Carrying Out Congruence Tests for Triangles

By the end of this lesson, learners should be able to

1. Distinguish between direct congruence and opposite congruence in paired geometric figures.
2. State and explain the four standard triangle congruence tests: SSS, SAS, AAS/ASA, and RHS.
3. Apply congruence tests to demonstrate mathematically that pairs of triangles are congruent.

Key vocabulary

Direct Congruence

Congruence where one shape superimposes onto another without needing to flip or turn over.

Triangles $JKL$ and $XYZ$ show direct congruence because sliding superimposes them without flipping.

Opposite Congruence

Congruence where one shape superimposes onto another only after flipping or turning over.

Reflecting triangle $JKL$ across a line produces a triangle having opposite congruence to it.

Included Angle

The angle located directly between two specified adjacent sides of a geometric polygon.

In triangle $ABC$, the included angle between sides $AB$ and $AC$ is $\mathrm{\angle }A$.

Corresponding Sides

Sides that occupy matching relative positions in two similar or congruent geometric figures.

Side $RS$ in triangle $RST$ and side $LM$ in triangle $LMN$ are corresponding sides.

Hypotenuse

The longest side of a right-angled triangle, opposite the ninety-degree right angle.

In right-angled triangle $PQR$, side $PR$ opposite the right angle is the hypotenuse.

Lesson preview

Slide 1 · Direct vs Opposite Congruence and Congruence CriteriaOpening · 12 min

Congruence tests allow us to verify if two triangles are identical in shape and size without measuring all six components. If a triangle can be slid onto another directly, they have direct congruence; if it must be turned over, they have opposite congruence. The primary criteria for congruence are Side-Side-Side ($SSS$), Side-Angle-Side ($SAS$), Angle-Angle-Side ($AAS$ or $ASA$), and Right angle-Hypotenuse-Side ($RHS$). Use Figure 1 to see how tick marks denote equal sides and angles for congruence tests.

• Two figures are directly congruent if one shifts onto the other without flipping.
• Opposite congruence occurs when a shape must be flipped over to fit the other.
• The SSS test requires all three corresponding sides of two triangles to be equal.
• The SAS test requires two corresponding sides and their included angle to be equal.

Worked example. Compare corresponding side lengths of triangles $JKL$ and $XYZ$: all three corresponding sides are equal ($4\text{ cm}$, $6\text{ cm}$, $7\text{ cm}$). Identify the criterion satisfied for the first pair as Side-Side-Side ($SSS$). Compare triangles $DEF$ and $PQR$: two sides ($5\text{ cm}$, $8\text{ cm}$) and the included angle (${40}^{\circ }$) match. Identify the criterion satisfied for the second pair as Side-Angle-Side ($SAS$). Answer: Triangles $JKL$ and $XYZ$ are congruent by $SSS$; triangles $DEF$ and $PQR$ are congruent by $SAS$..

Check question: Triangle $ABC$ has side $AB=5\text{ cm}$, $BC=7\text{ cm}$, and $\mathrm{\angle }B={50}^{\circ }$. Triangle $UVW$ has side $UV=5\text{ cm}$, $VW=7\text{ cm}$, and $\mathrm{\angle }V={50}^{\circ }$. If $UVW$ is a reflection of $ABC$, what type of congruence exists between them, and which test proves it?

1. Opposite congruence; $SAS$ criterion
2. Direct congruence; $SAS$ criterion
3. Opposite congruence; $SSS$ criterion
4. Direct congruence; $SSS$ criterion

Show answer

Opposite congruence; $SAS$ criterion. Since triangle $UVW$ is created by a reflection, it must be turned over to superimpose onto $ABC$, which means they have opposite congruence. The triangles match via two sides ($5\text{ cm}$ and $7\text{ cm}$) and their included angle (${50}^{\circ }$), which satisfies the $SAS$ criterion.

Slide 2 · Applying SSS, SAS, and AAS Congruence TestsPractice · 14 min

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This lesson, and every other Grade 1–10 CBE lesson, comes ready to project with the EAP ClassKit. It runs without electricity or internet.

• All 3 slides with diagrams and class activities
• Teacher notes, SNE adaptations and assessment checks
• Homework and the other 10 lessons in this sub-strand

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All 11 lessons in Reflection and Congruence

1. 1Identifying Lines of Symmetry in Plane FiguresIdentify lines of symmetry in plane figures
2. 2Identifying Lines of Symmetry in Plane FiguresIdentify lines of symmetry in plane figures
3. 3Properties of Reflection in Plane GeometryDetermine the properties of reflection in different situations
4. 4Properties of Reflection in Plane GeometryDetermine the properties of reflection in different situations
5. 5Reflecting Shapes on Plane Surfaces and Cartesian PlanesDraw an image given an object and a mirror line on plane surface and Cartesian plane
6. 6Reflecting Shapes on the Cartesian PlaneDraw an image given an object and a mirror line on plane surface and Cartesian plane
7. 7Determining the Equation of a Mirror LineDetermine the equation of the mirror line given an object and its image
8. 8Determining the Equation of a Mirror LineDetermine the equation of the mirror line given an object and its image
9. 9Direct and Opposite Congruence in Plane FiguresCarry out congruence tests for triangles
10. 10Carrying Out Congruence Tests for TrianglesCarry out congruence tests for triangles
11. 11Real-Life Applications of Reflection, Line Symmetry, and CongruencePromote use of reflection and congruence in real-life situations