Question

1 what does the included angle mean in triangle congruence?
a the angle opposite the given side
b the smallest angle in the triangle
c the largest angle in the triangle
d the angle formed by the two given sides
2 if ( a b = 7 mathrm { cm } , a c = 5 mathrm { cm } ), and ( angle b a c = 30 ^ { circ } ), which additional triangle could be congruent to ( \triangle a b c ) using sas?
a ( \triangle d e f ) with ( d e = 5 mathrm { cm } , d f = 7 mathrm { cm } , angle e d f = 40 )
b ( \triangle d e f ) with ( d e = 7 mathrm { cm } , d f = 5 mathrm { cm } , angle e d f = 30 ^ { circ } )
c ( \triangle d e f ) with ( d e = 7 mathrm { cm } , d f = 4 mathrm { cm } , angle e d f = 30 ^ { circ } )
d ( \triangle d e f ) with ( d e = 5 mathrm { cm } , d f = 5 mathrm { cm } , angle e d f = 30 ^ { circ } )
3 which of the following side lengths could not form a triangle congruent to one with sides 3 cm, 4 cm, and 5 cm by sss?
a 5 cm, 4 cm, 6 cm
b 5 cm, 4 cm, 3 cm
c 4 cm, 3 cm, 5 cm
d 4 cm, 5 cm, 3 cm
4 which is a correct measurement combination for proving triangles congruent using sas?
a two angles and a side
b two sides and the included angle
c one side and one angle
d three angles
5 if triangle mno and triangle pqr are congruent by sss criterion, and ( m n = 3 y + 2 , p q = 14 ), what is the value of y?
a 1
b 4
c 2
d 3
6 which of the following triangles are congruent to triangle abc with sides 4 cm, 5 cm, 6 cm?
a mno with sides 5 cm, 6 cm, 7 cm
b jkl with sides 3 cm, 4 cm, 5 cm
c ghi with sides 4 cm, 5 cm, 6 cm
d bct with sides 4 cm, 3 cm, 7 cm
7 which angle is crucial in the sas criterion?
a the largest angle
b any angle of the triangle
c the smallest angle
d the included angle

Explanation:

1. Question 1:

• Brief Explanations:
• In triangle congruence, the “included angle” is defined as the angle formed by the two given sides.
• Answer:
• d. The angle formed by the two given sides

1. Question 2:

• Brief Explanations:
• For SAS (Side - Angle - Side) congruence, we need two sides and the included angle. In \(\triangle ABC\), \(AB = 7\mathrm{cm}\), \(AC = 5\mathrm{cm}\), \(\angle BAC=20^{\circ}\). In \(\triangle DEF\), if \(DE = 7\mathrm{cm}\), \(DF = 5\mathrm{cm}\), \(\angle EDF = 20^{\circ}\), then by SAS, \(\triangle ABC\cong\triangle DEF\) (since the sides and the included angle match).
• Answer:
• b. \(\triangle DEF\) with \(DE = 7\mathrm{cm}\), \(DF = 5\mathrm{cm}\), \(\angle EDF = 20^{\circ}\)

1. Question 3:

• Brief Explanations:
• For SSS (Side - Side - Side) congruence, the side lengths must match. The given side lengths are \(3\mathrm{cm}\), \(4\mathrm{cm}\), \(5\mathrm{cm}\). Option a has side lengths \(3\mathrm{cm}\), \(4\mathrm{cm}\), \(6\mathrm{cm}\) which do not match the given set for SSS congruence.
• Answer:
• a. \(3\mathrm{cm}\), \(4\mathrm{cm}\), \(6\mathrm{cm}\)

1. Question 4:

• Brief Explanations:
• The SAS (Side - Angle - Side) criterion for proving triangle congruent requires two sides and the included angle.
• Answer:
• b. Two sides and the included angle

