062i - Congruence and Similarity Consolidation

Lesson 62 — Consolidation and Check: Congruence and Similarity

Strand: Space | Descriptor: AC9M8SP01 | Duration: 45 minutes

Learning Intentions

• To consolidate the conditions for congruence and similarity of triangles and other common shapes.
• To apply these conditions fluently to solve applied problems, including those involving transformations.

Success Criteria

I can:

1. State and apply SSS, SAS, AAS and RHS to prove triangle congruence.
2. State and apply AA, SSS and SAS to prove triangle similarity, and use a scale factor to find unknown lengths.
3. Explain the conditions for congruence or similarity of other common shapes (e.g. squares, regular polygons).
4. Identify which transformations preserve congruence, and which preserve similarity only.

Warmup

(6 minutes — “Always, sometimes, never”, pairs)

Decide whether each statement is always, sometimes, or never true. Give a supporting example or counterexample.

1. Two triangles with equal areas are congruent.
2. Two right-angled triangles with the same hypotenuse and one equal leg are congruent.
3. All squares are similar to one another.
4. A shape and its image under a dilation are congruent.

Answers: 1. Sometimes — congruent triangles always have equal area, but equal area alone does not force the same shape (e.g. a right triangle and a right triangle both have area but are not congruent). 2. Always — this is precisely the RHS condition. 3. Always — every square has four right angles and equal sides in the same ratio to any other square. 4. Sometimes — only when the scale factor is or ; any other factor changes the size.

Activities

Activity 1 — Mixed Fluency Review (13 min)

Rapid-fire, then pair check.

I do: Recap all four triangle congruence conditions (SSS, SAS, AAS, RHS) and all three similarity conditions (AA, SSS, SAS with proportional sides) on the board with a one-line diagram each.

We do: For each pair of triangles below, state whether they are congruent, similar (not congruent), or neither, and name the condition:

1. Sides and sides .
2. Sides and sides .
3. Sides and sides .
4. Angles and angles with no side information.

You do: Students complete a matching set of six further pairs independently, then swap and check with a partner.

Activity 2 — Applied Problems (20 min)

Pairs. Every answer must carry correct units and a one-sentence justification.

Problem 1. A map has a scale of . Two towns are cm apart on the map. What is the real distance, in kilometres?

Problem 2. A photograph measuring cm by cm is enlarged so that its longer side becomes cm. Find the new shorter side and the scale factor used.

Problem 3. Two right-angled triangular roof braces both have a hypotenuse of m and one leg of m. Explain, using a congruence condition, why the braces must be identical in shape and size.

Problem 4. A m flagpole casts a shadow of m. At the same time, a nearby tree casts a shadow of m. How tall is the tree?

Socratic scaffolding for Problem 4:

Prompt	Purpose
Understand: what is being asked?	The tree’s height — an unknown length, not a shadow length.
What makes the two triangles similar?	The sun’s rays are parallel (equal angle of elevation) and both objects meet the ground at — AA.
Devise a plan	Set up a ratio of corresponding sides: height to shadow, matched consistently.
Carry it out	, so .
Looking back	Does the answer make sense? The tree’s shadow is longer than the pole’s, so the tree should be taller than m — check the arithmetic against that expectation.

Answers: 1. cm km; 2. Scale factor ; new shorter side cm; 3. RHS — equal hypotenuse and one equal leg fixes a unique right-angled triangle, so the braces are congruent; 4. m.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. State the condition that proves two triangles with three pairs of equal sides congruent.
2. A triangle has sides . A second triangle has sides . Are they congruent, similar, or neither? Justify.
3. A m post casts a m shadow. Find the height of a tree casting an m shadow at the same time.
4. Name one transformation that always produces a congruent image, and one that can produce an image that is similar but not congruent.
5. Reasoning. Explain why AAA is enough to prove similarity but not congruence.

Answers: 1. SSS; 2. Similar, not congruent — scale factor throughout (), but not equal; 3. m; 4. Reflection, rotation or translation always give congruent images; a dilation (scale factor ) gives a similar but non-congruent image; 5. Equal angles fix the shape but not the size — any enlargement of a triangle keeps all three angles the same.

Common Misconceptions

Misconception	How to pre-empt it
”Similar” is used loosely to mean “looks a bit like.”	Insist on the precise meaning: same shape, angles equal, sides in a constant ratio.
Mixing up which triangle is the numerator and denominator in a scale factor.	Always write “new old” (or vice versa) consistently and label it before calculating.
Assuming a dilation always changes size.	Discuss the special case scale factor : the image is congruent (identical) to the original.
Believing congruent shapes can be different sizes “as long as they look similar.”	Return to the definition: congruent means identical in both shape and size.
Forgetting to check units are consistent (cm vs m) before setting up a ratio.	Require unit conversion as an explicit first step in every ratio problem.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). Two similar triangles have areas in the ratio . What is the ratio of their corresponding side lengths?

Answer

Since area scales with the square of the side ratio, .

E2 (Kangaroo style). A ladder m long leans against a wall, reaching m up the wall. A second, similar setup uses a ladder m long at the same angle. How far up the wall does it reach?

Answer

Scale factor , so the second ladder reaches m up the wall.

E3 (Challenge). and are congruent by SSS. has a perimeter of cm and one side of cm. What is the length of the corresponding side in , and why can you be certain?

Answer

cm — congruent triangles have identical corresponding sides by definition, regardless of which SSS values are known first.

E4 (Investigation). Explain why every two equilateral triangles are similar to each other, but not necessarily congruent.

Answer

Every equilateral triangle has three angles, so any two are similar by AAA. However, side lengths can differ (e.g. cm vs cm equilateral triangles), so they need not be congruent.

Homework

1. State whether each pair of triangles is congruent, similar (not congruent), or cannot be determined: (a) sides and (b) angles in both, no sides given (c) sides and .
2. A model car is built at a scale of . If the model is cm long, find the real car’s length in metres.
3. A right-angled triangle has hypotenuse cm and one leg cm. A second right-angled triangle has hypotenuse cm and one leg cm. Explain why they must be congruent.
4. A tree’s shadow is m long when a m student’s shadow is m long. Find the height of the tree.
5. A photo cm cm is enlarged to a poster with shorter side cm. Find the longer side of the poster.
6. Reasoning. Two students argue about whether “same shape” and “same size” are the same claim. Using the words congruent and similar, settle the argument in two or three sentences.
7. Challenge. A triangle is dilated by scale factor , then the image is dilated again by scale factor . Is the final shape congruent to the original? Justify using the combined scale factor.

Answers: 1. (a) similar, not congruent, scale factor (b) cannot be determined — AAA proves similarity only (c) congruent, SSS; 2. cm m; 3. RHS fixes a unique triangle shape and size, so equal hypotenuse and equal leg forces congruence; 4. m; 5. cm; 6. Congruent means identical in shape and size; similar means identical in shape only, with sizes possibly different (in a constant ratio); 7. Yes — combined scale factor , so the final shape is identical in size to the original, i.e. congruent.