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Grade 10 Mathematics Learners’ Book

INNODEMS

Subsection 2.2.7 Congruence Tests for Triangles

Activity 2.2.7.

Work in groups
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Conditions for Congruence in Triangles
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Materials
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Instructions
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Trace the following triangles on a construction paper.
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Identify pairs of congruent triangles.
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From the pairs of congruent triangles you have identified, which pairs fit the following criteria:
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  1. The three sides of one triangle is equal to the three sides of the corresponding triangle.
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  2. 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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  3. 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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  4. 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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Key Takeaway
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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.
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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.
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Example 2.2.10.

Check if the triangles below are congruent and state the test of congruence criterion.
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Solution.
From the figure, identify corresponding sides and angles.
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Sides \(AB = PR = 3\, cm \text{ and } BC = PQ = 8\,cm\)
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\(\angle B = \angle P = 60^\circ.\)
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Therefore, \(\Delta ABC \cong \Delta PQR\) by SAS criterion
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Exercises Exercises

1.

Check if the triangles below are congruent and state test of congruence criterion.
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2.

Show that \(\Delta ABC \cong \Delta ADB\) if \(AD = AE = BE = BC.\)
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3.

\(A(0,4)\, B(-3,0) \text{ and } C(0,2)\) 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.
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4.

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.
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Checkpoint 2.2.11.

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Checkpoint 2.2.12.

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