Ms. Nassif Mathematics. Guide. 2- Correction key. Example of an appropriate method. Area of the base. Volume of water

Transcription

1 orrection key Example of an appropriate method rea of the base Volume of water Mathematics Guide 2 ( 19 16)( 19 12)( 19 10) = m 19 = rea of the base height = m 3 ost of treating the water with chlorine nswer: Note: = $ It costs $11.98 to treat the water in this pool with chlorine. Students who use an appropriate method in order to determine the area of the base of the pool have shown that they have a partial understanding of the problem.

2 3 4 5 Ms. Example of an appropriate solution Statements Reasons 1. m EFG = Two parallel lines intercepted by a transversal determine two pairs of alternating interior congruent angles. 2. m FEG = In an isosceles triangle, the angles opposite the congruent sides are congruent. 3. m FGE = The sum of the interior angles of a triangle is m EG = Two adjacent angles whose exterior sides form a straight line are supplementary. nswer: The measure of angle EG is 100. Nassif

3 6 Example of an appropriate solution ngle E measures 108 because the interior angles of a regular pentagon are congruent. ngle HG measures 54 because F is an angle bisector. ngle HG measures 36 because vertically opposite angles are congruent and it is congruent to angle HF. Since the sum of the interior angles of a triangle is 180, m GH = 180 ( ). Hence, m GH = 90 G E 108 H 36 F

4 7 8 9 " " Statements " " ' ' ' O Justifications 1 2 E 3 4 '

5 10 11 Example of an appropriate method of solution 1. Since //, then alternate interior angles and are congruent. Hence, m = m = Since m = m, then triangle is an isosceles triangle. Hence, m = m. 3. Since the measures of the interior angles of triangle is 180 and m = 38, then m + m = = Hence, m = = 71. 2

6 Ms. Example of an appropriate method Since m = m, triangle is an isosceles triangle, where m = m. The sum of the measures of the interior angles of a triangle is 180 and angle measures Hence, m = = 61 2 Since // Hence, m GHE = 61. EF, corresponding angles and GHE are congruent. Triangles E and E are isometric because two triangles whose corresponding sides are congruent must be isometric. Nassif

7 17 Work : (example) Statements Justifications 1. m = m 1. The opposite sides of a parallelogram are congruent. 2. m = m 2. The opposite sides of a parallelogram are congruent. 3. m = m 3. The opposite angles of a parallelogram are congruent Two triangles are congruent if they have one congruent angle bounded by two congruent sides. or S--S

8 18 Example of an appropriate solution Perimeter : 6.3 m or 630 cm Measure of each side : Heron s Formula : 630 = 105 cm 6 S(area of a triangle) = p( p a)( p b)( p c) P = 2 1 (a + b + c) P = 2 1 (105 3) P = 315 = cm 2 S(area of a triangle) = ( ) ( ) ( ) rea of the hexagon = (52.5) (52.5) (52.5) = cm = cm 2 or m 2 Final answer The area of the table is m 2.

9 1 In the adjacent diagram, and intersect at O. and Mathematics Question ooklet ate : Which of the following statements could be used to prove that triangle O is congruent to triangle O? ) Two triangles whose corresponding sides are congruent must be congruent. (SSS) ) Two triangles whose corresponding angles are congruent must be congruent. () ) If two sides and the contained angle of one triangle are congruent to two sides and the contained angle of another triangle, then the triangles must be congruent. (SS) ) If two angles and the contained side of one triangle are congruent to two angles and the contained side of another triangle, then the triangles must be congruent. (S) O

10 2 swimming pool is in the shape of a right prism with a triangular base. The edges of the base measure 12 m, 10 m and 16 m respectively. The water in the pool is 2 m deep. 12 m 2 m 10 m 16 m Treating the water with chlorine costs $0.10 per cubic metre of water. How much does it cost to treat the water in this pool with chlorine? Show all your work.

11 3 road map shows four linear routes. Route 1 is parallel to route E Route 1 45 Route 4 Route 2 Route 3 The following is part of a procedure used to determine the measure of angle E. Step 1 m = 54 because the sum of the measures of the interior angles of triangle is equal to 180. Step 2 m E = m = 54 because Which one of the following statements correctly completes step 2 of this procedure? ) ngles E and are vertically opposite angles, and are therefore congruent. ) ngles E and are corresponding angles formed when a transversal intersects two parallel lines, and are therefore congruent.

12 4 ) ngles E and are alternate interior angles formed when a transversal intersects two parallel lines, and are therefore congruent. ) ngles E and are alternate exterior angles formed when a transversal intersects two parallel lines. In the figure on the right, and are parallel lines. In triangle EGF, EG is congruent to FG. The measure of angle EF is 50º. What is the measure of angle EG? Justify each step of your work. Show all your work. F G 50 E?

13 6 onsider the regular pentagon on the right. F and G are angle bisectors that intersect at H. lso, m = 108 and m HF = 36. Prove that angle GH measures 90. Explain each step of your reasoning. Show all your work. G E 108 H 36 F

14 8 In the figure below, is a parallelogram and FE is an isosceles triangle. Match each numbered statement used in determining the measure of angle FE with the appropriate lettered justification. Statements 1. m = m G = m F = m = m EF = 180 m F = m FE = (180 m EF) 2 m FE = 30 G 60 E Justifications ) djacent angles whose external sides are in a straight line are supplementary. ) In an isosceles triangle, the angles opposite the congruent sides are congruent. ) Vertically opposite angles are congruent. ) If a transversal intersects two parallel lines, the alternate interior (exterior) angles are congruent. E) If a transversal intersects two parallel lines, the F

15 corresponding angles are congruent.

16 10 In the figure on the right, // is a transversal m = m m = 38 Show that the measure of angle is 71. Justify each step of your work. 38

17 11 In triangles Y and X shown below, X Y and. X Which one of the following statements could be used to prove that triangle Y is congruent to triangle X? ) If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.(sss) ) If two sides and a contained angle of one triangle are congruent to two sides and the contained angle of another triangle, then the triangles are congruent.(ss) ) If two angles and a contained side of one triangle are congruent to two angles and the contained side of another triangle, then the triangles are congruent.(s) ) If two angles of one triangle are congruent to two angles of another triangle, then the triangles are congruent.() Z Y

18 12 13 In the figure on the right, // EF m = m m = 58 Show that angle GHE measures 61. Explain each step in your work. seamstress makes a quilt by assembling triangular pieces to form square illustrated on the right. In this square, triangles E and E are isosceles triangles. Why are triangles E and E isometric? E G H F

19 15 In the diagram below, E is the mid-point of segments and. Which statement can be used to prove that triangle E is congruent to triangle E? ) Two triangles having three corresponding sides congruent, must be congruent. ) Two triangles having two corresponding angles congruent, must be congruent. ) Two triangles having two corresponding sides and the contained angle congruent, must be congruent. ) Two triangles having two corresponding angles and the contained side congruent, must be congruent. E

20 16 In the figure below, M is the midpoint of segments and. Which of the following can be used to justify that triangles M and M are congruent? ) Two triangles are congruent if they have one congruent angle bounded by two corresponding congruent sides. (SS) ) Two triangles are congruent if they have three corresponding congruent sides. (SSS) ) Two triangles are congruent if they have one congruent side between two corresponding congruent angles. (S) ) Two triangles are congruent if they have two corresponding congruent angles. () M

21 17 18 Given parallelogram to the right, prove that triangle and are congruent. Justify your statements. table in a conference room, in the shape of a regular hexagon, has a perimeter of 6.3 m. alculate the area of this table. Show all the work needed to solve the problem.