Mathematics 10-3: Module 51Teacher S Guide

Mathematics 10-3: Module 51Teacher’s Guide

Unit 3: Geometry

Students were encouraged to add to a formula sheet in Module 3. You may want to provide this Formula Sheet to students as a handy reference as they proceed through the course or for review purposes as they conclude the course. As in Module 3, you may decide to allow students to adapt this formula sheet.

Overview

The theme of this unit is sports, games, art, and design, and there is a look at how each topic can be described by applying the principles of geometry. The geometry of this unit is presented in three modules. In Module 5, students explore the geometry of angles. As part of their exploration, they will review how angles are measured, estimated, classified, and related. One of the angle relationships involves angles formed by the intersection of parallel lines.

In Module 6 students examine polygons with an emphasis on the triangle and the relationships among the sides and angles of similar geometric figures. Proportional reasoning is used to solve a variety of practical problems involving similar right triangles. At the conclusion of this module is the proof and applications of the Pythagorean theorem.

Module 7 is the study of trigonometry—sine, cosine, and tangent ratios. Throughout this module, students solve a wide spectrum of problem situations, some of which will relate to the unit theme of art, design, games, and sport.

The unit theme is further reflected in the Unit 3 Project. This project focuses on how sports, games, art, and design can be understood through the principles of geometry.

Module 5: Angles

The solutions for the student checked questions in Module 5 are collected in the Suggested Answers PDFfor you to use as needed.

Lesson 1: Sketching and Measuring Angles

Some students may have a plastic protractor in which the scales are reversed from the protractor depicted in the lesson. For these students, the scale that increases in a counterclockwise direction is on the inside and the scale that increases in the clockwise direction is on the outside. These students may need your guidance in using their protractors.

Try This

You may find that Carnival Game (Canon Angle) helps students estimate angles. This game can be found on LearnAlberta.ca with this URL:

TT 1.

a.63°

b.114°

c.82°

d.161°

TT 2.

Diagrams should have the indicated sizes. When checking student diagrams, watch that the appropriate scale is used. A common error is using the wrong scale.

a.49°

b.149°

c.16°

d.127°

Answers will vary. Sample answers are given.

TT 3.Each angle is of one full rotation, or 360°.

That’s because

TT 4.Each angle

TT 5.Check 45°, 135°, 225°, and 315°on the final result.

TT 6.each angle

TT 7. Check for 22.5°, 67.5°, 112.5°, 157.5°, 202.5°, 247.5°, 292.5°, and 337.5°.

The student may need an explanation as to where the smallest angles are formed, their angular measure, and how they relate to referents already found.

The following illustration of the unfolded paper may help clarify the angular measures.

You could ask students if they recall other common angles. How are 30° and 60° related to the 360° in a circle? Sketch where these angles should be if you had a circle and a coordinating axis like this.

Lesson 1 Assignment (20 marks)

1.
This is a reflex angle. (2 marks)

2.Petra is correct. Since 231°+ 129°= 360° or one full rotation, a 231° angle drawn counterclockwise appears the same as a 129° angle drawn clockwise. The arms of the two angles would coincide.(3 marks)

3.
Place a square corner of a piece of paper at the vertex as shown. Then, place the 30° angle of the triangle on the left of the square. The angle is a little more than this angle. So, the angle is about 122°.(4 marks)

4.From observing collisions, the largest possible angle the ball can be deflected at is 90° (to the left or to the right). If the angle of deflection were greater than 90°, the second ball would be moving partly in a direction opposite to the direction of the cue ball. That is not possible, since the balls have the same mass.(2 marks)

Place the square corner of a sheet of paper next to the vertex as shown. The angle to the right of the square appears to be a little smaller than the referent angle of 22.5°. An estimate of the reflex angle is 270°+ 20°= 290°. (4 marks)

6.The angle measures 242°. (2 marks)

7.a.a straight angle (1 mark)

b.a reflex angle(1 mark)

c.an acute angle(1 mark)

Lesson 2: Constructing Congruent Angles

TT 1. The answer can be found in the Lesson 2 Assignment (question 3).

Note that answers may include angles found in the images of the snake‘s skin, the butterflies, and the snowflakes. In these images, congruent angles can be found readily through image symmetry.

