50+ congruent triangles sss sas and asa worksheets for 9th Grade ... - Free Printable

I'll work through each problem step by step to determine which congruence theorem applies.

Problem 1:
Looking at the parallelogram divided by a diagonal, I need to identify what we know about the two triangles formed.
- The diagonal is shared (common side) - that's one side
- The opposite sides of a parallelogram are marked equal (the tick marks show this)
- So we have: Side-Side-Side (two pairs of corresponding sides plus the shared diagonal)

Wait, let me reconsider. In a parallelogram with a diagonal:
- The diagonal creates two triangles
- Opposite sides of parallelogram are equal (shown by tick marks)
- The diagonal is common to both triangles
- We need to check if we have enough information

Actually, looking more carefully: The tick marks show that opposite sides are equal. When we draw a diagonal, we get:
- Two sides from the parallelogram (opposite sides are equal)
- One shared diagonal

But we need to be careful about which sides correspond. Let me think about this differently:
- If we have a parallelogram ABCD with diagonal AC
- Triangle ABC and triangle CDA share side AC
- AB = CD (opposite sides of parallelogram)
- BC = DA (opposite sides of parallelogram)
- This gives us SSS: three pairs of equal sides

Answer: D) SSS

Problem 2:
I see two triangles with angle markings.
- One angle is marked in each triangle (appears to be the same marking)
- Another angle is marked in each triangle (different marking)
- One side between these angles appears to be marked

Let me look more carefully at the markings:
- Each triangle has two angles marked
- There's a side marked in each triangle
- The side appears to be between the two marked angles

If we have Angle-Side-Angle (two angles and the included side), that would be ASA.

But wait, I need to check if the side is actually between the angles or not. Looking at the diagram:
- The side with the tick mark appears to be adjacent to only one of the marked angles
- This would make it AAS (Angle-Angle-Side) where the side is not between the two angles

Answer: B) AAS

Problem 3:
This shows a triangle with an altitude drawn from the top vertex to the base.
- The altitude creates two right triangles
- There's a right angle symbol at the base
- The altitude is shared (common side)
- There appear to be angle markings at the top vertex

Let me analyze:
- Both triangles are right triangles (90° angle shown)
- They share the altitude (one leg)
- There's an angle marked at the top vertex

For right triangles, we can use HL (Hypotenuse-Leg) if we have the hypotenuse and one leg.
Or we could use other methods if we have appropriate information.

Looking at the markings:
- Right angle at the base
- Shared altitude (leg)
- Angle at the top vertex is split/marked

If we have a right angle, one acute angle, and the side between them, that's ASA.
If we have a right angle, another angle, and a non-included side, that's AAS.

The altitude is a leg, not the hypotenuse. We have:
- Right angle (90°)
- An acute angle at the top
- The leg (altitude) between these angles

This is ASA for right triangles.

Actually, let me reconsider. For right triangles specifically:
- We have the right angle
- We have an acute angle marked
- We have the side between them (the altitude)

This is ASA. But since these are right triangles, we could also consider if HL applies. For HL, we'd need the hypotenuse and a leg. The altitude is a leg, but I don't see markings on the hypotenuses.

Given the angle markings and the shared altitude, ASA seems most appropriate.

Answer: D) ASA

Problem 4:
This shows two triangles that share a vertex (forming an X shape or vertical angles).
- There are tick marks on sides
- The triangles share a vertex where vertical angles are formed

Let me analyze:
- Vertical angles are always equal (this is a geometric fact)
- There are tick marks showing two pairs of sides are equal
- The vertical angles give us one pair of equal angles

So we have:
- Two sides marked equal
- The angle between them? Or not between them?

Looking at the configuration:
- The tick marks are on the outer sides
- The vertical angle is at the shared vertex
- The sides with tick marks appear to include the vertical angle

If the angle is between the two marked sides, that's SAS.

The vertical angles are at the intersection point, and the marked sides appear to radiate from that point, making the angle included between the sides.

Answer: B) SAS

Let me double-check all my answers:

1. Parallelogram with diagonal → SSS ✓
2. Two triangles with two angles and non-included side → AAS ✓
3. Right triangle with altitude, angle at vertex, and shared altitude → ASA ✓
4. Vertical angles with two marked sides including the angle → SAS ✓

Final Answer:
1. D) SSS
2. B) AAS
3. D) ASA
4. B) SAS