Geometric triangle problems showing angles and side lengths for comparison and analysis.

Problem Analysis

The task involves analyzing the given triangles and determining whether they are congruent. Congruent triangles have the same size and shape, meaning all corresponding sides and angles are equal. We will use the standard criteria for triangle congruence:
- SSS (Side-Side-Side): All three sides of one triangle are equal to the corresponding sides of another triangle.
- SAS (Side-Angle-Side): Two sides and the included angle of one triangle are equal to the corresponding parts of another triangle.
- ASA (Angle-Side-Angle): Two angles and the included side of one triangle are equal to the corresponding parts of another triangle.
- AAS (Angle-Angle-Side): Two angles and a non-included side of one triangle are equal to the corresponding parts of another triangle.
- HL (Hypotenuse-Leg): For right triangles, the hypotenuse and one leg of one triangle are equal to the corresponding parts of another triangle.

Solution

#### Problem 5
- Triangle on the left: Angles are \(80^\circ\), \(70^\circ\), and \(30^\circ\) (since the sum of angles in a triangle is \(180^\circ\)). Sides are 8 cm, 4 cm, and an unknown side.
- Triangle on the right: Angles are \(90^\circ\), \(60^\circ\), and \(30^\circ\). Sides are 6 cm, 4 cm, and 3 cm.
- Analysis: The angles do not match (e.g., \(80^\circ \neq 90^\circ\)), so these triangles are not congruent.
- Conclusion: Not congruent.

#### Problem 6
- Triangle on the left: Right triangle with angles \(90^\circ\), \(60^\circ\), and \(30^\circ\). Sides are 8 cm, 4 cm, and an unknown side.
- Triangle on the right: Right triangle with angles \(90^\circ\), \(60^\circ\), and \(30^\circ\). Sides are 6 cm, 4 cm, and 3 cm.
- Analysis: Both triangles are right triangles with the same angles (\(90^\circ\), \(60^\circ\), \(30^\circ\)). However, the side lengths do not match (e.g., 8 cm ≠ 6 cm). Therefore, these triangles are not congruent.
- Conclusion: Not congruent.

#### Problem 7
- Triangle on the left: Right triangle with sides 8 cm, 4 cm, and an unknown side.
- Triangle on the right: Right triangle with sides 6 cm, 2 cm, and an unknown side.
- Analysis: Both triangles are right triangles, but the side lengths do not match (e.g., 8 cm ≠ 6 cm). Therefore, these triangles are not congruent.
- Conclusion: Not congruent.

#### Problem 8
- Triangle on the left: Right triangle with sides 8 cm, 4 cm, and an unknown side.
- Triangle on the right: Right triangle with sides 6 cm, 2 cm, and an unknown side.
- Analysis: Both triangles are right triangles, but the side lengths do not match (e.g., 8 cm ≠ 6 cm). Therefore, these triangles are not congruent.
- Conclusion: Not congruent.

#### Problem 9
- Triangle on the left: Triangle with sides 4 cm, 6 cm, and 12 cm.
- Triangle on the right: Triangle with sides 4 cm, 6 cm, and 12 cm.
- Analysis: All corresponding sides are equal (4 cm = 4 cm, 6 cm = 6 cm, 12 cm = 12 cm). By the SSS criterion, these triangles are congruent.
- Conclusion: Congruent.

#### Problem 10
- Triangle on the left: Triangle with sides 4 cm, 6 cm, and 12 cm.
- Triangle on the right: Triangle with sides 4 cm, 6 cm, and 12 cm.
- Analysis: All corresponding sides are equal (4 cm = 4 cm, 6 cm = 6 cm, 12 cm = 12 cm). By the SSS criterion, these triangles are congruent.
- Conclusion: Congruent.

Final Answer

\[
\boxed{\text{Not congruent, Not congruent, Not congruent, Not congruent, Congruent, Congruent}}
\]