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Mathematics 10-3: Module 5
1
Teacher’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
PDF for 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:
http://www.learnalberta.ca/content/mec/flash/index.html?url=Data/5/A/A5A2.swf
TT 1.
a. 63°
Mathematics 10-3: Module 5
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Teacher’s Guide
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
That’s because
1
´
4
TT 4. Each angle =
1
4
of one full rotation, or 360°.
360° = 90°.
1
´
2
90°.
= 45°
TT 5. Check 45°, 135°, 225°, and 315° on the final result.
TT 6. each angle =
1
´
2
45°
= 22.5°
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.
Mathematics 10-3: Module 5
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Teacher’s Guide
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)
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Teacher’s Guide
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)
5.
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).
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Teacher’s Guide
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 ÐDEF .
TT 7. Answers will vary.
They should be equal in measure within a degree or two.
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Teacher’s Guide
Note: Allow some latitude in the measures.
Share
Go to the Share rubric in the Rubric Appendix for student assessment.
Lesson 2 Assignment (20 marks)
Show all your work when appropriate.
1.
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Step 1: Draw BC. 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 ÐA.
Step 2: Use your compass to draw circles with the same radius centred at A and at B.
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The first circle cuts through the arms of ÐA at P and Q. The second circle cuts across BC
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.
Mathematics 10-3: Module 5
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Teacher’s Guide
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Step 4: Draw BY . Use your protractor to check that ÐA is congruent to ÐB .
(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 purple stars 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.
ÐA @ÐB @ÐC @ÐD , so all angles are 90°.
TT 3.
Mathematics 10-3: Module 5
ÐA @ÐC and ÐB @ÐD
TT 4.
Ð1 @Ð3 and Ð2 @Ð4
TT 5.
ÐA @ÐB @ÐC @ÐD @ÐE
(4 marks)
8
Teacher’s Guide
Mathematics 10-3: Module 5
Teacher’s Guide
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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.
Share
Students may need help conducting research. Possibly suggest a web search with the keywords “angle
bisector construction.”
TT 3.
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Step 1: Fold the sheet of paper so that BA falls on BC.
uuur
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Step 2: Unfold the sheet. Draw BD between the arms of the angle and along the crease. BD is the
bisector.
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TT 4. BD is the bisector of ÐABC, since BD divides ÐABC into two congruent parts.
TT 5. The measures of the angles will vary, but ÐABD = ÐCBD =
1
ÐABC.
2
Share
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
Mathematics 10-3: Module 5
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Teacher’s Guide
Example
Use compasses and a straight edge to bisect ÐA.
Solution
Step 1: Draw any ÐA. 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 ÐA. Name the point where
these arcs intersect X.
Mathematics 10-3: Module 5
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Teacher’s Guide
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Step 3: Draw AX .
Use your protractor to measure ÐA, ÐPAX , and ÐQAX .
If the work is done carefully, ÐPAX is congruent to ÐQAX —that is, both angles are equal in measure.
And, ÐPAX = ÐPAX =
1
ÐA.
2
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)
Mathematics 10-3: Module 5
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Teacher’s Guide
2.
6w
(5 marks)
3.
(5 marks)
4. TT 1. The bisector is along the mirrored surface midway between the pencil and its image in the
mirror.
Mathematics 10-3: Module 5
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Teacher’s Guide
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
1
´
2
original angle. (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 3 since their rays coincide.
1 is 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.
2 is 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
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Teacher’s Guide
Note: Angles may be named differently from the noted listing e.g., ÐABC = ÐABD and so on.
(2 marks)
3. a. A and C (1 mark)
b. A and B (1 mark)
4. a = 23°
¬
opposite angles
b + 23° = 180° ¬ supplementary angles
b = 180° - 23°
b = 157°
a + c + 90° =
23° + c + 90° =
c + 113° =
c + 113° - 113° =
c=
(5 marks)
5.
180° ¬ triangle sum
180°
180°
180° - 113°
67°
a + 21° = 90° ¬ complementary angles
a + 21° - 21° = 90° - 21°
a = 69°
c + 21° = 180° ¬ supplementary angles
c + 21° - 21° = 180° - 21°
c = 159°
a+ b=
69 + b =
b + 69° - 69° =
b=
(6 marks)
6.
180° ¬ supplementary angles
180° ¬ supplementary angles
180° - 69°
111°
b + 50° + 90° =
b + 140° =
b + 140° - 140° =
b=
180° ¬ triangle sum
180°
180° - 140°
40°
Mathematics 10-3: Module 5
a+ b=
a + 40 =
a + 40° - 40° =
a=
(3 marks)
Teacher’s Guide
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180° ¬ supplementary angles
180° ¬ supplementary angles
180° - 40°
140°
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
perpendicular lines
lines that are the same distance apart everywhere
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
perpendicular lines
lines that are the same distance apart everywhere
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.
Mathematics 10-3: Module 5
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Teacher’s Guide
Share
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)
6.
x = 120° ¬
y = 120° ¬
(3 marks)
Alternate interior angles are congruent.
Corresponding angles are congruent OR x = y because they are opposite angles.
Lesson 6: Problems Involving Parallel and Perpendicular Lines
You may want to indicate to students that this lesson is considerably shorter than other lessons.
Mathematics 10-3: Module 5
Teacher’s Guide
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Try This
TT 1.
Sum of
Co-Interior Angles
Sum of
Co-Exterior Angles
Example 1
180°
180°
Example 2
180°
180°
Example 1
not equal to180°
not equal to180°
Example 2
not equal to180°
not equal to180°
Parallel Lines
Non-Parallel Lines
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)
2. a + 71° = 180° ¬ co-exterior angles
a = 180° - 71°
a = 109°
b= a ¬
= 109°
vertically opposite angles
c = 71° ¬
(6 marks)
vertically opposite angles
3. ÐB + 63° = 180° ¬ co-interior angles
ÐB = 180° - 63°
ÐB = 117°
(4 marks)
4. Line 1 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 Ð1+ Ð2 = 180°.
(2 marks)
Mathematics 10-3: Module 5
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b. The angles are co-exterior angles. (1 mark)
Teacher’s Guide