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A C E Answers | Investigation 2 Applications 1. Since these figures appear to be 8. Certainly congruent because the parallelograms, there are two possible correspondences of vertices that will pair congruent sides and angles: A → L, B → K, C → N, and D → M or A → N, B → M, C → L, and D → K. Pythagorean Theorem guarantees that the third sides are the same length. 9. There is not enough information to determine congruence. 2. The correspondence of vertices that seems right here is: E → R, F → P, and G → Q. 3. The correspondence of vertices that seems right here is: S → X, T → Z, and U → Y. 10. The two figures do not appear congruent, and the information marked on the drawing does not provide enough justification for invoking any of the standard congruence criteria. 4. The correspondence of vertices that seems right here is: A → R, B → S, C → T, D → P, and E → Q. 11. The two figures may or may not be 5. There are many pairs of congruent triangles in the given figure: ∆JNM ∆LNK, ∆JNM ∆KNL, ∆JNK ∆MNL, ∆JNK ∆LNM, and ∆JLM ∆KML ∆LJK ∆MKJ. Note: We’ve listed what are essentially the same triangles several times with vertices in different order since there is symmetry in the design that allows more than one correspondence of vertices consistent with congruence. 12. The two figures are not congruent. Three congruent. The corresponding sides share an angle of different measures. corresponding angles do not guarantee congruence. 13. In this figure it appears that ∆ABC ∆EBD, a congruence that could be established by a line reflection in the bisector of ABE or by a combination of a counterclockwise rotation of 90° centered at B and reflection in the line that contains points B and D. 14. In this figure it appears that ∆PQR 6. There are several different combinations of congruent triangles that can be paired up in response to this question. We give several responses to indicate the types of answers that could be given. There are essentially only two transformations needed—line reflections and half-turn rotations about N. ∆UTS, a congruence that could be established by translation that carries Q to T (also P to U and R to S). 15. Measurement evidence should suggest that the given triangles are not congruent. 16. In this figure it appears that ∆ABC ∆EDF, a congruence that could be established most easily by a line reflection in the perpendicular bisector of BD . ∆JNM ∆LNK, ∆KML ∆MKJ, can be shown by half-turn rotation centered at N. That point must be the midpoint of each diagonal due to the symmetry of the rectangle. 17. In this figure it appears that ∆JKL ∆TSR, a congruence that could be established in several sequences of transformations. One such sequence would slide L to R, rotate that image of ∆JKL counterclockwise by 90° centered at R, and reflection in the line that contains RT . ∆JNM ∆KNL, ∆JLM ∆KML, can be shown by reflection in a vertical line through point N, again due to the symmetry of the rectangle. 7. Certainly congruent by the Side-Side-Side criterion. Butterflies, Pinwheels, and Wallpaper 1 Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved. Investigation 2 A C E Answers | Investigation 2 18. In this figure it appears that ∆JKL ∆MNO, a congruence that could be established by a line reflection in JL and translation to the right. Of course, other sequences of transformations are possible. Note: For Exercises 19–22, the instructions do not imply that the parts of triangle ABC necessarily correspond to parts of triangle DEF by the order of the vertex labels; in other words, triangle ABC could be congruent to triangle DFE. 19. It is possible, but not certain, that the two triangles are congruent. One could be larger than the other. Three corresponding angles do not guarantee congruence. 20. It is not possible for the two triangles to be congruent. If they were, then DE = 3 cm and ∆DEF would have to be an isosceles triangle with congruent base angles. ∆ABC cannot be isosceles, because B has measure of 30°. 21. The two triangles must be congruent because the Pythagorean Theorem implies that the third sides of each must be 15.62 – 8.22 ≈ 13.3 cm. 22. These two triangles are congruent by the Side-Angle-Side criterion. 23. Without any information about side lengths, the directions are unlikely to produce a congruent copy of the given figure. 24. Although the given information is not in one of the standard congruence criterion forms, the recipient could deduce that the measure of A has to be 35° and thus he/she will have Angle-Side-Angle data to construct a triangle that must be congruent to Figure 2. 25. Definitely congruent by the Angle-SideAngle criterion. 