91Ó°ÊÓ

The postulates and theorems in this book represent Euclidean geometry. In spherical geometry, all points are points on the surface of a sphere. A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. In spherical geometry, do you think that two triangles are congruent if their corresponding sides are congruent? Justify your answer.

MATHEMATICAL CONNECTIONS Six statements are given about \(\Delta\) TUV and \(\Delta \mathrm{XYZ}.\) $$\mathrm{TU} \cong \overline{\mathrm{XY}} \qquad \overline{\mathrm{UV}} \cong \overline{\mathrm{YZ}} \qquad \overline{\mathrm{TV}} \cong \overline{\mathrm{XZ}}\\\ \angle \mathrm{T} \cong<\mathrm{X} \quad \angle \mathrm{U} \cong \angle \mathrm{Y}\angle \mathrm{V} \cong \angle \mathrm{Z} $$ a. List all combinations of three given statements that would provide enough information to prove that \(\Delta\) TUV is congruent to \(\Delta \mathrm{XYZ}\) . b. You choose three statements at random. What is the probability that the statements you choose provide enough information to prove that the triangles are congruent?

A boat is traveling parallel to the shore along \(\mathrm{RT}\). When the boat is at point \(\mathrm{R}\) , the captain measures the angle to the lighthouse as \(35^{\circ}\). After the boat has traveled 2.1 miles, the captain measures the angle to the lighthouse to be \(70^{\circ}\). a. Find \(\mathrm{SL}\) Explain your reasoning. b. Explain how to find the distance between the boat and the shoreline.

The postulates and theorems in this book represent Euclidean geometry. In spherical geometry, all points are points on the surface of a sphere. A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. In spherical geometry, do all equiangular triangles have the same angle measures? Justify your answer.

Find the coordinates of the midpoint of the line segment with the given endpoints. $$R(-5,-7) \text { and } S(2,-4)$$

Prove that the Corollary to the Base Angles Theorem (Corollary 5.2\()\) follows from the Base Angles Theorem (Theorem 5.6 ).

MODELING WITH MATHEMATICS You are bending a strip of metal into an isosceles triangle for a sculpture. The strip of metal is 20 inches long. The \(|\) first bend is made 6 inches from one end. Describe two ways you could complete the triangle.

Prove that the Corollary to the Converse of the Base Angles Theorem (Corollary 5.3 ) follows from the Converse of the Base Angles Theorem (Theorem 5.7\()\) .

PROVING A COROLLARY Prove the Corollary to the Triangle Sum Theorem (Corollary 5.1\()\) . Given \(\triangle A B C\) is a right triangle. Prove \(\angle\) A and \(\angle B\) are complementary.

PROVING A THEOREM Prove the Exterior Angle Theorem (Theorem 5.2\()\) . Given \(\triangle A B C\) , exterior \(\angle B C D\) Prove \(m \angle A+m \angle B=m \angle B C D\)