Lesson Notes By Weeks and Term v4 - JHS 3

Lesson Video

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Performance objectives

• Define ratios and proportions.
• Identify similar and congruent triangles.
• Solve problems involving ratios.
• Use ratios to compare quantities in real life.
• Apply knowledge of similar triangles to solve problems.

Lesson summary

This lesson introduces two very important concepts in geometry: congruence and similarity. Imagine a tailor cutting two identical triangular pieces of cloth for a dress – that's congruence. Now, imagine an artist drawing a small sketch of a triangular Adinkra symbol, and then painting a much larger version on a wall – that's similarity. Understanding these ideas helps us compare shapes, make accurate scale models, and even measure things that are too tall or far away to measure directly, like the height of a coconut tree using its shadow. The concept of similarity is directly built on the ideas of ratios and proportion, which we have studied before.

Lesson notes

Part 1: Congruent Triangles (Same Shape, Same Size)

Definition: Two triangles are congruent if they are exactly the same in every way. This means they have the same shape and the same size. If you could cut one out, it would fit perfectly on top of the other. All three corresponding sides are equal in length. All three corresponding angles are equal in measure.

The symbol for congruence is ≅. So, if triangle ABC is congruent to triangle PQR, we write: ΔABC ≅ ΔPQR.

Conditions for Congruence: We don't need to check all 6 parts (3 sides, 3 angles) to prove congruence. We can use one of these four "shortcuts": SSS (Side-Side-Side): If all three sides of one triangle are equal to the corresponding three sides of another triangle. *Example:* If AB = PQ, BC = QR, and AC = PR, then ΔABC ≅ ΔPQR. SAS (Side-Angle-Side): If two sides and the *included* angle (the angle between those two sides) of one triangle are equal to the corresponding parts of another triangle. *Example:* If AB = PQ, ∠B = ∠Q, and BC = QR, then ΔABC ≅ ΔPQR. ASA (Angle-Side-Angle): If two angles and the *included* side (the side between those two angles) of one triangle are equal to the corresponding parts of another triangle. *Example:* If ∠A = ∠P, AC = PR, and ∠C = ∠R, then ΔABC ≅ ΔPQR. RHS (Right-angle-Hypotenuse-Side): If two right-angled triangles have their hypotenuses equal and one other corresponding side equal. *Example:* If ∠B = ∠Q = 90°, AC = PR (hypotenuse), and BC = QR, then ΔABC ≅ ΔPQR.