Question

which of the following properties is always true for triangles proven congruent by sss?

their corresponding sides and angles are equal

their medians are equal

their perimeters are equal

their altitudes are equal

if triangle abc has sides of 5 cm, 7 cm, and 9 cm, and triangle def has sides of 5 cm, 7 cm, and 9 cm, what can be concluded?

the triangles are similar but not congruent.

the triangles are congruent by sas.

the triangles are congruent by sss

the triangles are not congruent.

a triangle has sides 5 cm, 7 cm, and 10 cm. which of the following cannot form a triangle congruent to it by sss?

7 cm, 5 cm, 10 cm

5 cm, 7 cm, 10 cm

10 cm, 7 cm, 5 cm

6 cm, 7 cm, 10 cm

what does sss stand for in geometry?

side-square-side

square-side-square

square-square-square

side-side-side

two triangles are congruent by sss. what does this imply about their angles?

their angles must be complementary.

their corresponding angles are equal.

their angles must add up to 180 degrees.

their angles must be supplementary.

if triangle pqr has sides pq = 5 cm, pr = 12 cm, and qr = 13 cm, which of the following must be true for triangle xyz to be congruent to triangle pqr by sss?

xyz must have all angles equal to triangle pqr

xyz must have two sides and the included angle equal to triangle pqr

xyz must have sides xy = 5 cm, xz = 12 cm, and yz = 13 cm.

xyz must have sides xy = 5 cm, xz = 11 cm, and yz = 12 cm

which of the following conditions must be met to use the sss triangle congruence rule?

Explanation:

First question:

- **Option a**: When two triangles are congruent (by any criterion including SSS), their corresponding sides and angles are equal. This is the definition of congruent triangles.
- **Option b**: Medians are line - segments from a vertex to the mid - point of the opposite side. Just because triangles are congruent by SSS, we cannot directly say medians are equal without further proof (though they will be, but it's not the most fundamental property).
- **Option c**: Perimeters are equal (since sides are equal). But this is a consequence of the sides being equal (a sub - property of the more fundamental side - angle equality).
- **Option d**: Altitudes (perpendicular distances from a vertex to the opposite side) are not immediately obvious to be equal just from the SSS congruence statement without further calculation (though they will be, but not the most basic property).

Second question:

- **Option a**: Similar triangles have proportional sides. Here, sides are equal (\(5 = 5\), \(7 = 7\), \(9 = 9\)), so they are congruent, not just similar.
- **Option b**: SAS (Side - Angle - Side) requires an included angle. We are given three sides, so SSS is the congruence criterion, not SAS.
- **Option c**: Since \(AB = DE=5\mathrm{cm}\), \(BC = EF = 7\mathrm{cm}\), \(AC=DF = 9\mathrm{cm}\), by the SSS (Side - Side - Side) congruence criterion, \(\triangle ABC\cong\triangle DEF\).
- **Option d**: They are congruent.

Third question:

- **Option a**: \(7\mathrm{cm},5\mathrm{cm},10\mathrm{cm}\) are the same set of side lengths as \(5\mathrm{cm},7\mathrm{cm},10\mathrm{cm}\) (order of sides does not matter for SSS congruence).
- **Option b**: \(5\mathrm{cm},7\mathrm{cm},10\mathrm{cm}\) is the same set of side lengths as the given triangle.
- **Option c**: \(10\mathrm{cm},7\mathrm{cm},5\mathrm{cm}\) is the same set of side lengths as the given triangle.
- **Option d**: \(6\mathrm{cm},7\mathrm{cm},10\mathrm{cm}\) has a different side length (\(6 eq5\)) compared to the given triangle (\(5\mathrm{cm},7\mathrm{cm},10\mathrm{cm}\)), so it cannot form a congruent triangle by SSS.

Fourth question:

- **Option a**: SSS does not stand for Side - Square - Side.
- **Option b**: SSS does not stand for Square - Side - Square.
- **Option c**: SSS does not stand for Square - Square - Square.
- **Option d**: In geometry, SSS (Side - Side - Side) is a congruence criterion for triangles where if three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.

Fifth question:

- **Option a**: Complementary angles add up to \(90^{\circ}\). Congruent triangles (by SSS) do not imply angle complementarity.
- **Option b**: By the CPCTC (Corresponding Parts of Congruent Triangles are Congruent) theorem, if \(\triangle ABC\cong\triangle DEF\) (by SSS), then \(\angle A=\angle D\), \(\angle B=\angle E\), \(\angle C=\angle F\).
- **Option c**: The sum of angles in any triangle is \(180^{\circ}\), but this is a property of triangles in general, not specifically for congruent triangles by SSS.
- **Option d**: Supplementary angles add up to \(180^{\circ}\). Congruent triangles (by SSS) do not imply angle - supplementarity.

Sixth question:

- **Option a**: While angles will be equal (by CPCTC) when triangles are congruent by SSS, the SSS criterion is based on side - lengths. The question asks for the SSS requirement.
- **Option b**: Two sides and an included angle is the SAS criterion.
- **Option c**: For \(\triangle PQR\cong\triangle XYZ\) by SSS, \(PQ = XY = 5\mathrm{cm}\), \(PR=XZ = 12\mathrm{cm}\), \(QR = YZ=13\mathrm{cm}\).
- **Option d**: \(XZ = 11\mathrm{cm} eq PR = 12\mathrm{cm}\), so these side - lengths will not make the triangles congruent by SSS.

Answer:

1. A. Their corresponding sides and angles are equal
2. C. The triangles are congruent by SSS
3. D. 6 cm, 7 cm, 10 cm
4. D. Side - Side - Side
5. B. Their corresponding angles are equal
6. C. XYZ must have sides \(XY = 5\mathrm{cm}\), \(XZ = 12\mathrm{cm}\), and \(YZ = 13\mathrm{cm}\)