are vertical. \(\quad\) A horizontal plane is represented by a figure like that shown, with two sides horizontal and no sides vertical. (GRAPH CANT COPY) Can two horizontal planes intersect?

The Ruler Postulate suggests that there are many ways to assign coordinates to a line. The Fahrenheit and Celsius temperature scales on a thermometer indicate two such ways of assigning coordinates. A Fahrenheit temperature of \(32^{\circ}\) corresponds to a Celsius temperature of \(0^{\circ} .\) The formula, or rule, for converting a Fahrenheit temperature \(F\) into a Celsius temperature \(C\) is $$C=\frac{5}{9}(F-32)$$ a. What Celsius temperatures correspond to Fahrenheit temperatures of \(212^{\circ}\) and \(98.6^{\circ} ?\) b. Solve the equation above for \(F\) to obtain a rule for converting Celsius temperatures to Fahrenheit temperatures. c. What Fahrenheit temperatures correspond to Celsius temperatures of \(-40^{\circ}\) and \(2000^{\circ} ?\)