Lesson Notes By Weeks and Term v5 - Grade 10

Lesson Video

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Performance objectives

• Prove the congruency of triangles using SSS, SAS, ASA, and RHS.
• Prove the similarity of triangles using AAA and proportionality of sides.
• Apply these theorems to solve geometric problems.
• Provide reasons for statements when solving geometry riders.

Lesson summary

Euclidean geometry is fundamental to understanding the world around us, from the structures of buildings and bridges to the layout of sports fields and even the patterns in traditional African art and beadwork. In this week, we will focus on applying the geometric theorems related to lines, angles and triangles. Understanding these theorems provides a powerful framework for solving problems related to shapes and space. This week’s focus will be on proving congruency and similarity in triangles using Euclidean Geometry.

Lesson notes

Congruency of Triangles Two triangles are congruent if they have the same shape and size. This means all corresponding sides and angles are equal. There are four conditions for proving congruency: SSS (Side-Side-Side): If all three sides of one triangle are equal to the corresponding three sides of another triangle, then the triangles are congruent.

Notation: If AB = DE, BC = EF, and CA = FD, then ∆ABC ≡ ∆DEF 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 two sides and included angle of another triangle, then the triangles are congruent.

Notation: If AB = DE, ∠BAC = ∠EDF, and AC = DF, then ∆ABC ≡ ∆DEF 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 two angles and included side of another triangle, then the triangles are congruent.

Notation: If ∠BAC = ∠EDF, AB = DE, and ∠ABC = ∠DEF, then ∆ABC ≡ ∆DEF RHS (Right angle-Hypotenuse-Side): If the hypotenuse and one side of a right-angled triangle are equal to the hypotenuse and corresponding side of another right-angled triangle, then the triangles are congruent.

Notation: If ∠B = ∠E = 90°, AC = DF, and AB = DE, then ∆ABC ≡ ∆DEF Example 1 (Congruency - SSS): Given: In the diagram below, AB = DE, BC = EF, and AC = D

F. Prove: ∆ABC ≡ ∆DEF [Diagram: Two separate triangles, ABC and DEF, with sides AB = DE, BC = EF, and AC = DF marked clearly.] Proof: | Statement | Reason | | :----------- | :-------------------------- | | AB = DE | Given | | BC = EF | Given | | AC = DF | Given | | ∆ABC ≡ ∆DEF | SSS Congruency Condition | Example 2 (Congruency - SAS): Given: In the diagram below, AB = DE, ∠BAC = ∠EDF, and AC = D

F. Prove: ∆ABC ≡ ∆DEF [Diagram: Two separate triangles, ABC and DEF, with sides AB = DE, AC = DF and angles ∠BAC = ∠EDF marked clearly.] Proof: | Statement | Reason | | :----------- | :-------------------------- | | AB = DE | Given | | ∠BAC = ∠EDF | Given | | AC = DF | Given | | ∆ABC ≡ ∆DEF | SAS Congruency Condition | Example 3 (Congruency - ASA): Given: In the diagram below, ∠BAC = ∠EDF, AB = DE, and ∠ABC = ∠DE

F. Prove: ∆ABC ≡ ∆DEF [Diagram: Two separate triangles, ABC and DEF, with side AB = DE and angles ∠BAC = ∠EDF, ∠ABC = ∠DEF marked clearly.] Proof: | Statement | Reason | | :----------- | :-------------------------- | | ∠BAC = ∠EDF | Given | | AB = DE | Given | | ∠ABC = ∠DEF | Given | | ∆ABC ≡ ∆DEF | ASA Congruency Condition | Example 4 (Congruency - RHS): Given: In the diagram below, ∠B = ∠E = 90°, AC = DF, and AB = D

E. Prove: ∆ABC ≡ ∆DEF [Diagram: Two separate right angled triangles, ABC and DEF, with right angles at B and E respectively, with sides AC = DF, and AB = DE marked clearly.] Proof: | Statement | Reason | | :----------- | :-------------------------- | | ∠B = ∠E = 90° | Given | | AC = DF | Given | | AB = DE | Given | | ∆ABC ≡ ∆DEF | RHS Congruency Condition | Similarity of Triangles Two triangles are similar if they have the same shape but not necessarily the same size. This means their corresponding angles are equal, and their corresponding sides are in proportion. There are three conditions for proving similarity: AAA (Angle-Angle-Angle): If all three angles of one triangle are equal to the corresponding three angles of another triangle, then the triangles are similar.

Notation: If ∠A = ∠D, ∠B = ∠E, and ∠C = ∠F, then ∆ABC ||| ∆DEF Proportionality of Sides: If the corresponding sides of two triangles are in the same proportion, then the triangles are similar.

Notation: If AB/DE = BC/EF = AC/DF, then ∆ABC ||| ∆DEF Sides in Proportion are Included in Equal Angles: If two sides of one triangle are proportional to the corresponding two sides of another triangle, and the included angles (the angle between those two sides) are equal, then the triangles are similar.

Notation: If AB/DE = AC/DF and ∠A = ∠D, then ∆ABC ||| ∆DEF Example 5 (Similarity - AAA): Given: In the diagram below, ∠A = ∠D, ∠B = ∠E, and ∠C = ∠

F. Prove: ∆ABC ||| ∆DEF [Diagram: Two separate triangles, ABC and DEF, with angles ∠A = ∠D, ∠B = ∠E, and ∠C = ∠F marked clearly.] Proof: | Statement | Reason | | :----------- | :----------------------- | | ∠A = ∠D | Given | | ∠B = ∠E | Given | | ∠C = ∠F | Given | | ∆ABC ||| ∆DEF | AAA Similarity Condition | Example 6 (Similarity - Proportionality of Sides): Given: In the diagram below, AB/DE = BC/EF = AC/D

F. Prove: ∆ABC ||| ∆DEF [Diagram: Two separate triangles, ABC and DEF, with sides AB/DE = BC/EF = AC/DF marked clearly.] Proof: | Statement | Reason | | :----------- | :--------------------------------- | | AB/DE = BC/EF = AC/DF | Given | | ∆ABC ||| ∆DEF | Proportionality of Sides | Example 7 (Similarity - Sides in Proportion with Equal Included Angle): Given: In the diagram below, AB/DE = AC/DF and ∠A = ∠D.