26. Again, this information satisfies the SideAngle-Side criterion for congruence. Connections 27. a. The most natural basic design element would be a 120° sector bounded by two ‘radii’ of the hexagon, though there are several ways to choose the sector boundaries. 30. a. Rectangle perimeter is 30 cm and area is 36 cm2; parallelogram perimeter is 34 cm and area is 36 cm 2. b. Congruence, area, and perimeter. i. No, you could have a square with sides 7.5 cm making perimeter of 30 cm and that figure would not be congruent to the given rectangle. b. Rotations of 120° and 240° will do the job. 28. a. The design will appear unmoved after slides left or right, up or down, or along slanted lines, all assuming the appropriate distance of slide is chosen thoughtfully. ii. No, the rectangle and parallelogram given have the same area but are not congruent. b. One needs a basic design element including at least two adjacent fish—one pointing in each direction. c. If flips are allowed, one fish can be replicated to fill in each spot of the wallpaper design. 29. The circles are congruent because one can imagine sliding or reflecting one center onto the other. Since the circles have the same radius and diameter, they are identical size (and, of course, shape). Butterflies, Pinwheels, and Wallpaper iii. No, the rectangle shown and the square with sides 7.5 cm have the same perimeter, but the rectangle area is 36 cm2 and the square area would be 56.25 cm2. iv. No, the given rectangle and parallelogram have the same area but not the same perimeter. 2 Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved. Investigation 2 A C E Answers | Investigation 2 31. a. No are congruent by the Angle-Side-Angle criterion (angles at A and D are right angles, sides AC = DC, and BCA ECD because they are vertical or opposite angles formed by intersecting lines). b. This is sufficient information because opposite sides and opposite angles of a parallelogram are congruent and consecutive angles are supplementary. b. DE = AB. 33. a. This congruence is true for a variety of c. There are several different combinations of sufficient information to determine the shape and size of a quadrilateral. Side-Angle-Side-Angle-Side is one; Angle-Side-Angle-Side-Angle is another. Less information won’t work. reasons. For instance, the angles at O are both right angles by construction of the coordinate grid, OB = OC because both are length 3, and AO = AO. b. AC AB , OC OB , OAC OAB , and ACO ABO . 34. a. PS and QR are hypotenuses of right triangles that have legs 2 and 3 units or one could reason that reflection in the y-axis would ‘move’ point R to S and point Q to P. b. The angles are congruent because they are images under the reflection in the y-axis. 35. a. To build a physical triangle you have to make sides of definite lengths. Those lengths determine exactly one triangular shape, so unless the anchors at each vertex break, the figure cannot be distorted into any other shape. b. Bracing a quadrilateral by connecting 32. a. It makes sense to locate point D on AC so that AC = DC. Then locate point E on a line that is perpendicular to CD so that points B, C, and E are collinear. This will mean that the two triangles opposite vertices with a diagonal essentially creates two triangles that are rigid figures (joined together by the common side). Extensions 36. Two sides and an angle of one triangle congruent to two sides and an angle of another triangle guarantee congruence only if the angle is include between the sides. 37. Yes; proof of this fact is beyond the level of this course, but some students might get hooked trying to find a counterexample! This question is equivalent to asking, “If you know the sum and the product of two numbers, do you know the individual numbers?” Butterflies, Pinwheels, and Wallpaper 38. a. All three angles are the same size. b. All three sides are the same length. 39. a. The angle measures of the pentagon must all be 108° because each triangle has one 72° and two others that add to 108°. b. The side lengths must all be the same. c. The segments from C to the vertices must all be the same length. 3 Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved. Investigation 2 A C E Answers | Investigation 2 40. In addition to the sides and angles given to be congruent, AD is a side of both triangles, and it sets up a Side-Angle-Side congruence condition for ∆ABD ∆DEA. This means that angle B angle E because these are corresponding angles in the two triangles just shown to be congruent. And we know that angle BCA angle ECD because these are vertical angles. Since we have two pairs of congruent corresponding angles and one pair of congruent corresponding sides in ∆ABC and ∆DEC, ∆ABC ∆DEC (using the Angle-AngleSide condition). Butterflies, Pinwheels, and Wallpaper 4 Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved. Investigation 2