Secondary 2
Chapter 5
Secondary II
Unit 5– Congruence Through Transformations
Date
Section Assignment
2014/2015
Concept
A: 10/13
B: 10/14
5.1/5.2
- Worksheet 5.1/5.2
Translating, Rotating, and
Reflecting Geometric Figures
Congruent Triangles
A: 10/15
5.3-5.6
- Worksheet 5.3-5.6
SSS, SAS, ASA, AAS Congruence
Theorems
Fall Break
10/16-17
B: 10/20
5.3-5.6
- Worksheet 5.3-5.6
SSS, SAS, ASA, AAS Congruence
Theorems
A: 10/21
B: 10/22
5.7
- Worksheet Proofs
Using Congruent Triangles
A: 10/23
B: 10/24
Review
A: 10/27
B: 10/28
Chapter 5 TEST
Late and absent work will be due on the day of the review (absences must be excused). The review
assignment must be turned in on test day. All required work must be complete to get the curve on the test.
Remember, you are still required to take the test on the scheduled day even if you miss the review, so come
prepared. If you are absent on test day, you will be required to take the test in class the day you return. You
will not receive the curve on the test if you are absent on test day unless you take the test prior to your
absence.
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Secondary 2
Chapter 5
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Secondary 2
Chapter 5
Chapter 5: Congruence through Transformations
5.1 & 5.2 – Translating, Rotating, and Reflecting Geometric Figures & Congruent Triangles
Example 1: Translating Geometric Figures on the Coordinate Plane
Graph trapezoid ABCD by plotting points A(3,9), B(3,4), C(11,4), and D(11,10).
a) Translate trapezoid ABCD 15 units left to form
trapezoid A’B’C’D’. List the coordinates.
b) Translate trapezoid ABCD 12 units down to form
trapezoid A”B”C”D”. List the coordinates.
c) How would you determine the image coordinates without graphing?
Example 2: Rotating Geometric Figures on the Coordinate Plane
Recall that a rotation is a rigid motion that turns a figure about a fixed point, called the point of
rotation. The figure is rotated in a given direction for a given angle is called the angle of rotation. The
angle of rotation is the measure of the amount the figure is rotated about the point of rotation. The
direction can either be clockwise or counterclockwise.
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Chapter 5
Let’s rotate point A about the origin. The origin will be the point of rotation and you will rotate point A
90°, 180°, and 270°.
First, let’s rotate point A 90° counterclockwise about the origin.
Step 1: Draw a “hook” from the origin to point A.
Step 2: Rotate the “hook” 90° counterclockwise.
Where is A’ located? _________
a) What do you notice about the coordinates of point A
and the coordinates of point A’?
b) Rotate point A’ about the origin 90° counterclockwise
about the origin. Label the point A”. List the coordinates.
c) What do you notice about the coordinates of point A and A”?
d) Rotate point A” about the origin 90° counterclockwise. Label the point A”’. List the coordinates.
e) What do you notice about the coordinates of point A and A”’?
You may have noticed that the values of the x-and y-coordinates seem to switch places for every 90°
rotation about the origin. You may have also noticed that the rotation from point A to A” is a 180°
counterclockwise rotation about the origin, and that the rotation from point A to A”’ is a 270°
counterclockwise rotation about the origin.
Example 3: Determine the coordinates of point (x,y) after rotations of 90°, 180°, 270°, and 360°.
Original Point
Coordinates After
a 90°
Counterclockwise
Rotation About
the Origin
Coordinates After
a 180°
Counterclockwise
Rotation About
the Origin
Coordinates After
a 270°
Counterclockwise
Rotation About
the Origin
Coordinates After
a 360°
Counterclockwise
Rotation About
the Origin
(x,y)
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Chapter 5
Example 4: Graph Trapezoid ABCD by plotting the points A(-12,9), B(-12,4), C(-4,4) and D(-4, 10).
a) Rotate Trapezoid ABCD 90° counterclockwise about
the origin to form trapezoid A’B’C’D’. List the
coordinates.
b) Rotate Trapezoid ABCD 180° counterclockwise
about the origin to form trapezoid A”B”C”D”. List
the coordinates.
c) Rotate Trapezoid ABCD 270° counterclockwise
about the origin to form trapezoid A’’’B’’’C’’’D’’’. List
the coordinates.
Example 5: Reflecting Geometric Figures on the Coordinate Plane.
Recall that figures that are mirror images of each other are called reflections. A reflection is a rigid
motion that reflects, or “flips,” a figure over a given line called a line of reflection. A line of reflection is a
line over which a figure is reflected so that corresponding points are the same distance from the line.
Graph trapezoid ABCD by plotting the points A (3, 9), B (3, 4), C (11, 4), and D (11, 10).
a) Reflect trapezoid ABCD across the y-axis to create
A’B’C’D’. List the coordinates.
b) What do you notice about the coordinates from
trapezoid ABCD and the image A’B’C’D’?
c) Reflect trapezoid ABCD across the x-axis to create
A’’B’’C’’D’’. List the coordinates.
d) What do you notice about the coordinates from
trapezoid ABCD and the image A’’B’’C’’D’’?
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Secondary 2
Chapter 5
Determine the coordinates of point (x, y) after reflections about the x-axis or y-axis.
Example 6: The vertices of parallelogram DEFG are D (29, 7), E (212, 2), F (23, 2), and G (0, 7).
a. Determine the vertex coordinates of image D’E’F’G’ if parallelogram DEFG is reflected over
the x-axis.
b. Determine the vertex coordinates of image D’’E’’F’’G’’ if parallelogram DEFG is reflected
over the y-axis.
