058i - Conditions for Similar Triangles

Lesson 58 — Conditions for Similar Triangles: Guided Practice

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

Learning Intentions

• To understand that similar triangles have the same shape (equal corresponding angles, proportional corresponding sides) but not necessarily the same size.
• To identify similar triangles using AA, SAS~ (two sides in ratio with the included angle equal) and SSS~ (three sides in the same ratio).

Success Criteria

I can:

1. State the three conditions for triangle similarity: AA, SAS~, SSS~.
2. Correctly match corresponding sides and angles between two similar triangles.
3. Calculate a scale factor between similar triangles and use it to find an unknown side.
4. Distinguish between conditions that prove similarity only and those that prove congruence.

Warmup

(5 minutes — quick judgement, mini whiteboards)

Recall Lesson 57. For each pair, decide: congruent, similar (not congruent), or neither.

1. Sides and sides .
2. Sides and sides .
3. Angles in both, with no side lengths given.
4. Sides and sides .

Answers: 1. Congruent — SSS; 2. Similar, not congruent — sides in constant ratio ; 3. Similar (by AAA/AA reasoning) — but “congruent” cannot be confirmed without side lengths; 4. Neither — one side differs and the ratio is not constant (, , ).

Discussion: “Q2 introduced today’s idea precisely: same shape, different size — a photocopier enlargement. Today we pin down exactly what conditions guarantee similarity.”

Activities

Activity 1 — Explicit Instruction: AA and SAS~ (12 min)

I do — AA (Angle-Angle): if two pairs of corresponding angles are equal, the triangles are similar. Why is this enough? Because the third angle is automatically equal too (angle sum in every triangle) — matching angles fixes the shape, but says nothing about size. Model: with , and with , — similar by AA, written .

I do — SAS~ (Side-Angle-Side, similarity version): if two pairs of corresponding sides are in the same ratio, and the included angle is equal, the triangles are similar. Model: with , , ; with , , .

Since the ratios match and the included angle is equal, with scale factor .

We do: Together, has , ; has , . Are they similar? (Check the third angle of each: ‘s third angle is , matching ‘s paired correctly — confirm which angles truly correspond before concluding AA applies.)

You do:

1. : , , . : , , . State the condition and scale factor.
2. : , . : , . Are they similar? Justify using all three angles.

Activity 2 — Explicit Instruction: SSS~ and Guided Practice Matching Correspondence (12 min)

I do — SSS~ (three sides in the same ratio): if all three pairs of corresponding sides are in the same ratio, the triangles are similar — no angle information is needed. Key technique: order each triangle’s sides from smallest to largest before comparing ratios, so you don’t accidentally compare non-corresponding sides.

Model: has sides ; has sides . Ordered: and .

We do: Together, test with sides against with sides . (Order first: and . Ratios: , , — not constant, so not similar. This is a deliberate near-miss to reinforce checking all three ratios, not just one.)

You do — guided matching practice:

1. Given , with , , : if , , and , find and .
2. Test whether (sides ) and (sides ) are similar by SSS~.
3. Test whether (sides ) and (sides ) are similar, and state the scale factor.

Activity 3 — Inquiry Task: the Overlapping Triangles Problem (11 min)

Pairs, then whole-class share.

In the diagram, and share vertex , with on and on , and . Given , , , explain why , and find and the scale factor.

Socratic scaffolding:

Prompt	Purpose
Understand: what does tell you about angles?	Corresponding angles on the parallel lines are equal: and .
What angle do both triangles already share?	is common to both and .
Devise a plan — which similarity condition applies?	Two equal angles ( common, plus one from the parallel lines) — AA.
What is , and what does that give you?	. Scale factor .
Carry out the plan for	, so .
Looking back — does the answer make sense?	should be larger than since and sit inside the larger triangle — check the scale factor is greater than .

Checks for Understanding

(5 minutes — exit ticket, collected)

1. State the condition that proves two triangles similar when only angle information is given.
2. has sides . has sides . State the condition, the scale factor, and whether the triangles are congruent.
3. has , , . has , , . State the condition and find the scale factor.
4. Reasoning. Explain why AA is sufficient to prove similarity even though only two of the three angles are actually checked.

Answers: 1. AA; 2. SSS~, scale factor , not congruent (different size); 3. SAS~, scale factor (check: ✓); 4. If two angles match, the third must also match since every triangle’s angles sum to — so AA guarantees all three angles are equal, i.e. the same shape.

Common Misconceptions

Misconception	How to pre-empt it
Matching sides in the wrong order when computing a ratio.	Always order each triangle’s sides smallest-to-largest before comparing (Activity 2’s key technique).
Believing AA alone is not “enough” information because a third angle wasn’t checked.	Explicitly derive the third angle using the angle sum every time, to show it is automatically fixed.
Confusing the direction of the scale factor (big ÷ small vs small ÷ big).	Always state which triangle is being mapped to which before dividing, and label the scale factor’s direction.
Assuming SSS~ requires equal sides, not just proportional sides.	Contrast explicitly with SSS (congruence), where sides must be equal, versus SSS~, where sides must be in a constant ratio.
Treating “similar” as a vague, informal word rather than a precise mathematical claim.	Require every claim of similarity to be backed by a named condition (AA, SAS~ or SSS~) and the matching ratio or angles.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). Two similar triangles have corresponding sides in the ratio . If the smaller triangle has a side of cm, what is the corresponding side in the larger triangle?

Answer

E2 (Kangaroo style). with scale factor (from to ). If cm, find .

Answer

E3 (Challenge). In the overlapping-triangles configuration from Activity 3, suppose instead , , . Find , given .

Answer

Homework

1. State the condition (AA, SAS~, SSS~) that proves each pair of triangles similar: (a) two matching angles only (b) three sides in the same ratio (c) two sides in the same ratio with the included angle equal.
2. has sides . has sides . Test whether they are similar using SSS~, and state the scale factor if so.
3. has , . has , . Find the missing angle in each triangle and decide whether the triangles are similar.
4. with , , . Given , , , find .
5. Reasoning. Explain why two equilateral triangles of any size are always similar, but explain what extra information would be needed to also prove them congruent.
6. Challenge. sits inside with on , on , and . If , , and the area of is , use the scale factor to find the area of . (Hint: area scales with the square of the side ratio.)

Answers: 1. (a) AA (b) SSS~ (c) SAS~; 2. Ratios , , — similar, scale factor ; 3. ‘s third angle is ; ‘s third angle is — both triangles have angles , so they are similar (AA); 4. Scale factor , so ; 5. Every equilateral triangle has three angles, so any two are similar by AA; to also be congruent, their side lengths would need to be equal, not just their angles; 6. Scale factor ; area scale factor ; area of .