Congruent Triangles With Solutions by Hassan Lakiss ... - Free Printable

Let’s solve each problem step by step.

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Problem 1: Which triangles are congruent? Give reasons.

We have six triangles labeled A through F, with some angles and side lengths given. Remember: congruent triangles have exactly the same size and shape — meaning all corresponding sides and angles are equal. We can use congruence rules like:

- SAS (Side-Angle-Side): Two sides and the included angle match.
- ASA (Angle-Side-Angle): Two angles and the included side match.
- AAS (Angle-Angle-Side): Two angles and a non-included side match.
- SSS (Side-Side-Side): All three sides match.
- RHS (Right-angle-Hypotenuse-Side): Only for right triangles.

Also note: The diagrams are NOT drawn to scale, so we must rely only on the numbers given.

Let’s list what we know for each triangle:

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Triangle A:
Angles: 60°, 50° → third angle = 180 - 60 - 50 = 70°
Side: 8 cm (between 60° and 50°? Let’s check diagram description — actually, in triangle A, the 8cm side is opposite the 70° angle? Wait — better to look at which sides are between which angles.)

Actually, let’s reconstruct carefully based on standard labeling:

In Triangle A:
- Angle at left vertex: 60°
- Angle at top vertex: 50°
- Side of 8 cm is on the right side — that would be between the 50° angle and the unknown bottom-right angle.

Wait — perhaps it's easier to compute all angles first, then see which sides are where.

But since diagrams aren’t drawn to scale, we need to assume the labels are placed as shown.

Alternative approach: For each triangle, write down the two known angles and the side, and see if they match another triangle using ASA or AAS.

Let me tabulate:

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Triangle A:
Angles: 60°, 50° → third angle = 70°
Side: 8 cm — looking at position: it’s adjacent to the 50° angle and opposite the 60° angle? Actually, from typical diagram layout:

Assume in Triangle A:
- Left angle: 60°
- Top angle: 50°
- Right side (vertical): 8 cm → this side is between top (50°) and bottom-right (which is 70°). So 8 cm is between 50° and 70°.

So sides/angles:
Angle 50°, side 8 cm, angle 70° → so ASA: 50° - 8cm - 70°

Wait — but we don’t have 70° labeled, we calculated it. But for congruence, we can use calculated angles.

Actually, let’s do this systematically.

For each triangle, find all three angles (if possible), and note which sides are given and between which angles.

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Triangle A:
Given: angles 60° and 50°, side 8 cm.
Third angle = 70°.
The 8 cm side is opposite the 60° angle? Or between 50° and 70°? From diagram description: in triangle A, the 8cm is on the right side, connecting the 50° angle (top) and the unlabeled bottom-right angle (which is 70°). So yes — 8 cm is between 50° and 70°.

So: angles 50° and 70° with included side 8 cm → ASA.

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Triangle B:
Angles: 60° and 50° → third angle 70°
Side: 8 cm — again, likely between 50° and 70°? In diagram, 8cm is on the right side, same as A? But angles are arranged differently? Wait — in B, 60° is at left, 50° at bottom, 8cm on right — so same as A? Then A and B might be congruent.

Wait — let’s compare A and B:

Triangle A:
- Angles: 60° (left), 50° (top), 70° (bottom right)
- Side 8 cm: between 50° and 70° → so side between top and bottom right.

Triangle B:
- Angles: 60° (left), 50° (bottom), 70° (top right?)
Wait — if 60° is left, 50° is bottom, then top angle is 70°.
Side 8 cm is on the right — so between top (70°) and bottom (50°). Same as A!

So both A and B have:
- Angles 50° and 70° with included side 8 cm → ASA → congruent.

But wait — in A, the 50° is at top, in B, 50° is at bottom — but as long as the side is between the same two angles, it’s fine. Congruence doesn’t care about orientation.

So A ≅ B by ASA.

Now check others.

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Triangle C:
Angles: 35°, 70° → third angle = 75°
Side: 6 cm — between 35° and 70°? Diagram says 6cm on left, 70° at top, 35° at bottom left → so 6cm is between 35° and 70°? Yes.

So: angles 35° and 70° with included side 6 cm → ASA.

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Triangle D:
Angles: 40° (left), side 6 cm (bottom), side 8 cm (left) — wait, no angles given except 40°? Wait, diagram shows: angle 40° at left, side 6 cm at bottom, side 8 cm on left — so actually, we have two sides and the included angle? Let’s see:

In D:
- Angle at left: 40°
- Side adjacent to it: 8 cm (going up-left?) and 6 cm (going down-right?) — actually, typically, if 40° is at left vertex, and sides 8cm and 6cm are the two sides forming that angle, then it’s SAS.

