061i - Proving Congruence and Similarity

Lesson 61 — Problem-Solving: Proving Congruence and Similarity

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

Learning Intentions

• To construct a structured, reasoned proof that two triangles are congruent using SSS, SAS, AAS or RHS.
• To use similar triangles to solve indirect-measurement problems.

Success Criteria

I can:

1. Select and justify the correct congruence condition to prove two triangles congruent.
2. Write a proof as a sequence of statements, each supported by a reason.
3. Use similarity (AA) to set up and solve a scale-factor problem in an applied context.
4. Use a congruence proof to derive a new geometric fact (e.g. that base angles of an isosceles triangle are equal).

Warmup

(5 minutes — “Spot the condition”, mini whiteboards)

For each pair of marked triangles, decide: SSS, SAS, AAS, RHS, or not enough information.

1. Two triangles share three pairs of equal sides.
2. Two triangles share two equal sides and the angle between them.
3. Two triangles share two equal angles and a non-included equal side.
4. Two right-angled triangles share the hypotenuse and one other side.
5. Two triangles share two equal sides and a non-included angle. (Trap.)
6. Two triangles share three equal angles. (Trap.)

Answers: 1. SSS 2. SAS 3. AAS 4. RHS 5. Not enough information — SSA is not a valid condition; the triangle is not fixed. 6. Not enough information for congruence — AAA only guarantees the triangles are similar, not the same size.

Teacher note: Questions 5 and 6 are deliberate traps. Push students to sketch a counterexample for each before accepting “not enough information.”

Activities

Activity 1 — Explicit Instruction: Proving the Isosceles Triangle Theorem (12 min)

I do: Model a formal proof, statement by statement, using a labelled diagram of with and drawn as the bisector of , meeting at .

Claim: The base angles of an isosceles triangle are equal, i.e. .

Statement	Reason
	Given (isosceles triangle)
	Given ( bisects )
	Common side
	SAS
	Corresponding angles in congruent triangles

We do: Extend the same diagram. Since and are corresponding angles in the congruent triangles, . Since they are also supplementary (, angles on a straight line), show together that each must equal . Conclude that the angle bisector from the apex of an isosceles triangle is also the altitude.

You do: Using the same congruent triangles, prove that is the midpoint of (i.e. is also the median). State the statement–reason pair that finishes the proof.

Activity 2 — Similar Triangles for Indirect Measurement (10 min)

I do: A flagpole casts a shadow m long. At the same time, a m student casts a shadow m long. The sun’s rays are parallel and both objects meet the ground at a right angle, so the two triangles are similar (AA: equal angle of elevation, equal right angle).

We do: A m fence post casts a m shadow. At the same time a tree casts an m shadow. Find the tree’s height.

You do: A mirror is placed on the ground m from the base of a wall. A student of height m stands m from the mirror and can just see the top of the wall reflected in it. Using similar triangles, find the height of the wall.

Activity 3 — Problem-solving: a Formal Proof (12 min)

Pairs, then whole-class share.

Line segments and bisect each other at (that is, and ). Prove that .

Socratic scaffolding (Polya’s cycle):

Prompt	Purpose
Understand: what are you given, and what must you show?	Given: is the midpoint of both segments. To prove: two lines are parallel — this needs an angle argument.
What do equal angles at suggest?	and are vertically opposite, so they are already equal — a free fact.
Devise a plan — can you find two congruent triangles?	Look at and . Which parts are already known to be equal?
Carry out the plan — which condition applies?	, , — that’s SAS.
What does the congruence give you?	(corresponding angles in congruent triangles).
How does an angle fact prove parallel lines?	and are alternate angles on transversal . Equal alternate angles mean .
Looking back — does this match a shape you know?	This is exactly why the diagonals of a parallelogram bisecting each other forces opposite sides to be parallel — a preview of Lesson 63.

Full proof:

Statement	Reason
,	Given
	Vertically opposite angles
	SAS
	Corresponding angles in congruent triangles
	Equal alternate angles on transversal

Checks for Understanding

(6 minutes — exit ticket, collected)

1. State the congruence condition proved by: two triangles with two equal angles and the side between them equal.
2. In , and is bisected by ( on ). Name the congruence condition that proves .
3. A m student casts a m shadow. A building casts a m shadow at the same time. How tall is the building?
4. Why is “SSA” not accepted as a valid congruence condition?
5. Reasoning. Two triangles have all three angles equal. Explain why this does not prove the triangles are congruent, and state what it does prove.

Answers: 1. ASA; 2. SAS; 3. m; 4. Given two sides and a non-included angle, more than one triangle shape can satisfy the conditions — it does not fix a unique triangle; 5. It proves the triangles are similar (same shape), not congruent, since size (side lengths) is not fixed by angles alone.

Common Misconceptions

Misconception	How to pre-empt it
SSA is treated as a valid congruence condition.	Sketch two different triangles satisfying the same SSA data side by side — a concrete counterexample.
AAA proves congruence.	Contrast with a photocopy enlarged to 150%: same angles, different size. AAA only proves similarity.
Any correspondence order is acceptable when naming congruent triangles.	Insist means , , — order encodes the matching.
A proof is “just the diagram” — no statements needed.	Require every proof to be written as statement–reason pairs, even in rough working.
In similar-triangle problems, students match the wrong sides in the ratio.	Always write the ratio with corresponding sides stacked directly above one another before solving.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). Two triangles are congruent by SAS. One has sides cm, cm and an included angle of . What is the included angle in the second triangle?

Answer

— congruent triangles have identical corresponding angles as well as identical corresponding sides.

E2 (Kangaroo style). A tree of unknown height casts a shadow of m. At the same moment, a m lamppost casts a shadow of m. What is the height of the tree, to the nearest metre?

Answer

E3 (Challenge proof). is isosceles with . Points and lie on and respectively such that . Prove that .

Answer

Consider and : (given), (given), (common angle ). So (SAS), hence (corresponding sides in congruent triangles).

Homework

1. State the condition (SSS, SAS, AAS, RHS) that proves each pair of triangles congruent: (a) three matching sides (b) two right-angled triangles with equal hypotenuse and one equal leg (c) two angles and the included side equal.
2. A m man casts a m shadow. At the same time a flagpole casts a m shadow. Find the height of the flagpole.
3. is isosceles with . The bisector of meets at . Name the two congruent triangles formed and the condition that proves them congruent.
4. Two triangles have two pairs of equal sides and a pair of equal angles that is not between those sides. Can you conclude they are congruent? Sketch to justify your answer.
5. Reasoning. Explain, using a diagram, why AA is sufficient to prove two triangles are similar even though only two of the three angles are checked.
6. Challenge. is the midpoint of . and are points on opposite sides of such that and . Prove that .

Answers: 1. (a) SSS (b) RHS (c) ASA; 2. m; 3. by SAS (, , common); 4. No — this is SSA, which does not guarantee a unique triangle; a counterexample diagram should show two different triangles fitting the same data; 5. If two angles match, the third must also match since angle sums are in every triangle — so AA guarantees the same angle set, i.e. the same shape; 6. In and : (given), , (vertically opposite) — AAS, so , giving .