Curriculum Alignment
- Strand
- 2.0 Measurements and Geometry
- Sub-Strand
- 2.2 Reflection and Congruence
- Specific Learning Outcomes
- Carry out congruence tests for triangles
Subsection 2.2.4 Congruence in Triangles
Teacher Resource 2.2.22.
To assist your teaching, we have prepared lesson resources, aligned with this textbook and the CBC. The Lesson Plan links to syllabus learning outcomes and provides suggest time allocations. The Step-by-Step Guide provides more detailed guidance on how to teach the content, including suggested questions to ask learners, and possible answers.
Learner Experience 2.2.6.
Work in groups
Conditions for Congruence in Triangles
Materials
Instructions
Trace the following triangles on a construction paper.
Identify pairs of congruent triangles.
From the pairs of congruent triangles you have identified, which pairs fit the following criteria:
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The three sides of one triangle is equal to the three sides of the corresponding triangle.
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Two sides and an included angle of one triangle is equal to the two corresponding sides and the included angle of the other triangle.
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One side and two included angles of one triangle is equal to the corresponding side and the two included angles of the other triangle.
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One side and the hypotenuse of a right-angled triangle are equal to the hypotenuse and the corresponding side of another right-angled triangle.
Exploration 2.2.7. Determining Congruence.
Two triangles are congruent if one can be translated, rotated, and/or reflected to obtain the other. The following interactive demonstrates this fact.
Key Takeaway 2.2.24.
Figure with the same size and shape are said to be congruent.
The symbol for congruence is \(\cong.\)
Figures with the same shape but different size are said to be similar.
Congruence can also be applied to line segments and angles.
Congruence in triangles depends on the measure of the sides and angles. Two triangles are said to be congruent if a pair of the corresponding sides and corresponding angles are equal.
Criteria for congruence tests in triangles include:
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Side-side-side (SSS): the three sides of one triangle is equal to the three sides of the corresponding triangle.
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Side-angle-side (SAS): two sides and an included angle of one triangle is equal to the two corresponding sides and the included angle of the other triangle.
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Angle-side-angle (ASA): one side and two included angles of one triangle is equal to the corresponding side and the two included angles of the other triangle.
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Right angle-hypotenuse-side (RHS): one side and the hypotenuse of a right-angled triangle are equal to the hypotenuse and the corresponding side of another right-angled triangle.
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Angle-angle-side (AAS): one side and two included angles of one triangle is equal to the corresponding side and the two included angles of the other triangle.
| Test | What You Need |
|---|---|
| SSS | Three pairs of equal sides |
| SAS | Two pairs of equal sides + included angle |
| ASA | Two pairs of equal angles + included side |
| RHS | Hypotenuse + one other side in right triangles |
| AAS | Two pairs of equal angles + non-included side |
Example 2.2.26.
Check if the triangles below are congruent and state the test of congruence criterion.
Example 2.2.27.
A surveyor needs to determine whether two triangular plots of land are equal in shape and size. She measures the sides of the first plot and finds them to be \(15\) m, \(20\) m, and \(25\) m. The second plot has sides measuring \(15\) m, \(20\) m, and \(25\) m.
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Are the two triangular plots congruent? Explain
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State the congruence criterion used
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If one side of the second plot was \(24\) m instead of \(25\) m, would the triangles still be congruent?
Solution.
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Yes, the two triangular plots are congruent because all three corresponding sides are equal
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The congruence criterion used is SSS
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No, the triangles would not be congruent because for SSS criterion, all three corresponding sides must be equal. Changing one side length would produce a different triangle.
Example 2.2.28.
A ladder of length \(5\) m leans against a vertical wall, reaching a height of \(4\) m. Another ladder of the same length leans against a different wall, also reaching a height of \(4\) m. The base of each ladder is on horizontal ground.
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Are the two triangles congruent
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State the congruence criterion used.
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If the second ladder reached a height of 3.5 m instead of 4 m, would the triangles still be congruent?
Solution.
From the diagram and the given information:
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Each ladder forms a right-angled triangle with the wall and the ground.\(\angle B = \angle E = 90^\circ\)
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The hypotenuse (ladder length) in both triangles is equal.\(AC = DF = 5\) m
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One corresponding side (height reached on the wall) is equal.\(AB = DE = 4\) m
Therefore, the two triangles are congruent by the RHS (Right angle-Hypotenuse-Side) criterion.
If the second ladder reached a height of \(3.5\) m instead of \(4\) m, then the corresponding sides would no longer be equal. Since one of the required equal sides would differ, the triangles would not be congruent.
Extended activity.
Work in groups
Trace and cut out the following shapes on a piece of paper.
1. Identify figures that are directly congruent.
Example: \(B \cong I.\) \(B\) fits directly on \(I\) without flipping, thus \(B \text{ and } I\) are said to be directly congruent.
2. Identify figures that have opposite congruence.
Example: \(E \text{ and } L\) do not fit directly but when you flip (lateral inversion) figure \(L\) and fit it on \(E,\) they align. This is called opposite congruence.
3. Identify:
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The figures with the same shape.
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The figures with the same size.
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The figure with same size and shape.
Checkpoint 2.2.29.
Checkpoint 2.2.30.
Checkpoint 2.2.31.
Exercises Exercises
1.
Identify figures that are congruent.
2.
Check if the triangles below are congruent and state test of congruence criterion.
3.
4.
\(A(0,4)\, B(-3,0) \text{ and } C(0,2)\) are the coordinates of \(\Delta ABC.\) Reflect the triangle over mirror line \(x = 0.\) Prove that the triangle and its image are congruent and state the test of congruence criterion.
5.
Construct an equilateral triangle \(UVW\) with sides \(6\,cm.\,X\) is the midpoint of \(UW\) and \(VX\) is perpendicular to \(UW.\) Show that \(\Delta UVX \, \cong \, \Delta VWX.\) State the test of congruence criterion.