TT 2. to TT 5. Answers can be found in the Lesson 2 Assignment (question 4).

TT 6.

Step 1: Fold the sheet of paper in half from top to bottom with the angle showing.

Step 2: Fold along AB and BC.

Step 3: Unfold the sheet and draw

TT 7. Answers will vary.

They should be equal in measure within a degree or two.

Note: Allow some latitude in the measures.

Go to the Share rubric in the Rubric Appendix for student assessment.

Lesson 2 Assignment (20 marks)

Show all your work when appropriate.

Step 1: Draw This will be the lower arm of the new angle. Notice that it does not have to point in the same direction as the lower arm of

Step 2: Use your compass to draw circles with the same radius centred at A and at B.

The first circle cuts through the arms of at Pand Q. The second circle cuts across
at X.

Step 3:With centre Q, draw a circle through P. With centre X and the same radius, draw a similar circle cutting the circle you drew in Step 2 at Y.

Step 4: Draw . Use your protractor to check that is congruent to .

(5 marks)

2.Answers will vary. Sample answers are given.

The right angles at the corners of the quilt are congruent angles.
The eight triangular sections meet at the centre of the stars at 45° angles. The angles at the inside corners are also 45° angles.
The points of the white stars are 45° angles.
The points of the purplestars are 45° angles.
The purple parallelograms forming the purple star contain 135° angles.
Trapezoidal pieces contain congruent points of 45° angles and 135° angles.

(4 marks)

3.Answers will vary. Sample answers are given.

angles at the vertices of stop signs
two 45°angles at the corners of picture frames where the frames meet to form right angles
the angles formed by the hinges of an open door
angles cut on a glass chandelier
the angles at the 11 vertices of a Canadian dollar coin

(5 marks)

4.TT 2.

,so all angles are 90°.
TT 3.

TT 4.

TT 5.

(4 marks)

5.
The angle of the roof line is about 23°. (2 marks)

Lesson 3: Bisecting Angles

TT 1. and TT 2. Answers can be found in the Lesson 3 Assignment.

Students may need help conducting research. Possibly suggest a web search with the keywords “angle bisector construction.”

TT 3.

Step 1: Fold the sheet of paper so that falls on

Step 2: Unfold the sheet. Draw between the arms of the angle and along the crease. is the bisector.

TT 4.is the bisector of since divides into two congruent parts.

TT 5. The measures of the angles will vary, but

Go to the Share rubric in the Rubric Appendix for student assessment.

If students are having difficulty bisecting an angle, you may want to walk students through an example like the following.

Bringing Ideas Together

Example

Use compasses and a straight edge to bisect

Solution

Step 1: Draw any Set your compasses to a suitable radius. With centre A, the vertex of the angle, draw an arc of a circle cutting the arms of the angle at points P and Q.

Step 2: Open your compass to a radius that is at least greater than one-half the distance between points P and Q. With centres P and Q, draw arcs between the arms of Name the point where these arcs intersect X.

Step 3: Draw

Use your protractor to measure

If the work is done carefully, is congruent to—that is, both angles are equal in measure.

And,

Lesson 3 Assignment(20 marks)

Show all your work when appropriate.

1.A baseball diamond is a square. The first and third base lines meet at 90°.The line formed by the pitcher’s mound and home plate bisects the angle formed by the first- and third-base lines into two 45°angles.(4 marks)

2.
6w

(5 marks)

4.TT 1. The bisector is along the mirrored surface midway between the pencil and its image in the mirror.
TT 2. The symmetry of the leaf and the airplane mimics the pencil in the mirror. The left side is a mirror image of the right side—as if a mirror were placed along the midline.(4 marks)

5.Measure the original angle. The bisector must divide that angle into halves.
Each smaller angle is(2 marks)

Lesson 4: Relationships Among Angles

TT 1. Fold the sheet (with a loose fold) in half roughly along the horizontal line through the point of intersection as shown.

Then adjust the fold so that the rays of 1 and 3 will fall on top of each other when the fold is flattened. Then flatten the fold so the rays coincide precisely.

1 coincides with3since their rays coincide.

1is congruent to 3 since the two angles coincide.