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Secondary 2
Chapter 5
Example 7: Understanding congruence.
Graph triangle ABC by plotting the points A (8, 10), B (1, 2), and C (8, 2).
a. Classify triangle ABC.
b. Calculate the length of side ̅̅̅̅
𝐴𝐵.
c. Translate triangle ABC 10 units to the left to form
triangle DEF. Graph triangle DEF and list the
coordinates of points D, E, and F.
d. Predict the side lengths of triangle DEF.
e.
Verify that the side lengths and angles are the same.
Example 8: Statements of Triangle Congruence.
Consider the congruence statement ∆JRB ≅ ∆MNS .
a. Identify the congruent angles.
b.
Identify the congruent sides.
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Secondary 2
Chapter 5
Example 9: Analyze the two triangles shown.
a. Determine the transformation used to create
triangle PMK.
b. Does the transformation preserve the size and
shape of the triangle in this problem situation?
Why or why not?
c. Write a triangle congruence statement for the triangles.
d. Identify the congruent angles and congruent sides.
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Secondary 2
Chapter 5
Additional Notes
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Secondary 2
Chapter 5
5.3/5.4/5.5/5.6 – SSS, SAS, ASA, AAS Congruence Theorems
The Side-Side-Side Congruence Theorem states: “If three sides of one triangle are congruent to the
corresponding sides of another triangle, then the triangles are congruent.”
Example 1: Graph triangle ABC by plotting the points A (8, 25), B (4, 212), and C (12, 28).
1. How can you determine the length of each side of this
triangle?
2. Calculate the length of each side of triangle ABC. Record the
measurements in the table.
3. Translate line segments ̅̅̅̅
𝐴𝐵, ̅̅̅̅
𝐵𝐶 , 𝑎𝑛𝑑 ̅̅̅̅
𝐴𝐶 up 7 units to form triangle A’B’C’.
4. Calculate the length of each side of triangle A’B’C’. Record the measurements in the table.
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Chapter 5
5. Are the corresponding sides of the pre-image and image congruent? Explain your reasoning.
6. Do you need to determine the measures of the angles to verify that the triangles are congruent?
Explain why or why not.
The Side-Angle-Side Congruence Theorem states: “If two sides and the included angle of one triangle
are congruent to the corresponding sides and the included angle of the second triangle, then the
triangles are congruent.”
Example 2: Use the Side-Angle-Side (SAS) Congruence Theorem and a protractor to determine if the
two triangles drawn on the coordinate plane shown are congruent. Use a protractor to determine the
measures of the included angles.
Example 3: Determine if there is enough information to prove that the two triangles are congruent
by SSS or SAS. Write the congruence statements to justify your reasoning.
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Secondary 2
Chapter 5
The Angle-Side-Angle Congruence Theorem states: “If two angles and the included side of one triangle
are congruent to the corresponding two angles and the included side of another triangle, then the
triangles are congruent.”
Example 4: Analyze triangles ABC and DEF.
a) Measure the angles and calculate the side lengths of both triangles.
b) Describe the possible transformation(s) that could have occurred to transform pre-image ABC
into image DEF.
c) Identify two pairs of corresponding angles and a pair of corresponding included sides that could
be used to determine congruence through the ASA Congruence Theorem.
d) Determine if triangles DEF and GHJ are congruent.
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Secondary 2
Chapter 5
The Angle-Angle-Side Congruence Theorem states: “If two angles and a non-included side of one
triangle are congruent to the corresponding angles and the corresponding non-included side of a
second triangle, then the triangles are congruent.”
Example 5: Use the previous graph to show that ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹 using AAS.
Example 6: Determine if there is enough information to prove that the two triangles are congruent
by ASA or AAS. Write the congruence statements to justify your reasoning.
a)
b)
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Secondary 2
Chapter 5
Example 7: This chapter focused on four methods that you can use to prove that two triangles are
congruent. Complete the graphic organizer by providing an illustration of each theorem.
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Chapter 5
Additional Notes
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Secondary 2
Chapter 5
5.7 – Proofs
Steps to complete a triangle congruence proof:
a.
b.
c.
d.
Mark the picture based on the given information.
Decide what else you know for a fact is congruent (reflexive, vertical angles, etc.)
Decide which theorem to use based on what is congruent (SSS, SAS, ASA, AAS)
Fill in the five lines of your proof.
Example 1: Fill in the missing information for the proofs.
a. Given: ̅̅̅̅
𝐴𝐷 ⊥ ̅̅̅̅
𝐷𝐶 ; ̅̅̅̅
𝐶𝐵 ⊥ ̅̅̅̅
𝐴𝐵; ∠𝐴𝐶𝐷 ≅ ∠𝐶𝐴𝐵
Prove: ∆𝐴𝐷𝐶 ≅ ∆𝐶𝐵𝐴
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Secondary 2
b.
Chapter 5
Given: C bisects ̅̅̅̅
𝐵𝐸 and ̅̅̅̅
𝐴𝐷
Prove: ∠𝐴 ≅ ∠𝐷
c. Given: ̅̅̅̅
𝐺𝐸 ⊥ ̅̅̅̅
𝐷𝐹 ; ̅̅̅̅
𝐷𝐺 ≅ ̅̅̅̅
𝐺𝐹
Prove: ∆𝐷𝐸𝐺 ≅ ∆𝐹𝐸𝐺
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Secondary 2
Chapter 5
Additional Notes
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