Similarity in Triangles

Grade 12 · Mathematics

Semester 2 | Period 5 | Week 28

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Subject: Mathematics

Semester: 2

Period: 5

Week: 28

WEEK 28

Class: Grade 12
Age: 17 years
Duration: 40 minutes × 5 periods
Subject: Mathematics
Topic: Similarity in Triangles
Focus: Properties of similar triangles, similarity criteria (AAA, SAS, SSS), relationship between sides and angles, and applications to areas and volumes.

SPECIFIC OBJECTIVES

By the end of the lesson, students should be able to:

  1. Define and explain the concept of similarity in triangles.
  2. Identify and prove properties of similar triangles.
  3. Establish similarity in triangles using AAA, SAS, and SSS
  4. Relate corresponding sides and angles of similar triangles.
  5. Solve problems involving areas and volumes of similar figures.

 

INSTRUCTIONAL TECHNIQUES

  • Guided discovery
  • Question and answer
  • Group work and discussion
  • Practical drawing and measurements
  • Problem-solving approach

 

INSTRUCTIONAL MATERIALS

  • Ruler, compass, protractor, mathematical set
  • Graph paper and plain sheets
  • Whiteboard and markers
  • Models of similar triangles (cutouts, cardboard)

 

PERIOD 1 & 2: Properties of Similar Triangles

PRESENTATION

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Introduces the idea of similar figures (shapes same but sizes different).

Students give examples (maps, photographs, models).

Step 2 - Explanation

Explains that similar triangles have equal corresponding angles and proportional corresponding sides.

Students listen attentively and take notes.

Step 3 - Demonstration

Draws two triangles with equal angles and shows side ratios.

Students verify with measurements.

Step 4 - Practice

Guides students in proving similarity in examples.

Students work in groups on exercises.

NOTE ON BOARD:

  • Similar triangles: Same shape, not necessarily same size.
  • Properties:
  1. Corresponding angles are equal.
  2. Corresponding sides are proportional.

EVALUATION (5 Exercises):

  1. Define similar triangles.
  2. State two properties of similar triangles.
  3. Are all equilateral triangles similar? Explain.
  4. If ΔABC ∼ ΔDEF, what is the relationship between ∠A and ∠D?
  5. If ΔABC ∼ ΔDEF, and AB = 4 cm, DE = 6 cm, find the ratio of similarity.

CLASSWORK (5 Questions):

  1. State two conditions that make two triangles similar.
  2. In ΔXYZ ∼ ΔPQR, if XY = 8 cm, PQ = 4 cm, XZ = 6 cm, find PR.
  3. Draw two triangles with sides in the ratio 2:3 and verify similarity.
  4. Which of these pairs are similar? (i) Isosceles triangles with equal vertex angle (ii) Right-angled triangles with same acute angle.
  5. Define corresponding sides.

HOMEWORK (5 Tasks):

  1. Prove that all isosceles right-angled triangles are similar.
  2. If ΔABC ∼ ΔDEF, AB = 5 cm, DE = 10 cm, AC = 7 cm, DF = 14 cm. Find the ratio of similarity.
  3. State two real-life applications of similarity in triangles.
  4. In two similar triangles, one side is 8 cm and the corresponding side is 12 cm. Find the ratio of their sides.
  5. Draw and label two pairs of similar triangles.

 

PERIOD 3: Establishing Similarity Criteria (AAA, SAS, SSS)

PRESENTATION

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Recalls properties of similar triangles.

Students respond to quick revision questions.

Step 2 - Explanation

Explains criteria:

  
  • AAA: All angles equal
  • SAS: One equal angle and proportional sides including the angle
  • SSS: Corresponding sides proportional | Students note and memorize criteria. |
    | Step 3 - Demonstration | Uses drawn triangles to illustrate AAA, SAS, and SSS criteria. | Students participate and verify. |
    | Step 4 - Practice | Assigns exercises on proving similarity. | Students solve guided problems. |

NOTE ON BOARD:

  • AAA Criterion: Equal angles → similar triangles.
  • SAS Criterion: One equal angle and two proportional sides → similar triangles.
  • SSS Criterion: All sides proportional → similar triangles.

