Question

9 multiple choice 1 point
the lengths of two sides of a triangle are 4 and 6. the third side cannot be _.
2
3
6
4
10 multiple choice 1 point
which congruency theorem is described below?
\in two triangles, if two pairs of sides and their included angles have equal measurement, then the triangles are congruent.\
sss
asa
sas
aaa

Explanation:

Question 9

Step1: Apply triangle inequality theorem

The triangle inequality theorem states that for a triangle with side lengths \(a\), \(b\), and \(c\), \(|a - b|\lt c\lt a + b\). Let \(a = 4\) and \(b=6\). Then \(|4 - 6|=2\) and \(4 + 6 = 10\). So \(2\lt c\lt10\).

Step2: Check each option

• For \(c = 2\), it does not satisfy \(2\lt c\) (since \(c = 2\) is not greater than \(2\)).
• For \(c = 3\), \(2\lt3\lt10\).
• For \(c = 6\), \(2\lt6\lt10\).
• For \(c = 4\), \(2\lt4\lt10\).

Question 10

Step1: Recall congruency theorems

• SSS (Side - Side - Side): If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.
• ASA (Angle - Side - Angle): If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent.
• SAS (Side - Angle - Side): If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent.
• AAA (Angle - Angle - Angle): Only shows similarity, not congruency.

The description “In two triangles, if two pairs of sides and their included angles have equal measurement, then the triangles are congruent” matches the SAS (Side - Angle - Side) theorem.

Answer:

Question 9: 2
Question 10: SAS