1. Question 5:

• Brief Explanations:
• If \(\triangle MNO\) and \(\triangle PQR\) are congruent by SSS, then \(MN = PQ\). Given \(MN=3y + 2\) and \(PQ = 14\). Set up the equation \(3y+2=14\). Subtract 2 from both sides: \(3y=14 - 2=12\). Divide both sides by 3: \(y = 4\).
• Answer:
• b. 4

1. Question 6:

• Brief Explanations:
• For SSS congruence, the side lengths must match. Triangle \(ABC\) has sides \(4\mathrm{cm}\), \(5\mathrm{cm}\), \(6\mathrm{cm}\). Option c has sides \(4\mathrm{cm}\), \(5\mathrm{cm}\), \(6\mathrm{cm}\) (assuming the order can be rearranged as SSS is not order - dependent for congruence).
• Answer:
• c. \(\triangle GHI\) with sides \(4\mathrm{cm}\), \(5\mathrm{cm}\), \(6\mathrm{cm}\)

1. Question 7:

• Brief Explanations:
• In the SAS criterion, the included angle is the angle between the two sides. It is not necessarily the largest, smallest, or any arbitrary angle.
• Answer:
• d. The included angle

Answer:

1. Question 1:

• Brief Explanations:
• In triangle congruence, the “included angle” is defined as the angle formed by the two given sides.
• Answer:
• d. The angle formed by the two given sides

1. Question 2:

• Brief Explanations:
• For SAS (Side - Angle - Side) congruence, we need two sides and the included angle. In \(\triangle ABC\), \(AB = 7\mathrm{cm}\), \(AC = 5\mathrm{cm}\), \(\angle BAC=20^{\circ}\). In \(\triangle DEF\), if \(DE = 7\mathrm{cm}\), \(DF = 5\mathrm{cm}\), \(\angle EDF = 20^{\circ}\), then by SAS, \(\triangle ABC\cong\triangle DEF\) (since the sides and the included angle match).
• Answer:
• b. \(\triangle DEF\) with \(DE = 7\mathrm{cm}\), \(DF = 5\mathrm{cm}\), \(\angle EDF = 20^{\circ}\)

1. Question 3:

• Brief Explanations:
• For SSS (Side - Side - Side) congruence, the side lengths must match. The given side lengths are \(3\mathrm{cm}\), \(4\mathrm{cm}\), \(5\mathrm{cm}\). Option a has side lengths \(3\mathrm{cm}\), \(4\mathrm{cm}\), \(6\mathrm{cm}\) which do not match the given set for SSS congruence.
• Answer:
• a. \(3\mathrm{cm}\), \(4\mathrm{cm}\), \(6\mathrm{cm}\)

1. Question 4:

• Brief Explanations:
• The SAS (Side - Angle - Side) criterion for proving triangle congruent requires two sides and the included angle.
• Answer:
• b. Two sides and the included angle

1. Question 5:

• Brief Explanations:
• If \(\triangle MNO\) and \(\triangle PQR\) are congruent by SSS, then \(MN = PQ\). Given \(MN=3y + 2\) and \(PQ = 14\). Set up the equation \(3y+2=14\). Subtract 2 from both sides: \(3y=14 - 2=12\). Divide both sides by 3: \(y = 4\).
• Answer:
• b. 4

1. Question 6:

• Brief Explanations:
• For SSS congruence, the side lengths must match. Triangle \(ABC\) has sides \(4\mathrm{cm}\), \(5\mathrm{cm}\), \(6\mathrm{cm}\). Option c has sides \(4\mathrm{cm}\), \(5\mathrm{cm}\), \(6\mathrm{cm}\) (assuming the order can be rearranged as SSS is not order - dependent for congruence).
• Answer:
• c. \(\triangle GHI\) with sides \(4\mathrm{cm}\), \(5\mathrm{cm}\), \(6\mathrm{cm}\)

1. Question 7:

• Brief Explanations:
• In the SAS criterion, the included angle is the angle between the two sides. It is not necessarily the largest, smallest, or any arbitrary angle.
• Answer:
• d. The included angle