EVALUATION (5 Exercises):

  1. State three criteria for similarity in triangles.
  2. If ΔABC and ΔDEF have equal corresponding angles, what criterion applies?
  3. What does SAS stand for?
  4. Two triangles have sides in ratio 3:4:5 and 6:8:10. What criterion shows they are similar?
  5. Prove that ΔPQR ∼ ΔXYZ if PQ/XY = QR/YZ = PR/XZ.

CLASSWORK (5 Questions):

  1. Write short notes on AAA, SAS, and SSS criteria of similarity.
  2. ΔABC has sides 6, 8, 10. ΔDEF has sides 9, 12, 15. Show that they are similar.
  3. State one difference between SSS similarity and congruence.
  4. Prove that two equilateral triangles are always similar (AAA).
  5. If ΔMNO and ΔPQR have ∠M = ∠P, and MN/PQ = NO/QR, prove they are similar.

HOMEWORK (5 Tasks):

  1. List two differences between congruent and similar triangles.
  2. Using AAA, prove that two right triangles with equal acute angles are similar.
  3. ΔABC has sides 4, 5, 6. ΔDEF has sides 8, 10, 12. Are they similar? Prove.
  4. State one real-life use of AAA similarity.
  5. In ΔXYZ and ΔPQR, if XY/PQ = YZ/QR = XZ/PR, prove they are similar.

 

PERIOD 4 & 5: Relationship Between Corresponding Sides, Areas, and Volumes

PRESENTATION

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Recalls similarity properties.

Students give quick examples.

Step 2 - Explanation

Explains:

  
  • Ratio of corresponding sides = ratio of similarity.
  • Ratio of areas = (ratio of sides)².
  • Ratio of volumes = (ratio of sides)³. | Students listen and take notes. |
    | Step 3 - Demonstration | Uses worked examples with numbers. | Students calculate alongside teacher. |
    | Step 4 - Practice | Provides guided problems. | Students work independently or in groups. |

NOTE ON BOARD:

  • If ΔABC ∼ ΔDEF,
    AB/DE = BC/EF = AC/DF
  • Ratio of Areas = (Ratio of sides)²
  • Ratio of Volumes = (Ratio of sides)³

EVALUATION (5 Exercises):

  1. If ΔPQR ∼ ΔXYZ, write one relationship between their sides.
  2. Ratio of similarity = 2:3. What is ratio of their areas?
  3. Ratio of similarity = 1:4. What is ratio of their volumes?
  4. If ΔABC ∼ ΔDEF, and AB/DE = 2/3, what is the ratio of areas?
  5. State the difference between side ratio and area ratio.

CLASSWORK (5 Questions):

  1. ΔABC ∼ ΔDEF, AB = 6, DE = 9. Find the ratio of areas.
  2. Two cubes are similar. Ratio of sides = 2:5. Find ratio of their volumes.
  3. ΔXYZ ∼ ΔPQR, XY/PQ = 3/4. Find ratio of their areas.
  4. If ratio of similarity is 5:7, find ratio of their perimeters.
  5. A cuboid has a volume of 64 cm³, another similar cuboid has side ratio 1:2. Find its volume.

HOMEWORK (5 Tasks):

  1. Two triangles are similar with side ratio 2:3. Find ratio of their areas.
  2. Two spheres are similar with radii ratio 1:5. Find ratio of their volumes.
  3. ΔABC ∼ ΔDEF, AB = 4 cm, DE = 6 cm. Find ratio of their areas.
  4. Explain the relationship between side ratio and area ratio.
  5. A cone has a height 3 cm. A similar cone has height 6 cm. Find the ratio of their volumes.