Question

proving and applying the sas and sss congruence criteria

1. in the figure shown, which composition of rigid transformations will map one triangle onto the other?

a a glide reflection
b a reflection followed by a translation
c two translations
d a rotation followed by a translation

1. which theorem shows that △abc≅△def?

a the triangles are not congruent.
b sas triangle congruence theorem
c isosceles triangle theorem
d sss triangle congruence theorem

1. in the figure shown, what additional information is needed to show that △abc≅△def by sss?

a m∠c
b (overline{ac}congoverline{df})
c (overline{ab}congoverline{de})
d m∠e

1. what are the necessary conditions to apply the sas triangle congruence theorem?

a one angle and two sides of one triangle are congruent to the corresponding parts of another triangle.
b two angles and the included side of one triangle are congruent to the corresponding parts of another triangle.
c an angle and the two sides collinear with the angle’s rays are congruent to the corresponding parts of another triangle.
d two sides and any angle of one triangle are congruent to the corresponding parts of another triangle.

1. what value of x will make the triangles congruent by sss?

Explanation:

Step1: Analyze triangle - mapping transformation

A glide - reflection is a combination of a translation and a reflection. Looking at the orientation of the triangles in the first question, a rotation followed by a translation can map one triangle onto the other. A reflection followed by a translation or two translations won't work as the orientation is not correct for just those. A glide - reflection also won't work in the given context. So for question 1, the answer is D.

Step2: Identify congruence theorem

In the second question, if we observe the markings on the triangles, we can see that two sides and the included angle of one triangle are congruent to the corresponding parts of the other triangle. So, by the SAS (Side - Angle - Side) Triangle Congruence Theorem, the triangles are congruent. The answer is B.

Step3: Determine SSS condition

For the SSS (Side - Side - Side) congruence criterion, we need all three pairs of corresponding sides to be congruent. In the third question, we already have some side - congruence markings. We need $\overline{AB}\cong\overline{DE}$ to complete the SSS condition. So the answer is C.

Step4: Recall SAS theorem conditions

The SAS Triangle Congruence Theorem states that two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle. So for question 4, the answer is A.

Step5: Solve for x in SSS congruence

For the triangles to be congruent by SSS in the fifth question, we set up an equation based on the side - length equality. If we assume that the side lengths are equal, we have $x - 4=19$. Solving for $x$ gives $x=19 + 4=23$.

Answer:

1. D. a rotation followed by a translation
2. B. SAS Triangle Congruence Theorem
3. C. $\overline{AB}\cong\overline{DE}$
4. A. One angle and two sides of one triangle are congruent to the corresponding parts of another triangle.
5. 23