Congruent Triangles Notes and Worksheets - Lindsay Bowden - Free Printable

I will solve this worksheet by examining each problem and determining if the triangles are congruent based on the given markings. I'll use the congruence postulates: SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle), AAS (Angle-Angle-Side), or HL (Hypotenuse-Leg for right triangles). If they are not congruent, I'll write "no".

Let me start with problem 1.

Problem 1:
- This shows a rectangle divided by a diagonal into two triangles.
- The markings indicate that opposite sides of the rectangle are equal (two tick marks on one pair of opposite sides, one tick mark on the other pair).
- The diagonal is shared by both triangles, so it's equal to itself.
- Therefore, we have three pairs of corresponding sides that are equal.
- Congruence Postulate: SSS

Problem 2:
- Two triangles share a common vertex and appear to be overlapping.
- Markings show that all three sides of one triangle are equal to the corresponding sides of the other triangle (one tick mark on one side, two tick marks on another, and three tick marks on the third).
- Congruence Postulate: SSS

Problem 3:
- An isosceles triangle with an altitude drawn from the apex to the base.
- The altitude creates two right triangles.
- The altitude is shared by both right triangles.
- The base is split into two equal parts by the altitude (indicated by the tick marks).
- Both right triangles have a right angle.
- So, we have two legs and the included right angle equal.
- Congruence Postulate: SAS (or HL, since they are right triangles)

Problem 4:
- Two right triangles.
- One leg of each triangle is marked with two tick marks, indicating they are equal.
- The hypotenuses are marked with one tick mark, indicating they are equal.
- Since these are right triangles, and we have a leg and hypotenuse equal, this fits the HL postulate.
- Congruence Postulate: HL

Problem 5:
- Two triangles sharing a common side.
- The arrows indicate that the sides are parallel, which implies that the alternate interior angles are equal.
- However, there are no specific markings for equal sides or angles between the two triangles.
- We cannot determine congruence without more information.
- Answer: no

Problem 6:
- Two triangles with one side marked equal (one tick mark) and one angle marked equal (the angle at the end of the marked side).
- However, the angle is not between the two marked sides; it's adjacent.
- We do not have enough information to apply any congruence postulate.
- Answer: no

Problem 7:
- Two triangles with two angles marked equal (one angle in each triangle has a single arc, and another angle in each has a double arc).
- Also, one side is marked equal (one tick mark on a side opposite one of the angles).
- This fits the AAS postulate (two angles and a non-included side).
- Congruence Postulate: AAS

Problem 8:
- Two triangles sharing a common side.
- Two sides are marked equal (one tick mark on each side of the shared side).
- The angle between these two sides is also marked equal (single arc).
- This fits the SAS postulate.
- Congruence Postulate: SAS

Problem 9:
- Two right triangles sharing a common side.
- Both have a right angle marked.
- One leg is marked equal (one tick mark), and the hypotenuse is marked equal (two tick marks).
- This fits the HL postulate for right triangles.
- Congruence Postulate: HL

Problem 10:
- Two triangles, ABC and EDC, sharing a common vertex C.
- Angles at A and E are marked equal (single arc).
- Angles at B and D are marked equal (double arc).
- The vertical angles at C are equal (though not marked, they are always equal when two lines intersect).
- We have two angles and the included side (AC and EC are not marked, but BC and DC are not marked either).
- Actually, looking again, we have two angles equal, and the side between them (AC and EC) is not marked, but the side opposite one of the angles is not marked either.
- Wait, let's look at the angles: angle A = angle E, angle B = angle D, and side BC and DC are not marked, but side AC and EC are not marked.
- Actually, we have two angles equal, and the side between them is not marked, but we can use AAS if we have a side.
- Looking at the diagram, we have angle A = angle E, angle B = angle D, and side AB and ED are not marked.
- But we have the vertical angles at C, which are equal, and side AC and EC are not marked.
- Actually, we have two angles equal, and the side between them is not marked, so we cannot apply ASA or AAS without a side.
- However, since we have two angles equal, the third angle must also be equal, and if we had a side, we could use AAS.
- But in this case, no sides are marked as equal.
- Answer: no

Wait, let me reconsider problem 10. The triangles are ABC and EDC. Angles at A and E are equal, angles at B and D are equal. The side between angles A and B is AB, and the side between angles E and D is ED. These are not marked. The side between angles A and C is AC, and between E and C is EC. These are not marked. The side between angles B and C is BC, and between D and C is DC. These are not marked. So, we have AAA, which is not a congruence postulate. Answer: no

Problem 11:
- Triangles JLK and NKM.
- Side JK is marked equal to side KM (two tick marks).
- Side JL is marked equal to side NM (one tick mark).
- Angle at J is marked equal to angle at M (single arc).
- The angle at J is between sides JL and JK, and the angle at M is between sides NM and KM.
- So, we have two sides and the included angle equal.
- Congruence Postulate: SAS

Problem 12:
- Quadrilateral QTRS with diagonal QT.
- Arrows indicate that QR is parallel to TS, and QT is parallel to RS.
- So, QTRS is a parallelogram.
- In a parallelogram, opposite sides are equal, and opposite angles are equal.
- Triangles QTR and RST share the diagonal QT.
- Side QR = side TS (opposite sides of parallelogram).
- Side QT = side RS (opposite sides of parallelogram).
- Side TR is common to both triangles.
- So, all three sides are equal.
- Congruence Postulate: SSS

Now, for problems 10, 11, and 12, I need to finish the congruency statement.

Problem 10:
- As determined, the triangles are not congruent.
- Answer: no

Problem 11:
- Triangle JLK is congruent to triangle NKM by SAS.
- The correspondence is J to N, L to K, K to M.
- So, ΔJLK ≅ ΔNKM

Problem 12:
- Triangle QTR is congruent to triangle RST by SSS.
- The correspondence is Q to R, T to S, R to T.
- So, ΔQTR ≅ ΔRST

Let me summarize the answers:

1. SSS
2. SSS
3. SAS (or HL)
4. HL
5. no
6. no
7. AAS
8. SAS
9. HL
10. no
11. ΔJLK ≅ ΔNKM
12. ΔQTR ≅ ΔRST

I think I have solved all the problems. Let me double-check problem 10. Upon closer inspection, triangles ABC and EDC have angles at A and E equal, angles at B and D equal, and the vertical angles at C are equal. But no sides are marked as equal. So, AAA is not sufficient for congruence. Therefore, "no" is correct.

Final Answer:
1. SSS
2. SSS
3. SAS
4. HL
5. no
6. no
7. AAS
8. SAS
9. HL
10. no
11. ΔJLK ≅ ΔNKM
12. ΔQTR ≅ ΔRST