Yes — so D has: side 8 cm, angle 40°, side 6 cm → SAS.

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Triangle E:
Angles: 35°, 70° → third angle 75°
Side: 6 cm — between 35° and 70°? Diagram: 35° at left, 70° at bottom, 6cm on bottom — so 6cm is between 35° and 70°? If 35° is left, 70° is bottom, then the side between them is the bottom side — which is labeled 6cm. Yes.

So E: angles 35° and 70° with included side 6 cm → same as C → so C ≅ E by ASA.

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Triangle F:
Angles: 40° (bottom right), sides: 6 cm (bottom), 8 cm (right) — so similar to D?

In F:
- Angle 40° at bottom right
- Side 6 cm at bottom (adjacent to 40°)
- Side 8 cm on right (also adjacent to 40°)
→ So two sides and included angle: 6 cm, 40°, 8 cm → same as D.

D had: 8 cm, 40°, 6 cm — order doesn’t matter for SAS, as long as the angle is between the two sides.

So D and F both have: sides 6 cm and 8 cm with included angle 40° → SAS → congruent.

Now, let’s summarize:

- A and B: both have angles 50° and 70° with included side 8 cm → ASA → congruent.
- C and E: both have angles 35° and 70° with included side 6 cm → ASA → congruent.
- D and F: both have sides 6 cm and 8 cm with included angle 40° → SAS → congruent.

Are there any others? For example, is A congruent to anyone else? A has angles 50,60,70 and side 8 between 50 and 70. No other triangle has that combination.

Similarly, C and E have 35,70,75 and side 6 between 35 and 70 — unique.

D and F have 40° between 6 and 8 — unique.

So pairs: A≅B, C≅E, D≅F.

But the question is “Which triangles are congruent?” — probably expecting to list the pairs.

Also, note: in triangle A and B, we used calculated angles — that’s allowed because sum of angles is always 180°.

So final answer for Problem 1:

Triangles A and B are congruent (ASA: 50°, 8 cm, 70°).
Triangles C and E are congruent (ASA: 35°, 6 cm, 70°).
Triangles D and F are congruent (SAS: 6 cm, 40°, 8 cm).

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Problem 2: Prove that triangles ABC and DEF are congruent.

Given:

Triangle ABC:
- Angle at A: 70°
- Angle at C: 50°
- Side BC: 10 cm (since it’s opposite A? Wait — label: points A, B, C. Side between B and C is 10 cm. So side BC = 10 cm.

Angles:
At A: 70°, at C: 50° → so angle at B = 180 - 70 - 50 = 60°.

So triangle ABC has:
Angles: A=70°, B=60°, C=50°
Side BC = 10 cm — which is between angles B and C → so between 60° and 50°.

Triangle DEF:
Points D, E, F.
Side ED = 10 cm (labeled between E and D)
Angle at E: 60°
Angle at F: 70°
So angle at D = 180 - 60 - 70 = 50°.

So triangle DEF has:
Angles: D=50°, E=60°, F=70°
Side ED = 10 cm — which is between E and D → so between 60° and 50°.

Compare to ABC:
ABC: angles at B=60°, C=50°, side BC=10 cm (between them)
DEF: angles at E=60°, D=50°, side ED=10 cm (between them)

So corresponding parts:
Angle B (60°) corresponds to angle E (60°)
Angle C (50°) corresponds to angle D (50°)
Side BC (between B and C) corresponds to side ED (between E and D) — both 10 cm.

Therefore, by ASA (Angle-Side-Angle), triangles ABC and DEF are congruent.

Note: The correspondence is A↔F, B↔E, C↔D, but for congruence proof, we just need to show matching angles and included side.

So proof:

In triangle ABC:
∠A = 70°, ∠C = 50° ⇒ ∠B = 60°
Side BC = 10 cm (included between ∠B and ∠C)

In triangle DEF:
∠E = 60°, ∠F = 70° ⇒ ∠D = 50°
Side ED = 10 cm (included between ∠E and D)

Thus, ∠B = ∠E = 60°, BC = ED = 10 cm, ∠C = D = 50°
⇒ By ASA, △ABC ≅ △DEF

(Note: The order of vertices matters for correspondence, but the question just asks to prove they are congruent, not specify correspondence.)

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Final Answer:

1. Triangles A and B are congruent (ASA: 50°, 8 cm, 70°).
Triangles C and E are congruent (ASA: 35°, 6 cm, 70°).
Triangles D and F are congruent (SAS: 6 cm, 40°, 8 cm).

2. In △ABC: ∠B = 60°, ∠C = 50°, side BC = 10 cm.
In △DEF: ∠E = 60°, ∠D = 50°, side ED = 10 cm.
Since two angles and the included side are equal, △ABC ≅ △DEF by ASA.