Then unfold the sheet of paper and follow the same folding method as before. But this time, fold along a vertical line. So fold the sheet roughly along a vertical line through the point of intersection as shown so that the rays of 2 and 4 coincide.

2 coincides with4 since their rays coincide.

2is congruent to4 since the two angles coincide.

Lesson 4 Assignment (20 marks)

Show all your work when appropriate.

1.The angles are complementary angles because they form a right angle.(2 marks)

2.ABC and CBF
DCE and ECB
BFG and HFG

Note: Angles may be named differently from the noted listing e.g., and so on.
(2 marks)

3.a.A and C(1 mark)

b.A and B (1 mark)

4.

(5 marks)

5.

(6 marks)

6.

(3 marks)

Lesson 5: Parallel and Perpendicular Lines

TT 1. Students will point out a variety of lines.

TT 2. This question is the first one in the series dealing with parallel and perpendicular.Student definitions for parallel and perpendicular may be somewhat rudimentary at this stage. The intent is for students to arrive at definitions equivalent to the following.

Term / Definition
parallel lines / lines that are the same distance apart everywhere
perpendicular lines / lines that meet at right angles

TT 3. This question is the final one in the series dealing with parallel and perpendicular. Student definitions for parallel and perpendicular may vary but should be similar to the following.

Term / Definition
parallel lines / lines that are the same distance apart everywhere
perpendicular lines / lines that meet at right angles

TT 4.,TT 5., and TT 6. The answers to these questions can be found in the key to the lesson assignment.

TT 7.

Angle / Congruent Angles From 1, 2, 3, and 4
5 / 1, 3
6 / 2, 4
7 / 1, 3
8 / 2, 4

TT 8. There are exactly two sets of congruent angles.

1, 3, 5, 7

2, 4, 6, 8

Only two different colours are needed.

Go to the Share rubric in the Rubric Appendix for student assessment.

Lesson 5 Assignment (20 marks)

Students should show all their work when appropriate.

1.There are three sets of parallel lines (edges). In each set there are four lines (edges). (2 marks)

2.The perpendicular lines (edges) meet at the corners. There are three perpendicular lines (edges) at each corner. (2 marks)

3. Answers will vary. Be generous.

lodgepole pines in a stand of trees (parallel)

the trunk of a tree and level ground (perpendicular)

rays of light from the sun (parallel)

the sides of a cell in a honeycomb—the cells are the shape of a hexagon, so there are three pairs of parallel sides.

the ribs in some leaves (parallel)

(5 marks)

4.The hash marks are set in 24 yards from the sidelines. The sidelines are parallel, because the field is rectangular. Therefore, the two sets of hash marks run parallel to each other. (2 marks)

5. a.Line 3 is a transversal.(1 mark)

b.co-exterior angles (1 mark)

c.co-interior angles(1 mark)

d.vertically opposite angles(1 mark)

e.Since the two co-exterior angles 60° and 105° are not supplementary, line 1 and line 2 are not parallel.(2 marks)

(3 marks)

Lesson 6: Problems Involving Parallel and Perpendicular Lines

You may want to indicate to students that this lesson is considerably shorter than other lessons.

Try This

TT 1.

Sum of
Co-Interior Angles / Sum of
Co-Exterior Angles
Parallel Lines
Example 1 / 180° / 180°
Example 2 / 180° / 180°
Non-Parallel Lines
Example 1 / not equal to180° / not equal to180°
Example 2 / not equal to180° / not equal to180°

Lesson 6 Assignment (20 marks)

Students should show all your work when appropriate.

1.a.The set of angles at each line are identical sets. As the centre red line and the blue line are parallel, the corresponding angles are congruent. (2 marks)

b.If the centre line and the blue lines are perpendicular to the boards, the lines are parallel. The boards are a “transversal” cutting the lines at congruent corresponding angles. All right angles are equal in measure. (2 marks)

(6 marks)

(4 marks)

4.Line1 and line 2 are not parallel. The 91°angle and the 89° angle are alternate exterior angles.
If the lines were parallel, those two angles would be congruent, not supplementary. (3 marks)

5.a.2 could be any measure, except 136°. The lines would be parallel if
(2 marks)

b.The angles are co-exterior angles. (1 mark)