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Math 362 - Section 001 Winter 2006 Test 2 -Key

Closed Book / Closed Note. Write your answers on the test itself. Take the test in one sitting. It should take you no more than two hours.

Part I: Circle T if the statement is true, and F if the statement is false. (2 points each)

1. In neutral geometry it is possible that there are two lines l and m and F two points P and Q not on the lines so that there is only one line parallel to l through P and more than one line parallel to m through Q. 2. The Alternate Interior Angles Theorem is equivalent to the Euclidean F Parallel Postulate 3. The defect of a convex parallelogram GABCD is always the sum of the T defects of two triangles ÎABD and ÎCBD 4. The scalene inequality is used to compare relative sizes of angles or T sides in a single triangle. 5. The hinge theorem does not apply to right triangles. F

Comments:

1. No, it’s all or nothing – either every line and point have exactly one parallel, or every line and point have more than one.

2. No, the converse of the Alternate Interior Angles Theorem is equivalent to the Euclidean Parallel Postulate.

5. Sure it does – it applies to every pair of triangles that satisfy the criteria of the theorem. Section II: Matching (2 points each)

Match the Postulate or Theorem on the left with the description on the right by writing the letter of the correct description in the blank after each statement.

6. Alternate Interior A. Can be used to compare the length of one Angles Theorem D side of a triangle with the sum of lengths of the other two sides. 7. Euclid’s postulate V B. States that if two lines are cut be a transversal in such a way that the sum of B measures of two interior angles on the same side of the transversal is less that 180, the lines are not parallel. 8. Saccheri-Legendre E C. Can be used to compare the sizes of Theorem corresponding sides in two triangles. 9. Hinge theorem D. States that when two lines are cut by a transversal in such a way that alternate C interior angles are congruent, the lines are parallel. 10. Triangle Inequality E. States that the angle sum for any triangle is A less than or equal to 180. Section III: Short answer. (4 points each)

1. Given , R is interior to pPQS, and W is interior to pTVK.

a. List all non-identical angles that are congruent by definition.

The right angles pPQR and pTVW

b. List the additional angles that may be proven to be congruent.

pKVW &pSQR, as well as pSQP & pKVT

2. In the figure below, the angles at A and C are right angles. Find the missing values x and y in the figure, and state which triangle congruence criterion you used, including the two triangles. Write the triangle names so that the associated correspondence is clear.

x = 3, y = 2.2. ASA on . LA or SsA would also work.

NOTE: SAS does not work because you don’t have corresponding sides congruent in the two triangles. 3. Which of the following triangles cannot exist in absolute geometry (there may be more than one correct answer).

Numbers 1 cannot exist by Saccheri-Legendre; Number 3 cannot exist by exterior angle inequality

4. Use the exterior angle inequality to find an upper and lower bound for x if

40 < x < 100

(x > 40 by exterior angle inequality; also, 180 - x > 80 so 100 > x by exterior angle inequality. 5. Which of the angles A, B, and C shown below has the greatest measure, and why? (A sentence or two of explanation is all that is needed.)

B is bigger than A by hinge; in the third figure, the hypotenuse of the bottom triangle with legs of length 1 and 3 would be greater than 3, so C is bigger than B by hinge theorem as well.

6. Given that pACB is not obtuse, name the smallest angle in the figure.

In the angle at B is opposite the smallest side so is smaller than the angles at A or C. So we just need to compare it with the two angles at C. In B is still opposite the smallest side, so is the smallest angle. Since is isosceles, pDCA is congruent to pCAD so B is smaller than both of them. B is the smallest. 7. In the quadrilateral GRSTW shown in below, all four angles (at R, S, Tand W) are right angles. RW = ST = 3, WT = 4, and, and x = RS.

a. Find x. x = 4

b. What congruence criteria on which triangles justifies your answer in part a?

WS = RT by SAS on . Then by HL or SsA.

8. Point B varies on ray with A*C*D, :(pEAD) = 90, and :(pECD) = 135. What can be said about the range of values possible for y = :(pBEC) if

a. A*B*C (see first figure) 0 < y < 45

b. A*C*B (see second figure) 0 < y < 45

Note: write your answers in the form N < y < M for appropriate numbers N and M. 9. Draw two non congruent triangles in which two corresponding sides have the same length and one corresponding angle has the same measure. (Remember: We talked about this “ambiguous case” in class and in the notes.)

Any pair of triangle satisfying this must be something like is drawn below, with AC = XZ, CB = ZY, the angles at A and X congruent, and the angles at B and Y supplementary.

10. List two of the four equivalent statements that we have taken as the definition that a quadrilateral GABCD is convex.

Any two of the following four:

1. Each side of the quadrilateral is on a halfplane of the opposite side of the quadrilateral.

2. The vertex of each angle of the quadrilateral is in the interior of its opposite angle.

3. The diagonals intersect each other (at interior points).

4. The diagonals lie between opposite vertices (i.e. opposite vertices are on opposite sides of the diagonal).